ProblemIdx int64 3 918 | SystemPrompt stringclasses 5
values | UserPrompt stringlengths 63 73.4k | ExpectedOutput stringlengths 47 45.8k | FinalAnswer stringlengths 1 51 | Phase float64 1 12 | __index_level_0__ int64 5 7.09k | messages listlengths 3 3 |
|---|---|---|---|---|---|---|---|
3 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
Let $m$ be a constant not equal to $0$ or $1.$ Then the graph of \[x^2 + my^2 = 4\]is a conic section with two foci. Find all values of $m$ such that the foci both lie on the circle $x^2+y^2=16.$
Enter all possible values of $m,$ separated by commas.
</Question>
<Plan_1>
Okay, so I need to find all values ... | <Plan_3>
Based on execution_2.1:
Wait, this is getting a bit confusing. Let me take it step by step.
Case 1: m > 0. Then the equation x² + my² = 4 is an ellipse.
<prompt> Calculate the foci of the ellipse for m > 0 and verify if they lie on the circle x² + y² = 16. </prompt>
Based on execution_2.2:
Case 2: m < 0. Th... | \frac{1}{5}, -\frac{1}{3} | 3 | 5 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
3 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
Let $m$ be a constant not equal to $0$ or $1.$ Then the graph of \[x^2 + my^2 = 4\]is a conic section with two foci. Find all values of $m$ such that the foci both lie on the circle $x^2+y^2=16.$
Enter all possible values of $m,$ separated by commas.
</Question>
<Plan_1>
Okay, so I need to find all values ... | <Plan_4>
Based on execution_3.2:
But wait, let me check if this is correct. Let me verify the steps again.
For the hyperbola case, when m is negative, we have the equation x² + my² = 4. Let's write m as -k where k > 0.
<prompt> Verify and correct the calculation of the foci for the hyperbola when m < 0. </prompt>
---... | \frac{1}{5}, -\frac{1}{3} | 4 | 9 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
3 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
Let $m$ be a constant not equal to $0$ or $1.$ Then the graph of \[x^2 + my^2 = 4\]is a conic section with two foci. Find all values of $m$ such that the foci both lie on the circle $x^2+y^2=16.$
Enter all possible values of $m,$ separated by commas.
</Question>
<Plan_1>
Okay, so I need to find all values ... | <Plan_7>
Based on execution_6.1, execution_3.3:
Therefore, the possible values of m are 1/5 and -1/3.
Wait, let me confirm this. Let's check m = 1/5.
<prompt> Confirm all possible solutions by verifying the calculations for both m = 1/5 and m = -1/3. </prompt>
</Plan_7> | \frac{1}{5}, -\frac{1}{3} | 7 | 18 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
3 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
Let $m$ be a constant not equal to $0$ or $1.$ Then the graph of \[x^2 + my^2 = 4\]is a conic section with two foci. Find all values of $m$ such that the foci both lie on the circle $x^2+y^2=16.$
Enter all possible values of $m,$ separated by commas.
</Question>
<Plan_1>
Okay, so I need to find all values ... | <Final_answer>
Therefore, the possible values of $m$ are $m = \boxed{\frac{1}{5}, -\frac{1}{3}}.
$
</Final_answer> | \frac{1}{5}, -\frac{1}{3} | 8 | 20 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
4 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
Let $m$ be a constant not equal to $0$ or $1.$ Then the graph of \[x^2 + my^2 = 4\]is a conic section with two foci. Find all values of $m$ such that the foci both lie on the circle $x^2+y^2=16.$
Enter all possible values of $m,$ separated by commas.
</Question>
<Plan_1>
Okay, so I need to find all values ... | <Plan_5>
Based on execution_4.2, execution_4.3:
Therefore, this is the only solution.
Therefore, the only value of m is -1/3. So, the answer should be \boxed{-\dfrac{1}{3}}.
<prompt> Perform a final verification and conclude the possible values of m. </prompt>
</Plan_5> | \frac{1}{5}, -\frac{1}{3} | 5 | 34 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
4 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
Let $m$ be a constant not equal to $0$ or $1.$ Then the graph of \[x^2 + my^2 = 4\]is a conic section with two foci. Find all values of $m$ such that the foci both lie on the circle $x^2+y^2=16.$
Enter all possible values of $m,$ separated by commas.
</Question>
<Plan_1>
Okay, so I need to find all values ... | <Final_answer>
Therefore, the possible values of $m$ are $m = \boxed{\frac{1}{5}, -\frac{1}{3}}.
$
</Final_answer> | \frac{1}{5}, -\frac{1}{3} | 6 | 36 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
6 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
The real number $x$ satisfies
\[3x + \frac{1}{2x} = 3.\]Find
\[64x^6 + \frac{1}{729x^6}.\]
</Question>
<Plan_1>
Okay, let's see. I need to find the value of 64x⁶ + 1/(729x⁶) given that 3x + 1/(2x) = 3. Hmm, this seems a bit tricky, but maybe I can work through it step by step.
First, let me write down the ... | <Final_answer>
&= \boxed{\frac{416}{27}}.
\end{align*}
</Final_answer> | \frac{416}{27} | 11 | 67 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
17 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
A street has 50 houses on each side, for a total of 100 houses. The addresses on the south side of the street form an arithmetic sequence, as do the addresses on the north side of the street. On the south side, the addresses are 1, 5, 9, etc., and on the north side they are 3, 7, 11, etc. A sign painter pain... | <Plan_7>
Based on execution_6.1:
So for the south side: addresses are 1,5,9,13,...., 4*50 -3 = 197.
For the north side: addresses are 3,7,11,....,4*50 -1 =199.
<prompt> Calculate the total number of digits in the house numbers and determine the painter's earnings. </prompt>
</Plan_7> | 245 | 7 | 88 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
17 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
A street has 50 houses on each side, for a total of 100 houses. The addresses on the south side of the street form an arithmetic sequence, as do the addresses on the north side of the street. On the south side, the addresses are 1, 5, 9, etc., and on the north side they are 3, 7, 11, etc. A sign painter pain... | <Plan_8>
Based on execution_7.1:
Wait, but wait. Let me verify the number of terms:
<prompt> Verify the calculations for the total number of digits and the painter's earnings. </prompt>
</Plan_8> | 245 | 8 | 90 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
17 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
A street has 50 houses on each side, for a total of 100 houses. The addresses on the south side of the street form an arithmetic sequence, as do the addresses on the north side of the street. On the south side, the addresses are 1, 5, 9, etc., and on the north side they are 3, 7, 11, etc. A sign painter pain... | <Final_answer>
Thus, the total revenue is $1\times5 + 2\times 45 + 3\times 50 = \boxed{245}$.
</Final_answer> | 245 | 9 | 92 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
21 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
The set of integers $\{2,4,10,x\}$ has the property that the sum of any three members of the set plus 1 yields a prime number. What is the smallest possible value of $x$ greater than 10?
</Question>
<Plan_1>
Okay, let's try to figure out this problem. So, we have a set of integers {2, 4, 10, x}, and the pro... | <Final_answer>
So the smallest possible value of $x$ is $\boxed{16}$.
</Final_answer> | 16 | 8 | 121 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
25 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
At a certain university, the division of mathematical sciences consists of the departments of mathematics, statistics, and computer science. There are two male and two female professors in each department. A committee of six professors is to contain three men and three women and must also contain two profess... | <Plan_2>
Based on execution_1.1:
Since each department has two males and two females, the number of ways to choose one male and one female from a single department is the product of the combinations for each gender.
For one department, the number of ways to choose one male out of two is C(2,1) = 2, and similarly for on... | 88 | 2 | 124 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
25 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
At a certain university, the division of mathematical sciences consists of the departments of mathematics, statistics, and computer science. There are two male and two female professors in each department. A committee of six professors is to contain three men and three women and must also contain two profess... | <Plan_3>
Based on execution_2.1:
Wait, but actually, the generating function approach here would give the total number of ways to choose professors such that the total number of men is 3. But since each department's contribution is already considering the number of ways to choose the professors (i.e., the coefficients ... | 88 | 3 | 130 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
25 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
At a certain university, the division of mathematical sciences consists of the departments of mathematics, statistics, and computer science. There are two male and two female professors in each department. A committee of six professors is to contain three men and three women and must also contain two profess... | <Plan_4>
Based on execution_3.1:
Let's compute it.
