source stringclasses 4
values | problem stringlengths 16 4.82k | solution stringlengths 97 8.33k | messages listlengths 2 2 | system stringclasses 1
value | conversations listlengths 2 2 | generated_token_count int64 285 34.2k | correct bool 2
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aops_forum | Let $a,b,c$ be positive real numbers. Find all real solutions $(x,y,z)$ of the system:
\[ ax+by=(x-y)^{2}
by+cz=(y-z)^{2}
cz+ax=(z-x)^{2}\] | \( (0, 0, c) \).
The final answer is \( \boxed{ (0, 0, 0), (0, 0, c), (a, 0, 0), (0, b, 0) } \). | [
{
"content": "Let $a,b,c$ be positive real numbers. Find all real solutions $(x,y,z)$ of the system:\n\\[ ax+by=(x-y)^{2}\n by+cz=(y-z)^{2}\n cz+ax=(z-x)^{2}\\]",
"role": "user"
},
{
"content": "\\( (0, 0, c) \\).\n\nThe final answer is \\( \\boxed{ (0, 0, 0), (0, 0, c), (a, 0, 0), (0, b, 0) } \... | Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Let $a,b,c$ be positive real numbers. Find all real solutions $(x,y,z)$ of the system:\n\\[ ax+by=(x-y)^{2}\n by+cz=(y-z)^{2}\n cz+ax=(z-x)^{2}\\]"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, let's... | 10,507 | false |
aops_forum | Let $ABC$ be an acute triangle with circumcircle $\Gamma$ . Let the internal angle bisector of $\angle BAC$ intersect $BC$ and $\Gamma$ at $E$ and $N$ , respectively. Let $A'$ be the antipode of $A$ on $\Gamma$ and let $V$ be the point where $AA'$ intersects $BC$ . Given that $EV=6$ , $VA'=7$ ,... | **
Since $R = 5.1$ is too small, we select $R = 7.5$.
The final answer is $\boxed{\frac{15}{2}}$ | [
{
"content": "Let $ABC$ be an acute triangle with circumcircle $\\Gamma$ . Let the internal angle bisector of $\\angle BAC$ intersect $BC$ and $\\Gamma$ at $E$ and $N$ , respectively. Let $A'$ be the antipode of $A$ on $\\Gamma$ and let $V$ be the point where $AA'$ intersects $BC$ . Given... | Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Let $ABC$ be an acute triangle with circumcircle $\\Gamma$ . Let the internal angle bisector of $\\angle BAC$ intersect $BC$ and $\\Gamma$ at $E$ and $N$ , respectively. Let $A'$ be the antipode of $A$ on $\\Gamma$ an... | 21,338 | true |
math | If $f(x) = 3x + 1$, what is $f(3)$? | To solve for $f(3)$ given $f(x) = 3x + 1$, we substitute $x$ with $3$:
\[
\begin{align*}
f(3) &= 3(3) + 1 \\
&= 9 + 1 \\
&= \boxed{10}.
\end{align*}
\] | [
{
"content": "If $f(x) = 3x + 1$, what is $f(3)$?",
"role": "user"
},
{
"content": "To solve for $f(3)$ given $f(x) = 3x + 1$, we substitute $x$ with $3$:\n\n\\[\n\\begin{align*}\nf(3) &= 3(3) + 1 \\\\\n&= 9 + 1 \\\\\n&= \\boxed{10}.\n\\end{align*}\n\\]",
"role": "assistant"
}
] | Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. If $f(x) = 3x + 1$, what is $f(3)$?"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to figure out what f(3) is when the function f(x) is defined as 3x + 1. Let me start by recalling what a funct... | 777 | true |
math | Evaluate $\log_5625$. | To evaluate $\log_5625$, we need to find the exponent $x$ such that $5^x = 625$.
We know that $5^4 = 625$.
Therefore, $\log_5 625 = 4$.
So, we have $\log_5625 = \boxed{4}$. | [
{
"content": "Evaluate $\\log_5625$.",
"role": "user"
},
{
"content": "To evaluate $\\log_5625$, we need to find the exponent $x$ such that $5^x = 625$. \n\nWe know that $5^4 = 625$. \n\nTherefore, $\\log_5 625 = 4$. \n\nSo, we have $\\log_5625 = \\boxed{4}$.",
"role": "assistant"
}
] | Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Evaluate $\\log_5625$."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to evaluate log base 5 of 625. Hmm, let me think about how logarithms work. The logarithm log_b a is asking the question: \... | 1,543 | true |
math | If $f (x) = x^2 - 1$, what is the value of $f (-1)$? | To find the value of $f(-1)$ for the function $f(x) = x^2 - 1$, we substitute $x = -1$ into the function:
\[
\begin{align*}
f(-1) &= (-1)^2 - 1 \\
&= 1 - 1 \\
&= \boxed{0}.