First, expand (1 + 4x + x^2)^3.
<prompt> Expand the generating function and find the coefficient of x^3. </prompt>
-----
Based on execution_3.2:
The final answer is \boxed{88}.
</Plan_4> | 88 | 4 | 133 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
25 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
At a certain university, the division of mathematical sciences consists of the departments of mathematics, statistics, and computer science. There are two male and two female professors in each department. A committee of six professors is to contain three men and three women and must also contain two profess... | <Final_answer>
Summing these two values yields the final answer: $64 + 24 = \boxed{88}$.
</Final_answer> | 88 | 8 | 141 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
26 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
At a certain university, the division of mathematical sciences consists of the departments of mathematics, statistics, and computer science. There are two male and two female professors in each department. A committee of six professors is to contain three men and three women and must also contain two profess... | <Plan_5>
Based on execution_4.1:
Wait, let's check the first approach again. The first approach considered that for each department, the number of ways to choose two professors with m men is C(2, m)*C(2, 2 - m).
<prompt> Re-examine the first approach to understand why it gave a different result. </prompt>
-----
Based... | 88 | 5 | 156 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
26 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
At a certain university, the division of mathematical sciences consists of the departments of mathematics, statistics, and computer science. There are two male and two female professors in each department. A committee of six professors is to contain three men and three women and must also contain two profess... | <Final_answer>
Summing these two values yields the final answer: $64 + 24 = \boxed{88}$.
</Final_answer> | 88 | 6 | 159 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
27 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
At a certain university, the division of mathematical sciences consists of the departments of mathematics, statistics, and computer science. There are two male and two female professors in each department. A committee of six professors is to contain three men and three women and must also contain two profess... | <Final_answer>
Summing these two values yields the final answer: $64 + 24 = \boxed{88}$.
</Final_answer> | 88 | 7 | 179 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
30 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
A polynomial with integer coefficients is of the form
\[x^4 + a_3 x^3 + a_2 x^2 + a_1 x + 4 = 0.\]Enter all the possible integer roots of this polynomial, separated by commas.
</Question>
<Plan_1>
Okay, so I need to find all possible integer roots of the polynomial equation given by x⁴ + a₃ x³ + a₂ x² + a₁ ... | <Final_answer>
By the Integer Root Theorem, the possible integer roots are all the divisors of 4 (including negative divisors), so they are $\boxed{-4,-2,-1,1,2,4}.
$
</Final_answer> | -4,-2,-1,1,2,4 | 5 | 190 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
31 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
A polynomial with integer coefficients is of the form
\[x^4 + a_3 x^3 + a_2 x^2 + a_1 x + 4 = 0.\]Enter all the possible integer roots of this polynomial, separated by commas.
</Question>
<Plan_1>
Okay, so I need to find all possible integer roots of the polynomial equation given by x⁴ + a₃ x³ + a₂ x² + a₁ ... | <Final_answer>
By the Integer Root Theorem, the possible integer roots are all the divisors of 4 (including negative divisors), so they are $\boxed{-4,-2,-1,1,2,4}.
$
</Final_answer> | -4,-2,-1,1,2,4 | 4 | 202 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
34 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
A license plate consists of two letters followed by two digits; for example, $MP78$. Neither the digits nor the letters may be repeated, and neither the letter $O$ nor the digit $0$ may be used. When reading from left to right, the letters must be in alphabetical order and the digits must be in increasing or... | <Plan_1>
Okay, let's try to figure out how many different license plate combinations are possible under these rules. The license plate format is two letters followed by two digits. The letters must be in alphabetical order, and the digits must be in increasing order. Also, neither the digits nor the letters can be repe... | 10800 | 1 | 203 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
34 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
A license plate consists of two letters followed by two digits; for example, $MP78$. Neither the digits nor the letters may be repeated, and neither the letter $O$ nor the digit $0$ may be used. When reading from left to right, the letters must be in alphabetical order and the digits must be in increasing or... | <Plan_3>
Based on execution_2.1:
But let me check with another approach. Suppose we model this as choosing 2 letters out of 26, but since they must be in order, it's like combinations with repetition allowed, but since they can't be the same, it's just combinations without repetition.
The number of such pairs is C(26,... | 10800 | 3 | 212 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
34 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
A license plate consists of two letters followed by two digits; for example, $MP78$. Neither the digits nor the letters may be repeated, and neither the letter $O$ nor the digit $0$ may be used. When reading from left to right, the letters must be in alphabetical order and the digits must be in increasing or... | <Plan_4>
Based on execution_3.2:
Let's confirm once again.
Yes.
<prompt> Perform a final confirmation of the calculations and the answer. </prompt>
</Plan_4> | 10800 | 4 | 215 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
34 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
A license plate consists of two letters followed by two digits; for example, $MP78$. Neither the digits nor the letters may be repeated, and neither the letter $O$ nor the digit $0$ may be used. When reading from left to right, the letters must be in alphabetical order and the digits must be in increasing or... | <Final_answer>
Thus the total number of license plates is $\binom{25}{2}\binom{9}{2} = \boxed{10800}$.
</Final_answer> | 10800 | 5 | 217 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
41 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
Two standard six-faced dice are rolled. Jean wins if the product of the two numbers rolled is odd or a multiple of three, otherwise Allen wins. What is the probability that Jean wins? Express your answer as a common fraction.
</Question>
<Plan_1>
Okay, let's try to figure out this probability problem. So, t... | <Plan_2>
Based on execution_1.1, execution_1.2:
But now, we need to find the overlap between these two sets: outcomes where the product is odd and outcomes where the product is a multiple of three. Wait, let's think. For a product to be odd and a multiple of three, it must be odd and divisible by three. So, the product... | \frac{2}{3} | 2 | 221 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
41 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
Two standard six-faced dice are rolled. Jean wins if the product of the two numbers rolled is odd or a multiple of three, otherwise Allen wins. What is the probability that Jean wins? Express your answer as a common fraction.
</Question>
<Plan_1>
Okay, let's try to figure out this probability problem. So, t... | <Plan_3>
Based on execution_2.1:
But let me cross-verify this with another approach. Maybe instead of inclusion-exclusion, compute the probability directly.
Let's see.
<prompt> Verify the overlap calculation using an alternative approach by directly computing the number of pairs where the product is odd and a multipl... | \frac{2}{3} | 3 | 223 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
41 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
Two standard six-faced dice are rolled. Jean wins if the product of the two numbers rolled is odd or a multiple of three, otherwise Allen wins. What is the probability that Jean wins? Express your answer as a common fraction.
</Question>
<Plan_1>
Okay, let's try to figure out this probability problem. So, t... | <Final_answer>
Then, the probability that Jean wins is $1-1/3=\boxed{\frac{2}{3}}$.
</Final_answer> | \frac{2}{3} | 4 | 232 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
46 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
In triangle $ABC$, let angle bisectors $BD$ and $CE$ intersect at $I$. The line through $I$ parallel to $BC$ intersects $AB$ and $AC$ at $M$ and $N$, respectively. If $AB = 17$, $AC = 24$, and $BC = 33$, then find the perimeter of triangle $AMN$.
</Question>
<Plan_1>
Okay, so I need to find the perimeter ... | <Plan_3>
Based on execution_2.3:
But 592 divided by 17: 17*34 = 578, 592 - 578 = 14, so 34 and 14/17. So, 34.8235. But the problem asks for the perimeter, which should be an integer? Wait, 592/17 is 34.8235... which is not an integer. But the given sides are integers, but the perimeter might not be. Wait, but let me ch... | 41 | 3 | 240 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
46 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
In triangle $ABC$, let angle bisectors $BD$ and $CE$ intersect at $I$. The line through $I$ parallel to $BC$ intersects $AB$ and $AC$ at $M$ and $N$, respectively. If $AB = 17$, $AC = 24$, and $BC = 33$, then find the perimeter of triangle $AMN$.
</Question>
<Plan_1>
Okay, so I need to find the perimeter ... | <Plan_4>
Based on execution_3.1, execution_3.2, execution_3.3:
But wait, this contradicts the previous result where the perimeter was 592/17 ≈ 34.8235. Which is correct?
There's a discrepancy here. Let me check where the mistake is.
<prompt> Compare different approaches and identify discrepancies in the results. </pr... | 41 | 4 | 245 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
46 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
In triangle $ABC$, let angle bisectors $BD$ and $CE$ intersect at $I$. The line through $I$ parallel to $BC$ intersects $AB$ and $AC$ at $M$ and $N$, respectively. If $AB = 17$, $AC = 24$, and $BC = 33$, then find the perimeter of triangle $AMN$.
</Question>
<Plan_1>
Okay, so I need to find the perimeter ... | <Plan_5>
Based on execution_4.1, execution_4.3:
Alternatively, since we have coordinates, perhaps using the ratio or parametric equations could help.
But earlier, we found the ratio t for M on AB is 33/74. Therefore, the length AM is t * AB, since AB is 17. Therefore, AM = (33/74)*17 = 561/74 ≈ 7.58.
Similarly, for N... | 41 | 5 | 249 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
46 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
In triangle $ABC$, let angle bisectors $BD$ and $CE$ intersect at $I$. The line through $I$ parallel to $BC$ intersects $AB$ and $AC$ at $M$ and $N$, respectively. If $AB = 17$, $AC = 24$, and $BC = 33$, then find the perimeter of triangle $AMN$.
</Question>
<Plan_1>
Okay, so I need to find the perimeter ... | <Plan_7>
Based on execution_6.1:
But where did the coordinate geometry approach go wrong? Because when I computed the perimeter using coordinates, I got approximately 39.18, but according to this ratio, it should be 41. Therefore, perhaps there was an error in the coordinate calculations.
Let me check the coordinates ... | 41 | 7 | 253 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
46 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
In triangle $ABC$, let angle bisectors $BD$ and $CE$ intersect at $I$. The line through $I$ parallel to $BC$ intersects $AB$ and $AC$ at $M$ and $N$, respectively. If $AB = 17$, $AC = 24$, and $BC = 33$, then find the perimeter of triangle $AMN$.
</Question>
<Plan_1>
Okay, so I need to find the perimeter ... | <Final_answer>
&= \boxed{41}.
\end{align*}
</Final_answer> | 41 | 8 | 255 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
52 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
Let $\mathbf{v}_0$ be a vector. The vector $\mathbf{v}_0$ is rotated about the origin by an angle of $42^\circ$ counter-clockwise, taking it to vector $\mathbf{v}_1.$ The vector $\mathbf{v}_1$ is then reflected over the line with direction vector $\begin{pmatrix} \cos 108^\circ \\ \sin 108^\circ \end{pmatr... | <Final_answer>
\end{align*}Thus, $\theta = 174^\circ/2 = \boxed{87^\circ}.
$
</Final_answer> | 87^\circ | 11 | 284 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
54 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
For integers $n$, let \[f(n) = \left\{
\begin{array}{cl}
n^2 + 1 & \text{ if }n\text{ is odd}, \\
n^2 - 3n + 1 & \text{ if }n\text{ is even}.
\end{array}
\right.\]Find $f(f(f(f(f(f(2))))))$.
</Question>
<Plan_1>
Okay, let's tackle this problem step by step. The function f(n) is defined differently for odd a... | <Plan_3>
Based on execution_2.1:
Wait, no: starting from 2, each f is applied once. So:
f(2) = -1 (1st)
<prompt> Re-evaluate the function applications and conclude the result of the sixth application. </prompt>
-----
Based on execution_2.3:
Therefore, the initial detailed calculation where we went through 5, 26, etc... | 2 | 3 | 293 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
54 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
For integers $n$, let \[f(n) = \left\{
\begin{array}{cl}
n^2 + 1 & \text{ if }n\text{ is odd}, \\
n^2 - 3n + 1 & \text{ if }n\text{ is even}.
\end{array}
\right.\]Find $f(f(f(f(f(f(2))))))$.
</Question>
<Plan_1>
Okay, let's tackle this problem step by step. The function f(n) is defined differently for odd a... | <Final_answer>
$f(f(f(f(f(f(2))))))=f(f(f(f(f(-1)))))=f(f(f(f(2))))=f(f(f(-1)))=f(f(2))=f(-1)=\boxed{2}.
$
</Final_answer> | 2 | 5 | 297 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
57 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
In the diagram below, $\overline{AB}\parallel \overline{CD}$ and $\angle AXF= 118^\circ$. Find $\angle FYD$.
[asy]
unitsize(1inch);
pair A,B,C,D,X,Y,EE,F;
A = (0,0);
B=(1,0);
C = (0,0.8);
D=(1,0.8);
EE = (0.35,-0.3);
F = (0.8,1.1);
draw(EE--F);
draw(A--B);
draw(C--D);
dot(A);
dot(B);
dot(C);
dot(D);
dot(EE... | <Plan_2>
Based on execution_1.4:
Wait, in the Asymptote code, the coordinates are given, so maybe I can calculate the angles using coordinates. Let me try that as a verification.
<prompt> Calculate angle AXF using the coordinates provided and compare it with the given angle. </prompt>
-----
Based on execution_1.1, ex... | 62^\circ | 2 | 303 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
57 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
In the diagram below, $\overline{AB}\parallel \overline{CD}$ and $\angle AXF= 118^\circ$. Find $\angle FYD$.
[asy]
unitsize(1inch);
pair A,B,C,D,X,Y,EE,F;
A = (0,0);
B=(1,0);
C = (0,0.8);
D=(1,0.8);
EE = (0.35,-0.3);
F = (0.8,1.1);
draw(EE--F);
draw(A--B);
draw(C--D);
dot(A);
dot(B);
dot(C);
dot(D);
dot(EE... | <Plan_4>
Based on execution_3.1:
y = -0.3 + 1.4t = 0
1.4t = 0.3
<prompt> Re-calculate the coordinates and verify the angle using a more accurate approach. </prompt>
-----
Based on execution_3.2:
Wait, maybe I made a mistake in determining the type of angles.
Let me recall:
<prompt> Clarify angle definitions and de... | 62^\circ | 4 | 309 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
57 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
In the diagram below, $\overline{AB}\parallel \overline{CD}$ and $\angle AXF= 118^\circ$. Find $\angle FYD$.
[asy]
unitsize(1inch);
pair A,B,C,D,X,Y,EE,F;
A = (0,0);
B=(1,0);
C = (0,0.8);
D=(1,0.8);
EE = (0.35,-0.3);
F = (0.8,1.1);
draw(EE--F);
draw(A--B);
draw(C--D);
dot(A);
dot(B);
dot(C);
dot(D);
dot(EE... | <Plan_5>
Based on execution_4.2:
Let me imagine the diagram again. The transversal EF is going from E (lower left) to F (upper right).
<prompt> Re-evaluate the positions and types of angles to determine the correct measure of angle FYD. </prompt>
</Plan_5> | 62^\circ | 5 | 312 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
57 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
In the diagram below, $\overline{AB}\parallel \overline{CD}$ and $\angle AXF= 118^\circ$. Find $\angle FYD$.
[asy]
unitsize(1inch);
pair A,B,C,D,X,Y,EE,F;
A = (0,0);
B=(1,0);
C = (0,0.8);
D=(1,0.8);
EE = (0.35,-0.3);
F = (0.8,1.1);
draw(EE--F);
draw(A--B);
draw(C--D);
dot(A);
dot(B);
dot(C);
dot(D);
dot(EE... | <Plan_7>
Based on execution_6.1:
Therefore, according to the problem's data, angle AXF is 118 degrees, so angle FYD would also be 118 degrees as alternate exterior angles.
Wait, but in the coordinate calculation, angle FYD would be the angle at Y between FY and DY.
<prompt> Address discrepancies between coordinate-ba... | 62^\circ | 7 | 316 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
57 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
In the diagram below, $\overline{AB}\parallel \overline{CD}$ and $\angle AXF= 118^\circ$. Find $\angle FYD$.