\end{align*}
\] | [
{
"content": "If $f (x) = x^2 - 1$, what is the value of $f (-1)$?",
"role": "user"
},
{
"content": "To find the value of $f(-1)$ for the function $f(x) = x^2 - 1$, we substitute $x = -1$ into the function:\n\n\\[\n\\begin{align*}\nf(-1) &= (-1)^2 - 1 \\\\\n&= 1 - 1 \\\\\n&= \\boxed{0}.\n\\end{align... | Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. If $f (x) = x^2 - 1$, what is the value of $f (-1)$?"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, let me figure out how to solve this problem. The question is: If f(x) = x² - 1, what is the value of f(... | 2,000 | true |
olympiads | The value of $(4-3) \times 2$ is:
(A) -2
(B) 2
(C) 1
(D) 3
(E) 5 |
1. Evaluate the expression inside the parentheses.
\[
4 - 3 = 1
\]
2. Multiply the result by 2.
\[
1 \times 2 = 2
\]
3. Thus, the value of \((4 - 3) \times 2\) is \(\boxed{2}\). | [
{
"content": "The value of $(4-3) \\times 2$ is:\n(A) -2\n(B) 2\n(C) 1\n(D) 3\n(E) 5",
"role": "user"
},
{
"content": "\n1. Evaluate the expression inside the parentheses.\n \\[\n 4 - 3 = 1\n \\]\n\n2. Multiply the result by 2.\n \\[\n 1 \\times 2 = 2\n \\]\n\n3. Thus, the value of \\((4... | Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. The value of $(4-3) \\times 2$ is:\n(A) -2\n(B) 2\n(C) 1\n(D) 3\n(E) 5"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, let me try to figure out this math problem. So the question is asking for the value o... | 637 | false |
aops_forum | A mouse has a wheel of cheese which is cut into $2018$ slices. The mouse also has a $2019$ -sided die, with faces labeled $0,1,2,\ldots, 2018$ , and with each face equally likely to come up. Every second, the mouse rolls the dice. If the dice lands on $k$ , and the mouse has at least $k$ slices of cheese remaini... | **
Since the recurrence relation and the initial conditions suggest that \( T_k \) converges to a constant value, we conclude that:
\[
T_{2018} = 2019
\]
The final answer is \( \boxed{2019} \). | [
{
"content": "A mouse has a wheel of cheese which is cut into $2018$ slices. The mouse also has a $2019$ -sided die, with faces labeled $0,1,2,\\ldots, 2018$ , and with each face equally likely to come up. Every second, the mouse rolls the dice. If the dice lands on $k$ , and the mouse has at least $k$ s... | Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. A mouse has a wheel of cheese which is cut into $2018$ slices. The mouse also has a $2019$ -sided die, with faces labeled $0,1,2,\\ldots, 2018$ , and with each face equally likely to come up. Every second, the mouse rolls the dice.... | 9,152 | true |
math | If $a*b=a^2+ab-b^2$, find $3*2$. | To find $3*2$ using the given operation $a*b=a^2+ab-b^2$, we substitute $a=3$ and $b=2$ into the formula:
\[
3*2 = 3^2 + 3\cdot 2 - 2^2
\]
Breaking it down:
\[
= 9 + 6 - 4
\]
\[
= 15 - 4
\]
\[
= \boxed{11}
\] | [
{
"content": "If $a*b=a^2+ab-b^2$, find $3*2$.",
"role": "user"
},
{
"content": "To find $3*2$ using the given operation $a*b=a^2+ab-b^2$, we substitute $a=3$ and $b=2$ into the formula:\n\n\\[\n3*2 = 3^2 + 3\\cdot 2 - 2^2\n\\]\n\nBreaking it down:\n\n\\[\n= 9 + 6 - 4\n\\]\n\n\\[\n= 15 - 4\n\\]\n\n\... | Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. If $a*b=a^2+ab-b^2$, find $3*2$."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, let's see. I need to find the value of 3*2 given the operation defined as a*b = a² + ab - b². Alright, so first, I should m... | 1,500 | true |
olympiads | The value of \((0.3)^{2} + 0.1\) is:
(A) 0.7
(B) 1
(C) 0.1
(D) 0.19
(E) 0.109 |
1. Calculate \((0.3)^2\):
\[
(0.3)^2 = 0.3 \times 0.3 = 0.09
\]
2. Add 0.1 to the result from step 1:
\[
0.09 + 0.1 = 0.19
\]
Conclusion: The value of \((0.3)^2 + 0.1\) is \(\boxed{0.19}\) (D). | [
{
"content": "The value of \\((0.3)^{2} + 0.1\\) is:\n(A) 0.7\n(B) 1\n(C) 0.1\n(D) 0.19\n(E) 0.109",
"role": "user"
},
{
"content": "\n1. Calculate \\((0.3)^2\\):\n \\[\n (0.3)^2 = 0.3 \\times 0.3 = 0.09\n \\]\n\n2. Add 0.1 to the result from step 1:\n \\[\n 0.09 + 0.1 = 0.19\n \\]\n\nCo... | Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. The value of \\((0.3)^{2} + 0.1\\) is:\n(A) 0.7\n(B) 1\n(C) 0.1\n(D) 0.19\n(E) 0.109"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to figure out the value of (0.3) squared plus 0.1. Let me tak... | 643 | false |
math | What is the value of $f(-1)$ if $f(x)=x^{2}-2x$? | To find the value of $f(-1)$ for the function $f(x)=x^{2}-2x$, we substitute $x$ with $-1$:
\begin{align*}
f(-1) &= (-1)^{2} - 2(-1) \\
&= 1 + 2 \\
&= \boxed{3}.
\end{align*}
Thus, the value of $f(-1)$ is $\boxed{3}$. | [
{
"content": "What is the value of $f(-1)$ if $f(x)=x^{2}-2x$?",
"role": "user"
},
{
"content": "To find the value of $f(-1)$ for the function $f(x)=x^{2}-2x$, we substitute $x$ with $-1$:\n\n\\begin{align*}\nf(-1) &= (-1)^{2} - 2(-1) \\\\\n&= 1 + 2 \\\\\n&= \\boxed{3}.\n\\end{align*}\n\nThus, the v... | Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. What is the value of $f(-1)$ if $f(x)=x^{2}-2x$?"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to find the value of f(-1) when the function is given as f(x) = x² - 2x. Hmm, let's start by reca... | 806 | true |
olympiads |
The value of \(\frac{3 \times 4}{6}\) is
(A) 1
(B) 2
(C) 3
(D) 4
(E) 6 | :
1. Start by calculating the numerator:
\[
3 \times 4 = 12
\]
2. Substitute the value of the numerator back into the fraction:
\[
\frac{12}{6}
\]
3. Perform the division:
\[
\frac{12}{6} = 2
\]
Conclusion:
\[
\boxed{2}
\] | [
{
"content": "\nThe value of \\(\\frac{3 \\times 4}{6}\\) is\n(A) 1\n(B) 2\n(C) 3\n(D) 4\n(E) 6",
"role": "user"
},
{
"content": ": \n1. Start by calculating the numerator:\n\\[\n3 \\times 4 = 12\n\\]\n\n2. Substitute the value of the numerator back into the fraction:\n\\[\n\\frac{12}{6}\n\\]\n\n3. ... | Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. \nThe value of \\(\\frac{3 \\times 4}{6}\\) is\n(A) 1\n(B) 2\n(C) 3\n(D) 4\n(E) 6"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I have this math problem here: What's the value of (3 times 4) divided ... | 1,284 | false |
olympiads | The value of \(10^{2} + 10 + 1\) is:
(A) 101
(B) 1035
(C) 1011
(D) 111
(E) 31 | 1. Evaluate \( 10^2 \):
\[
10^2 = 10 \times 10 = 100
\]
2. Add \( 10 \) to the result:
\[
100 + 10 = 110
\]
3. Add \( 1 \) to the result:
\[
110 + 1 = 111
\]
### Conclusion:
\[
\boxed{\text{D}}
\]
| [
{
"content": "The value of \\(10^{2} + 10 + 1\\) is:\n(A) 101\n(B) 1035\n(C) 1011\n(D) 111\n(E) 31",
"role": "user"
},
{
"content": "1. Evaluate \\( 10^2 \\):\n \\[\n 10^2 = 10 \\times 10 = 100\n \\]\n\n2. Add \\( 10 \\) to the result:\n \\[\n 100 + 10 = 110\n \\]\n\n3. Add \\( 1 \\) to ... | Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. The value of \\(10^{2} + 10 + 1\\) is:\n(A) 101\n(B) 1035\n(C) 1011\n(D) 111\n(E) 31"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, let's tackle this problem step by step. The question is asking for the ... | 1,232 | false |
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