[asy]
unitsize(1inch);
pair A,B,C,D,X,Y,EE,F;
A = (0,0);
B=(1,0);
C = (0,0.8);
D=(1,0.8);
EE = (0.35,-0.3);
F = (0.8,1.1);
draw(EE--F);
draw(A--B);
draw(C--D);
dot(A);
dot(B);
dot(C);
dot(D);
dot(EE... | <Final_answer>
Since $\overline{AB} \parallel \overline{CD}$, we know that $\angle AXF + \angle FYD = 180^\circ$, so $\angle FYD = 180^\circ - 118^\circ = \boxed{62^\circ}$.
</Final_answer> | 62^\circ | 9 | 320 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
61 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
A tribe of eight Native Americans sits in a circle around a campfire. If the chief must sit between his mother and his wife, how many different seating arrangements are possible? Two seating arrangements are considered the same if each individual has the same person to the right and the same person to the le... | <Plan_2>
Based on execution_1.1:
But here, there's a restriction: the chief must be between his mother and his wife.
Alright, let's think about the constraints.
<prompt> Identify and explain the constraint involving the chief, his mother, and his wife. </prompt>
-----
But let me think of another way. Suppose all pos... | 240 | 2 | 323 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
61 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
A tribe of eight Native Americans sits in a circle around a campfire. If the chief must sit between his mother and his wife, how many different seating arrangements are possible? Two seating arrangements are considered the same if each individual has the same person to the right and the same person to the le... | <Plan_3>
Based on execution_2.1:
I remember that in circular permutations with specific adjacency constraints, sometimes we can treat the group as a single unit. For example, if three people must sit together, we might consider them as one entity, which reduces the problem's size. But in this case, it's not exactly thr... | 240 | 3 | 326 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
61 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
A tribe of eight Native Americans sits in a circle around a campfire. If the chief must sit between his mother and his wife, how many different seating arrangements are possible? Two seating arrangements are considered the same if each individual has the same person to the right and the same person to the le... | <Plan_4>
Based on execution_3.2:
But wait, let me confirm again.
Alternatively, if we didn't fix the chief's position, the total arrangements would be (8-1)!
<prompt> Verify the calculation by considering the constraint and its effect on the total arrangements. </prompt>
-----
Based on execution_3.2, execution_3.3:
... | 240 | 4 | 331 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
61 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
A tribe of eight Native Americans sits in a circle around a campfire. If the chief must sit between his mother and his wife, how many different seating arrangements are possible? Two seating arrangements are considered the same if each individual has the same person to the right and the same person to the le... | <Final_answer>
=\boxed{240}$ possible arrangements.
</Final_answer> | 240 | 6 | 338 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
74 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
Find the minimum of the function
\[\frac{xy}{x^2 + y^2}\]in the domain $\frac{2}{5} \le x \le \frac{1}{2}$ and $\frac{1}{3} \le y \le \frac{3}{8}.$
</Question>
<Plan_1>
Okay, so I need to find the minimum of the function f(x, y) = (xy)/(x² + y²) in the domain where x is between 2/5 and 1/2, and y is between... | <Plan_5>
Based on execution_4.1:
But wait, let me check the denominator computation again for (1/2, 1/3):
Denominator is (1/2)^2 + (1/3)^2 = 1/4 + 1/9.
<prompt> Re-check the denominator computation for (1/2, 1/3) as per the above plan. </prompt>
</Plan_5> | \frac{6}{13} | 5 | 355 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
74 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
Find the minimum of the function
\[\frac{xy}{x^2 + y^2}\]in the domain $\frac{2}{5} \le x \le \frac{1}{2}$ and $\frac{1}{3} \le y \le \frac{3}{8}.$
</Question>
<Plan_1>
Okay, so I need to find the minimum of the function f(x, y) = (xy)/(x² + y²) in the domain where x is between 2/5 and 1/2, and y is between... | <Plan_7>
Based on execution_4.1, execution_6.1, execution_2.7, execution_2.8, execution_2.10:
Now, compiling the minima from each edge:
Left edge: 0.4918 at (2/5, 1/3)
Right edge: 0.4615 at (1/2, 1/3)
<prompt> Compile the results from all edges and corners to determine the overall minimum as per the above plan. </pr... | \frac{6}{13} | 7 | 359 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
74 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
Find the minimum of the function
\[\frac{xy}{x^2 + y^2}\]in the domain $\frac{2}{5} \le x \le \frac{1}{2}$ and $\frac{1}{3} \le y \le \frac{3}{8}.$
</Question>
<Plan_1>
Okay, so I need to find the minimum of the function f(x, y) = (xy)/(x² + y²) in the domain where x is between 2/5 and 1/2, and y is between... | <Final_answer>
\]Thus, the minimum value is $\boxed{\frac{6}{13}}.
$
</Final_answer> | \frac{6}{13} | 8 | 361 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
75 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
Find the minimum of the function
\[\frac{xy}{x^2 + y^2}\]in the domain $\frac{2}{5} \le x \le \frac{1}{2}$ and $\frac{1}{3} \le y \le \frac{3}{8}.$
</Question>
<Plan_1>
Okay, so I need to find the minimum of the function f(x, y) = (xy)/(x² + y²) in the domain where x is between 2/5 and 1/2, and y is between... | <Plan_4>
Based on execution_3.1, execution_3.2, execution_2.5:
As calculated earlier, the minimum value is 12/25 at (1/2, 3/8).
But wait, earlier I saw that f(1/2, 3/8) = 12/25 = 0.48, which is higher than f(1/2, 1/3) = 6/13 ≈ 0.4615.
<prompt> Re-evaluate the minimum value among the corners and edges as per the above... | \frac{6}{13} | 4 | 374 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
75 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
Find the minimum of the function
\[\frac{xy}{x^2 + y^2}\]in the domain $\frac{2}{5} \le x \le \frac{1}{2}$ and $\frac{1}{3} \le y \le \frac{3}{8}.$
</Question>
<Plan_1>
Okay, so I need to find the minimum of the function f(x, y) = (xy)/(x² + y²) in the domain where x is between 2/5 and 1/2, and y is between... | <Plan_5>
Based on execution_4.1:
But let me check once more by evaluating another point along the bottom edge. Suppose x=0.45 (which is within [2/5=0.4, 0.5]), y=1/3=0.333.
<prompt> Confirm the minimum value by evaluating another point along the bottom edge as per the above plan. </prompt>
</Plan_5> | \frac{6}{13} | 5 | 376 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
75 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
Find the minimum of the function
\[\frac{xy}{x^2 + y^2}\]in the domain $\frac{2}{5} \le x \le \frac{1}{2}$ and $\frac{1}{3} \le y \le \frac{3}{8}.$
</Question>
<Plan_1>
Okay, so I need to find the minimum of the function f(x, y) = (xy)/(x² + y²) in the domain where x is between 2/5 and 1/2, and y is between... | <Final_answer>
\]Thus, the minimum value is $\boxed{\frac{6}{13}}.
$
</Final_answer> | \frac{6}{13} | 6 | 378 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
82 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
Consider the six by six grid of unit squares below. How many rectangles of area 3 square units can be formed using only the line segments of the grid as the sides of the rectangles?
[asy]unitsize(0.1inch);
draw((0,0)--(6,0)--(6,6)--(0,6)--cycle);
draw((1,0)--(1,6));
draw((2,0)--(2,6));
draw((3,0)--(3,6));
d... | <Final_answer>
Therefore, the total number of rectangles of area 3 which can be formed is $\boxed{48}$.
</Final_answer> | 48 | 8 | 402 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
83 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
How many integers $m$ are there such that $0<m<100$ and $\gcd(m,100)$ is a single-digit number?
</Question>
<Plan_1>
Okay, so I need to figure out how many integers m between 1 and 99 inclusive have a greatest common divisor (gcd) with 100 that is a single-digit number. Hmm, let's start by understanding the... | <Final_answer>
So, that leaves $99-9-2=\boxed{88}$ integers whose greatest common divisor with $100$ has a single digit.
</Final_answer> | 88 | 7 | 428 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
86 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
The exam grades of a pre-algebra class were arranged in a stem and leaf plot as illustrated. What is the arithmetic mean of the median and the mode of the given data?
\begin{tabular}{ c | c c c ccc c c c}
4&1&&&&&&&&\\
5&2&&&&&&&&\\
6&7&8&8&&&&&&\\
7&1&1&2&3&3&3&5&6&8\\
8&0&4&4&6&6&6&6&8&\\
9&1&3&5&5&7&&&&\... | <Plan_4>
Based on execution_3.2, execution_3.3:
Now, the arithmetic mean of the median and the mode is (76 + 86)/2 = (162)/2 = 81.
Based on execution_3.3:
Wait, but hold on. Let me double-check the counts for each number to make sure I got the mode right.
Looking back at the stem and leaf plot:
<prompt> Verify the m... | 82 | 4 | 437 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
86 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
The exam grades of a pre-algebra class were arranged in a stem and leaf plot as illustrated. What is the arithmetic mean of the median and the mode of the given data?
\begin{tabular}{ c | c c c ccc c c c}
4&1&&&&&&&&\\
5&2&&&&&&&&\\
6&7&8&8&&&&&&\\
7&1&1&2&3&3&3&5&6&8\\
8&0&4&4&6&6&6&6&8&\\
9&1&3&5&5&7&&&&\... | <Plan_6>
Based on execution_5.1:
That changes things. Let's correct that.
So, median is 78, mode is 86.
<prompt> Finalize the solution by calculating the correct arithmetic mean with the corrected median. </prompt>
</Plan_6> | 82 | 6 | 442 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
86 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
The exam grades of a pre-algebra class were arranged in a stem and leaf plot as illustrated. What is the arithmetic mean of the median and the mode of the given data?
\begin{tabular}{ c | c c c ccc c c c}
4&1&&&&&&&&\\
5&2&&&&&&&&\\
6&7&8&8&&&&&&\\
7&1&1&2&3&3&3&5&6&8\\
8&0&4&4&6&6&6&6&8&\\
9&1&3&5&5&7&&&&\... | <Final_answer>
The arithmetic mean of the median and mode of the data is $\dfrac{1}{2}(78+86),$ or $\boxed{82}.
$
</Final_answer> | 82 | 7 | 444 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
87 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
The exam grades of a pre-algebra class were arranged in a stem and leaf plot as illustrated. What is the arithmetic mean of the median and the mode of the given data?
\begin{tabular}{ c | c c c ccc c c c}
4&1&&&&&&&&\\
5&2&&&&&&&&\\
6&7&8&8&&&&&&\\
7&1&1&2&3&3&3&5&6&8\\
8&0&4&4&6&6&6&6&8&\\
9&1&3&5&5&7&&&&\... | <Plan_4>
Based on execution_3.2, execution_3.3:
Now, the arithmetic mean of the median and the mode is (76 + 86)/2.
Let's compute that.
76 + 86 = 162
162 divided by 2 is 81.
Based on execution_3.1:
Wait, but hold on. Let me verify the data points again in case I miscounted.
<prompt> Verify the total number of data p... | 82 | 4 | 453 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
87 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
The exam grades of a pre-algebra class were arranged in a stem and leaf plot as illustrated. What is the arithmetic mean of the median and the mode of the given data?
\begin{tabular}{ c | c c c ccc c c c}
4&1&&&&&&&&\\
5&2&&&&&&&&\\
6&7&8&8&&&&&&\\
7&1&1&2&3&3&3&5&6&8\\
8&0&4&4&6&6&6&6&8&\\
9&1&3&5&5&7&&&&\... | <Plan_5>
Based on execution_4.1:
Therefore, the median is the (27+1)/2 = 14th term? Wait, wait! Wait, when there are an odd number of data points, the median is the middle one.
So, for 27 data points, it's the 14th term. Wait, no, the formula is (n+1)/2 th term. So, yes, (27 +1)/2=14th term.
-----
Based on execution_4... | 82 | 5 | 456 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
87 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
The exam grades of a pre-algebra class were arranged in a stem and leaf plot as illustrated. What is the arithmetic mean of the median and the mode of the given data?
\begin{tabular}{ c | c c c ccc c c c}
4&1&&&&&&&&\\
5&2&&&&&&&&\\
6&7&8&8&&&&&&\\
7&1&1&2&3&3&3&5&6&8\\
8&0&4&4&6&6&6&6&8&\\
9&1&3&5&5&7&&&&\... | <Plan_6>
Based on execution_4.2, execution_5.1:
So listing all 27 numbers in order:
1.
<prompt> Perform a final verification and correction of the median and calculate the mean. </prompt>
</Plan_6> | 82 | 6 | 458 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
87 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
The exam grades of a pre-algebra class were arranged in a stem and leaf plot as illustrated. What is the arithmetic mean of the median and the mode of the given data?
\begin{tabular}{ c | c c c ccc c c c}
4&1&&&&&&&&\\
5&2&&&&&&&&\\
6&7&8&8&&&&&&\\
7&1&1&2&3&3&3&5&6&8\\
8&0&4&4&6&6&6&6&8&\\
9&1&3&5&5&7&&&&\... | <Final_answer>
The arithmetic mean of the median and mode of the data is $\dfrac{1}{2}(78+86),$ or $\boxed{82}.
$
</Final_answer> | 82 | 7 | 460 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
89 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
In the $19^\text{th}$ century, Britain used a money system which included pence, farthings, shillings and pounds. The following conversions were used:
$\bullet$ $4 \text{ farthings} = 1 \text{ pence},$
$\bullet$ $12 \text{ pence} = 1 \text{ shilling},$
$\bullet$ $20 \text{ shillings} = 1 \text{ pound}.$
... | <Plan_1>
Okay, let's see. The problem is asking how many farthings are equivalent to 1 pound and 5 pence in the old British money system. Hmm, I need to convert pounds and pence into farthings. Alright, let me start by recalling the conversion factors given.
First, the conversions:
- 4 farthings = 1 pence
- 12 pence ... | 980 | 1 | 461 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
89 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
In the $19^\text{th}$ century, Britain used a money system which included pence, farthings, shillings and pounds. The following conversions were used:
$\bullet$ $4 \text{ farthings} = 1 \text{ pence},$
$\bullet$ $12 \text{ pence} = 1 \text{ shilling},$
$\bullet$ $20 \text{ shillings} = 1 \text{ pound}.$
... | <Final_answer>
\] The remaining $5$ pence is equal to $20$ farthings, so the total is $\boxed{980}$ farthings.
</Final_answer> | 980 | 4 | 472 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
91 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
Polyhedron $P$ is inscribed in a sphere of radius $36$ (meaning that all vertices of $P$ lie on the sphere surface). What is the least upper bound on the ratio $$\frac{\text{volume of }P}{\text{surface area of }P}~?$$In other words, what is the smallest real number $t$ such that $$\frac{\text{volume of }P}{\... | <Plan_2>
Based on execution_1.1:
But wait, the problem says "the least upper bound on the ratio", which is the smallest t such that for all polyhedra P, the ratio is ≤ t. So even if the sphere's ratio is not attained by any polyhedron, it's still the least upper bound because you can approach it arbitrarily closely by ... | 12 | 2 | 508 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
91 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
Polyhedron $P$ is inscribed in a sphere of radius $36$ (meaning that all vertices of $P$ lie on the sphere surface). What is the least upper bound on the ratio $$\frac{\text{volume of }P}{\text{surface area of }P}~?$$In other words, what is the smallest real number $t$ such that $$\frac{\text{volume of }P}{\... | <Final_answer>
So, the answer is $\boxed{12}$.
</Final_answer> | 12 | 3 | 511 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
96 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
If $x+y=\frac{7}{12}$ and $x-y=\frac{1}{12}$, what is the value of $x^2-y^2$? Express your answer as a common fraction.
</Question>
<Plan_1>
Okay, let's see. I need to find the value of x² - y² given the equations x + y = 7/12 and x - y = 1/12. Hmm, x² - y²... Oh, right, that's a difference of squares. So, ... | <Plan_4>
Based on execution_3.1:
Wait a second, but earlier method gave 7/144. Hmm, there's a contradiction here.
<prompt> Identify and resolve the contradiction between the two methods used to find x² - y². </prompt>
</Plan_4> | \frac{7}{144} | 4 | 519 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
96 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
If $x+y=\frac{7}{12}$ and $x-y=\frac{1}{12}$, what is the value of $x^2-y^2$? Express your answer as a common fraction.
</Question>
<Plan_1>
Okay, let's see. I need to find the value of x² - y² given the equations x + y = 7/12 and x - y = 1/12. Hmm, x² - y²... Oh, right, that's a difference of squares. So, ... | <Plan_5>
Based on execution_4.1:
Wait, no. Let me check that again.
<prompt> Re-evaluate the calculations to resolve the contradiction between the two methods. </prompt>
But earlier when I tried solving for x and y in a different way, I thought:
Wait, perhaps I made a miscalculation there. Let me check again.
<prom... | \frac{7}{144} | 5 | 521 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
96 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
If $x+y=\frac{7}{12}$ and $x-y=\frac{1}{12}$, what is the value of $x^2-y^2$? Express your answer as a common fraction.
</Question>
<Plan_1>
Okay, let's see. I need to find the value of x² - y² given the equations x + y = 7/12 and x - y = 1/12. Hmm, x² - y²... Oh, right, that's a difference of squares. So, ... | <Final_answer>
$x^2 - y^2$ factors into $(x+y)(x-y)$, so, to obtain the value of $x^2 - y^2$, simply multiply $\frac{7}{12}\cdot \frac{1}{12}$ to get $\boxed{\frac{7}{144}}$.
</Final_answer> | \frac{7}{144} | 6 | 527 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
97 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
If $x+y=\frac{7}{12}$ and $x-y=\frac{1}{12}$, what is the value of $x^2-y^2$? Express your answer as a common fraction.
</Question>
<Plan_1>
Okay, let's see. I need to find the value of x² - y² given the equations x + y = 7/12 and x - y = 1/12. Hmm, x² - y²... Oh, right, that's a difference of squares. So, ... | <Plan_3>
Based on execution_2.1:
Let me just recap to make sure I didn't skip any steps.
First method: Recognize x² - y² as (x + y)(x - y), substitute the given values directly. Second method: Solve for x and y by adding and subtracting the equations, then compute x² - y².
Both methods lead to the same answer. Therefo... | \frac{7}{144} | 3 | 535 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
97 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
If $x+y=\frac{7}{12}$ and $x-y=\frac{1}{12}$, what is the value of $x^2-y^2$? Express your answer as a common fraction.
</Question>
<Plan_1>
Okay, let's see. I need to find the value of x² - y² given the equations x + y = 7/12 and x - y = 1/12. Hmm, x² - y²... Oh, right, that's a difference of squares. So, ... | <Plan_4>
Based on execution_3.1:
Wait, but earlier when I thought of another method where the user might have got confused, perhaps. Wait, the user's question is written as:
"If +y=7/12 and -y=1/12, what is the value of ^2 - y^20"?
Wait, maybe the original problem has a typo, but in the user's input, it's written as ... | \frac{7}{144} | 4 | 537 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
97 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
If $x+y=\frac{7}{12}$ and $x-y=\frac{1}{12}$, what is the value of $x^2-y^2$? Express your answer as a common fraction.
</Question>
<Plan_1>
Okay, let's see. I need to find the value of x² - y² given the equations x + y = 7/12 and x - y = 1/12. Hmm, x² - y²... Oh, right, that's a difference of squares. So, ... | <Final_answer>
$x^2 - y^2$ factors into $(x+y)(x-y)$, so, to obtain the value of $x^2 - y^2$, simply multiply $\frac{7}{12}\cdot \frac{1}{12}$ to get $\boxed{\frac{7}{144}}$.
</Final_answer> | \frac{7}{144} | 5 | 539 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
98 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
3 cards are chosen at random from a standard 52-card deck. What is the probability that they form a pair? (A 3-card hand is a 'pair' if two of the cards match in rank but the third card is different. For example, 668 is a pair, but 999 is not.)
</Question>
<Plan_1>
Okay, so I need to find the probability... | <Final_answer>
So the probability that a randomly drawn hand is a pair is $\dfrac{3744}{22100} = \boxed{\dfrac{72}{425}}$.
</Final_answer> | \dfrac{72}{425} | 4 | 552 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
100 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
3 cards are chosen at random from a standard 52-card deck. What is the probability that they form a pair? (A 3-card hand is a 'pair' if two of the cards match in rank but the third card is different. For example, 668 is a pair, but 999 is not.)
</Question>
<Plan_1>
Okay, so I need to find the probability... | <Final_answer>
So the probability that a randomly drawn hand is a pair is $\dfrac{3744}{22100} = \boxed{\dfrac{72}{425}}$.
</Final_answer> | \dfrac{72}{425} | 9 | 573 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
101 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
Let $a$ and $b$ be real constants such that
\[x^4 + ax^3 + 3x^2 + bx + 1 \ge 0\]for all real numbers $x.$ Find the largest possible value of $a^2 + b^2.$
</Question>
<Plan_1>
Okay, so I need to find the largest possible value of \( a^2 + b^2 \) given that the polynomial \( x^4 + ax^3 + 3x^2 + bx + 1 \geq 0... | <Plan_7>
Based on execution_6.1:
This suggests that larger values of \( a^2 + b^2 \) are possible. However, this approach might require finding the global maximum of \( a^2 + b^2 \) as a function of \( c \), which is a challenging calculus problem.
Let me consider another approach.
<prompt> Conclude the maximum value... | 40 | 7 | 594 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
101 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
Let $a$ and $b$ be real constants such that
\[x^4 + ax^3 + 3x^2 + bx + 1 \ge 0\]for all real numbers $x.$ Find the largest possible value of $a^2 + b^2.$
</Question>
<Plan_1>
Okay, so I need to find the largest possible value of \( a^2 + b^2 \) given that the polynomial \( x^4 + ax^3 + 3x^2 + bx + 1 \geq 0... | <Final_answer>
\]Hence, the maximum value of $a^2 + b^2$ is $\boxed{40}.
$
</Final_answer> | 40 | 8 | 596 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
106 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
In the diagram below, $AB = AC = 115,$ $AD = 38,$ and $CF = 77.$ Compute $\frac{[CEF]}{[DBE]}.$
[asy]
unitsize(0.025 cm);
pair A, B, C, D, E, F;
B = (0,0);
C = (80,0);
A = intersectionpoint(arc(B,115,0,180),arc(C,115,0,180));
D = interp(A,B,38/115);
F = interp(A,C,(115 + 77)/115);
E = extension(B,C,D,F);... | <Plan_6>
Based on execution_5.1:
Therefore, EC = (19/96) BE. Wait, but BE + EC = BC = 80.
<prompt> Calculate the segments BE and EC using the ratio found. </prompt>
-----
Based on execution_5.2:
Therefore, perhaps the coordinates need to be recalculated correctly.
Alternate approach: Let's ignore the Asymptote code ... | \frac{19}{96} | 6 | 611 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
106 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
In the diagram below, $AB = AC = 115,$ $AD = 38,$ and $CF = 77.$ Compute $\frac{[CEF]}{[DBE]}.$
[asy]
unitsize(0.025 cm);
pair A, B, C, D, E, F;
B = (0,0);
C = (80,0);
A = intersectionpoint(arc(B,115,0,180),arc(C,115,0,180));
D = interp(A,B,38/115);
F = interp(A,C,(115 + 77)/115);
E = extension(B,C,D,F);... | <Final_answer>
&= \boxed{\frac{19}{96}}.
\end{align*}
</Final_answer> | \frac{19}{96} | 8 | 617 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
110 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
Find the minimum value of
\[3x^2 + 12y^2 + 27z^2 - 4xy - 6xz - 12yz - 8y - 24z\]over all real numbers $x,$ $y,$ and $z.$
</Question>
<Plan_1>
Okay, so I need to find the minimum value of this quadratic expression:
3x² + 12y² + 27z² - 4xy - 6xz - 12yz - 8y - 24z.
Hmm, quadratic in three variables. I remem... | <Plan_9>
Based on execution_8.1:
Let's see if this can be written as a mixed number or something else.
4064 divided by 529: 529*7=3703, 4064-3703=361.
<prompt> Express the fraction -4064/529 as a mixed number or decimal. </prompt>
Let me check if this can be simplified. 4064 and 529.
<prompt> Verify if the fraction... | -14 | 9 | 637 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
110 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
Find the minimum value of
\[3x^2 + 12y^2 + 27z^2 - 4xy - 6xz - 12yz - 8y - 24z\]over all real numbers $x,$ $y,$ and $z.$
</Question>
<Plan_1>
Okay, so I need to find the minimum value of this quadratic expression:
3x² + 12y² + 27z² - 4xy - 6xz - 12yz - 8y - 24z.
Hmm, quadratic in three variables. I remem... | <Final_answer>
\]Thus, the minimum value is $\boxed{-14},$ which occurs when $x - \frac{2y + 3z}{3} = y - \frac{6z + 3}{8} = z - \frac{5}{6} = 0,$ or $x = \frac{3}{2},$ $y = 1,$ and $z = \frac{5}{6}.
$
</Final_answer> | -14 | 10 | 642 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
112 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
In the figure below, the smaller circle has a radius of two feet and the larger circle has a radius of four feet. What is the total area of the four shaded regions? Express your answer as a decimal to the nearest tenth.
[asy]
fill((0,0)--(12,0)--(12,-4)--(4,-4)--(4,-2)--(0,-2)--cycle,gray(0.7));
draw((0,0)-... | <Plan_7>
Based on execution_6.1:
But I recall that in the Asymptote code, the circles are filled white, which means they are subtracted from the shaded area. So the figure is filled gray, then the two circles are filled white.
<prompt> Confirm the final solution and provide the answer. </prompt>
</Plan_7> | 8.6 | 7 | 659 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
112 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
In the figure below, the smaller circle has a radius of two feet and the larger circle has a radius of four feet. What is the total area of the four shaded regions? Express your answer as a decimal to the nearest tenth.
[asy]
fill((0,0)--(12,0)--(12,-4)--(4,-4)--(4,-2)--(0,-2)--cycle,gray(0.7));
draw((0,0)-... | <Final_answer>
[(4)(8) - \dfrac{1}{2}\pi \cdot(4)^2] = 40 - 10\pi \approx \boxed{8.6}$.
Equivalently, we could notice that since the right side of the figure is scaled up from the left side by a factor of 2, areas will be scaled by a factor of $2^2 = 4$, and the right shaded region will be 4 times the size of the lef... | 8.6 | 8 | 661 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
113 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
In the figure below, the smaller circle has a radius of two feet and the larger circle has a radius of four feet. What is the total area of the four shaded regions? Express your answer as a decimal to the nearest tenth.
[asy]
fill((0,0)--(12,0)--(12,-4)--(4,-4)--(4,-2)--(0,-2)--cycle,gray(0.7));
draw((0,0)-... | <Final_answer>
[(4)(8) - \dfrac{1}{2}\pi \cdot(4)^2] = 40 - 10\pi \approx \boxed{8.6}$.
Equivalently, we could notice that since the right side of the figure is scaled up from the left side by a factor of 2, areas will be scaled by a factor of $2^2 = 4$, and the right shaded region will be 4 times the size of the lef... | 8.6 | 6 | 689 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
114 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
Let $S$ be the set of all points $(x,y)$ in the coordinate plane such that $0\leq x\leq \frac{\pi}{2}$ and $0\leq y\leq
\frac{\pi}{2}$. What is the area of the subset of $S$ for which \[
\sin^2 x -\sin x \sin y +\sin^2 y \leq\frac{3}{4}\,?
\]
</Question>
<Plan_1>
Okay, so I need to find the area of the subs... | <Final_answer>
2\cdot\frac{1}{2}\cdot\displaystyle\left(\frac{\pi}{6}\displaystyle\right)^2=\boxed{\frac{\pi^2}{6}}.
\][asy]
unitsize(3cm);
draw((0,0)--(1,0)--(1,1)--(0,1)--cycle,dashed);
fill((0,0.66)--(0.33,1)--(1,0.33)--(0.66,0)--(0,0)--cycle,gray(0.7));
dot((0,0));
dot((0,1));
dot((1,1));
dot((1,0));
dot(... | \frac{\pi^2}{6} | 3 | 716 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
117 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
Determine the value of the infinite sum
\[\sum_{n = 17}^\infty \frac{\binom{n}{15}}{\binom{n}{17}}.\]
</Question>
<Plan_1>
Okay, so I need to find the value of this infinite sum: \(\sum_{n = 17}^\infty \frac{\binom{n}{15}}{\binom{n}{17}}\). Hmm, infinite sums can be tricky, but maybe there's a pattern or a ... | <Final_answer>
&= \boxed{272}.
\end{align*}
</Final_answer> | 272 | 9 | 737 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
122 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
For each $x$ in $[0,1]$, define
\[\begin{cases} f(x) = 2x, \qquad\qquad \mathrm{if} \quad 0 \leq x \leq \frac{1}{2};\\ f(x) = 2-2x, \qquad \mathrm{if} \quad \frac{1}{2} < x \leq 1. \end{cases}\]Let $f^{[2]}(x) = f(f(x))$, and $f^{[n + 1]}(x) = f^{[n]}(f(x))$ for each integer $n \geq 2$. Then the number o... | <Final_answer>
$ The final answer is $2 + 2005 = \boxed{2007}.
$
</Final_answer> | 2007 | 6 | 751 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
125 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
What is the coefficient of $x^5$ in the expansion of $(2x+3)^7$?
</Question>
<Plan_1>
Okay, so I need to find the coefficient of x^5 in the expansion of (2x + 3)^7. Hmm, let me think. I remember something about binomial expansions. The general formula for expanding (a + b)^n is the sum from k=0 to n of (n c... | <Plan_4>
Based on execution_2.1, execution_3.1:
Alternatively, maybe I can use the general term formula again but write it out:
The general term is T_k = C(7, k) * (2x)^(7 - k) * 3^k.
So, the coefficient part is C(7, k) * 2^(7 - k) * 3^k, and the power of x is (7 - k). We need (7 - k) = 5, so k = 2. Therefore, the co... | 6048 | 4 | 778 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
125 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
What is the coefficient of $x^5$ in the expansion of $(2x+3)^7$?
</Question>
<Plan_1>
Okay, so I need to find the coefficient of x^5 in the expansion of (2x + 3)^7. Hmm, let me think. I remember something about binomial expansions. The general formula for expanding (a + b)^n is the sum from k=0 to n of (n c... | <Final_answer>
Using the Binomial Theorem, we know that the $x^5$ term of the expansion is $\binom{7}{5}(2x)^5(3)^{7-5}=(21)(32x^5)(9)=(21)(32)(9)x^5=\boxed{6048}x^5$.
</Final_answer> | 6048 | 5 | 781 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
129 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
When Marika bought her house she paid $80\%$ of the purchase price of the house with a loan. She paid the remaining $\$49,400$ of the purchase price with her savings. What was the purchase price of her house?
</Question> | <Plan_1>
Okay, let's see. Marika bought a house and paid 80% of the purchase price with a loan. The rest, which is 20%, she paid with her savings, and that amount was $49,400. I need to find the total purchase price of the house.
Hmm, so if 20% equals $49,400, then the whole 100% should be... Well, let me think. If 20... | 247000 | 1 | 782 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
129 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
When Marika bought her house she paid $80\%$ of the purchase price of the house with a loan. She paid the remaining $\$49,400$ of the purchase price with her savings. What was the purchase price of her house?
</Question>
<Plan_1>
Okay, let's see. Marika bought a house and paid 80% of the purchase price with... | <Plan_2>
Based on execution_1.1:
Let me check if that makes sense.
If the total price is $247,000, then 80% of that would be the loan amount. Let me compute 80% of 247,000.
0.8 × 247,000 = 197,600. Then the remaining 20% is 247,000 - 197,600 = 49,400, which matches the amount she paid with her savings.
-----
Based on... | 247000 | 2 | 786 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
129 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
When Marika bought her house she paid $80\%$ of the purchase price of the house with a loan. She paid the remaining $\$49,400$ of the purchase price with her savings. What was the purchase price of her house?
</Question>
<Plan_1>
Okay, let's see. Marika bought a house and paid 80% of the purchase price with... | <Final_answer>
Thus, if we call $x$ the total price of the house, we have: $\frac{x}{5} = 49400 \rightarrow x = \boxed{247000}$.
</Final_answer> | 247000 | 3 | 788 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
131 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
A square is drawn such that one of its sides coincides with the line $y = 7$, and so that the endpoints of this side lie on the parabola $y = 2x^2 + 8x + 4$. What is the area of the square?
</Question>
<Plan_1>
Okay, let's see. I need to find the area of a square where one of its sides coincides with the li... | <Plan_4>
Based on execution_3.5:
Wait, but let me check once more. Suppose the square is drawn such that the side coincides with y=7, and the endpoints are on the parabola.
<prompt> Re-evaluate the side length and confirm the area of the square. </prompt>
The distance between these two points is the length of the sid... | 22 | 4 | 801 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
131 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
A square is drawn such that one of its sides coincides with the line $y = 7$, and so that the endpoints of this side lie on the parabola $y = 2x^2 + 8x + 4$. What is the area of the square?
</Question>
<Plan_1>
Okay, let's see. I need to find the area of a square where one of its sides coincides with the li... | <Final_answer>
Thus, the area of the square is $\boxed{22}$.
</Final_answer> | 22 | 5 | 805 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
136 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
A circle is centered at $(5,15)$ and has a radius of $\sqrt{130}$ units. Point $Q = (x,y)$ is on the circle, has integer coordinates, and the value of the $x$-coordinate is twice the value of the $y$-coordinate. What is the maximum possible value for $x$?
</Question> | <Plan_1>
Okay, so I need to find the maximum possible x-coordinate for a point Q = (x, y) on a circle centered at (5, 15) with radius √130. The point Q has integer coordinates, and the x-coordinate is twice the y-coordinate. Hmm, let me break this down step by step.
First, the equation of the circle is (x - 5)² + (y -... | 12 | 1 | 806 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
136 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
A circle is centered at $(5,15)$ and has a radius of $\sqrt{130}$ units. Point $Q = (x,y)$ is on the circle, has integer coordinates, and the value of the $x$-coordinate is twice the value of the $y$-coordinate. What is the maximum possible value for $x$?
</Question>
<Plan_1>
Okay, so I need to find the max... | <Plan_3>
Based on execution_2.2:
But wait, let me check if these integer solutions are valid.
Wait, let me solve the quadratic equation we had earlier:
<prompt> Verify the integer solutions obtained from the quadratic equation y² - 10y + 24 = 0. </prompt>
Hmm, perhaps a miscalculation.
Wait, wait. Let me re-examine... | 12 | 3 | 814 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
136 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
A circle is centered at $(5,15)$ and has a radius of $\sqrt{130}$ units. Point $Q = (x,y)$ is on the circle, has integer coordinates, and the value of the $x$-coordinate is twice the value of the $y$-coordinate. What is the maximum possible value for $x$?
</Question>
<Plan_1>
Okay, so I need to find the max... | <Plan_4>
Based on execution_3.1, execution_3.2:
But wait, the user initially thought y=10 and y=5 leading to x=20 and x=10, but that contradicts the quadratic solution. So where did the mistake happen?
Wait, maybe calculation errors. Let me verify:
<prompt> Verify the validity of the solutions (y=6, y=4) and confirm ... | 12 | 4 | 817 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
136 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
A circle is centered at $(5,15)$ and has a radius of $\sqrt{130}$ units. Point $Q = (x,y)$ is on the circle, has integer coordinates, and the value of the $x$-coordinate is twice the value of the $y$-coordinate. What is the maximum possible value for $x$?
</Question>
<Plan_1>
Okay, so I need to find the max... | <Plan_5>
Based on execution_4.1:
So the actual problem is: circle centered at (5,15), radius √130, point Q(x,y) on circle with integer coordinates, x=2y. Find maximum x.
Then according to the equations done earlier, x=12 and x=8.
<prompt> Finalize the solution based on the confirmed problem statement and determine th... | 12 | 5 | 819 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
136 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
A circle is centered at $(5,15)$ and has a radius of $\sqrt{130}$ units. Point $Q = (x,y)$ is on the circle, has integer coordinates, and the value of the $x$-coordinate is twice the value of the $y$-coordinate. What is the maximum possible value for $x$?
</Question>
<Plan_1>
Okay, so I need to find the max... | <Final_answer>
Therefore, the possible $x$-coordinates are 12 and 8, the largest of which is $\boxed{12}$.
</Final_answer> | 12 | 6 | 821 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
138 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
While finding the sine of a certain angle, an absent-minded professor failed to notice that his calculator was not in the correct angular mode. He was lucky to get the right answer. The two least positive real values of $x$ for which the sine of $x$ degrees is the same as the sine of $x$ radians are $\frac{m... | <Plan_5>
Based on execution_4.1:
Wait, but let me check if x=180π/(180 + π) is indeed a solution. Let's plug into sin(x degrees) and sin(x radians).
<prompt> Verify that x = 180π/(180 + π) is a solution by checking sin(x degrees) = sin(x radians). </prompt>
Similarly, for x=360π/(180 - π):
x degrees is 360π/(180 - π... | 900 | 5 | 833 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
138 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
While finding the sine of a certain angle, an absent-minded professor failed to notice that his calculator was not in the correct angular mode. He was lucky to get the right answer. The two least positive real values of $x$ for which the sine of $x$ degrees is the same as the sine of $x$ radians are $\frac{m... | <Plan_6>
Based on execution_5.1, execution_5.2:
Therefore, the two least positive real solutions are indeed 180π/(180 + π) and 360π/(180 - π), so m=360, n=180, p=180, q=180. The sum is 360+180+180+180=900.
But wait, the problem states "the two least positive real values of $ for which the sine of $ degrees is the same... | 900 | 6 | 836 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
138 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
While finding the sine of a certain angle, an absent-minded professor failed to notice that his calculator was not in the correct angular mode. He was lucky to get the right answer. The two least positive real values of $x$ for which the sine of $x$ degrees is the same as the sine of $x$ radians are $\frac{m... | <Final_answer>
\]The least positive values with these forms are $\frac{360 \pi}{180 - \pi}$ and $\frac{180 \pi}{180 + \pi},$ so $m + n + p + q = \boxed{900}.
$
</Final_answer> | 900 | 7 | 838 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
139 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
While finding the sine of a certain angle, an absent-minded professor failed to notice that his calculator was not in the correct angular mode. He was lucky to get the right answer. The two least positive real values of $x$ for which the sine of $x$ degrees is the same as the sine of $x$ radians are $\frac{m... | <Plan_7>
Based on execution_6.1:
But wait, is that correct?
Wait, let's check x=180π/(180 + π). Let's compute x in radians and x in degrees.
<prompt> Verify the first solution x = 180π/(180 + π) in detail. </prompt>
Similarly, for x = 360π/(180 - π). Let's check:
x radians = 360π/(180 - π).
<prompt> Verify the sec... | 900 | 7 | 854 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
139 | You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is generated over multiple ... | <Question>
While finding the sine of a certain angle, an absent-minded professor failed to notice that his calculator was not in the correct angular mode. He was lucky to get the right answer. The two least positive real values of $x$ for which the sine of $x$ degrees is the same as the sine of $x$ radians are $\frac{m... | <Plan_8>
Based on execution_7.1, execution_7.2:
So, this confirms that both solutions x = 180π/(180 + π) and x = 360π/(180 - π) indeed satisfy sin(x degrees) = sin(x radians). These are the two least positive solutions.
Now, according to the problem statement, these two solutions are given in the form mπ/(n - π) and p... | 900 | 8 | 857 | [
{
"content": "You are the planner of an AI system that follows a systematic long thinking process to arrive at the precise and accurate answer to a math question specified within <Question> and </Question> tags. The system consists of a planner and multiple executors that can run in parallel. The solution is ge... |
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