id stringlengths 19 19 | question stringlengths 16 1.98k | gold stringlengths 1 189 | r1 stringlengths 0 29.6k | r2 stringlengths 337 39.4k | completion stringlengths 97 51.1k | structural bool 1
class | victim stringclasses 1
value | provider stringclasses 2
values | teacher_model stringclasses 1
value | answer_correct bool 1
class |
|---|---|---|---|---|---|---|---|---|---|---|
openthoughts_000003 | Let \( p = 2^{3009}, q = 3^{2006}, \) and \( r = 5^{1003} \). Which of the following statements is true?
(A) \( p < q < r \)
(B) \( p < r < q \)
(C) \( q < p < r \)
(D) \( r < p < q \)
(E) \( q < r < p \) | D | **Analyzing Exponents Relationships**
I'm now focusing on the exponents and their relationships. I've broken down the exponents 3009, 2006, and 1003. It's clear that 1003 is a common factor, specifically that 3009 is three times 1003, and 2006 is two times 1003. Now that I have this, I can begin to compare them.
| <|channel>thought
The question asks us to compare the three numbers $p = 2^{3009}$, $q = 3^{2006}$, and $r = 5^{1003}$.
Let's look at the exponents: 3009, 2006, and 1003.
Notice that:
$3009 = 3 \times 1003$
$2006 = 2 \times 1003$
$1003 = 1 \times 1003$
So we can rewrite $p$, $q$, and $r$ with the same exponent, 1003:... | <think>
**Analyzing Exponents Relationships**
I'm now focusing on the exponents and their relationships. I've broken down the exponents 3009, 2006, and 1003. It's clear that 1003 is a common factor, specifically that 3009 is three times 1003, and 2006 is two times 1003. Now that I have this, I can begin to compare the... | true | gemini-3.5-flash | Google AI Studio | gemini-3.5-flash | true |
openthoughts_000004 | When \( a < -1 \), the nature of the roots for the equation
$$
\left(a^{3}+1\right) x^{2}+\left(a^{2}+1\right) x-(a+1)=0
$$
is:
(A) Two negative roots.
(B) One positive root and one negative root, with the absolute value of the negative root being larger.
(C) One positive root and one negative root, with the absolute v... | C | $\text{ }$
```bash
$ cat reasoning_trace.txt
The given quadratic equation is:
$$
(a^3 + 1) x^2 + (a^2 + 1) x - (a + 1) = 0
$$
where $a < -1$.
Let the coefficients of the quadratic equation $Ax^2 + Bx + C = 0$ be:
$A = a^3 + 1$
$B = a^2 + 1$
$C = -(a + 1)$
Since $a < -1$:
1. $a^3 < -1 \implies A = a^3 + 1 < 0$.
2. $a^... | <think>
</think>
The given quadratic equation is:
$$
(a^3 + 1) x^2 + (a^2 + 1) x - (a + 1) = 0
$$
where $a < -1$.
Let the coefficients of the quadratic equation $Ax^2 + Bx + C = 0$ be:
$A = a^3 + 1$
$B = a^2 + 1$
$C = -(a + 1)$
Since $a < -1$:
1. $a^3 < -1 \implies A = a^3 + 1 < 0$.
2. $a^2 > 1 \implies B = a^2 + 1... | true | gemini-3.5-flash | Google | gemini-3.5-flash | true | |
openthoughts_000006 | Given that \(1 \leq x, y, z \leq 6\), how many cases are there in which the product of natural numbers \(x, y, z\) is divisible by 10? | 72 | An elegant way to solve this problem is by using the properties of divisibility and the principle of inclusion-exclusion.
For the product of three natural numbers $x, y, z$ (where $1 \leq x, y, z \leq 6$) to be divisible by $10$, the product $xyz$ must be divisible by both $2$ and $5$.
Within the set of possible val... | <think>
</think>
The question asks for the number of triples of natural numbers $(x, y, z)$ such that $1 \leq x, y, z \leq 6$ and the product $xyz$ is divisible by 10.
Since $x, y, z$ are natural numbers between 1 and 6, the possible values for each of $x, y, z$ are $\{1, 2, 3, 4, 5, 6\}$.
The total number of possib... | true | gemini-3.5-flash | Google | gemini-3.5-flash | true | |
openthoughts_000011 | Write the first $n$ natural numbers in decimal form on a (very long) strip of paper, then cut the strip so that each piece contains exactly one digit. Place these pieces in a box, mix them up, and draw one at random. Let $p_{n}$ denote the probability that the drawn piece of paper has the digit 0 on it. Determine the l... | \frac{1}{10} | **Exploring Sequence Limits**
I'm currently analyzing the probability $p_n$ of drawing a '0' from the first $n$ natural numbers' digits. My focus is on how this probability behaves as $n$ grows infinitely large, aiming to determine the limiting value.
**Calculating Digit Distributions**
I'm focusing on the distribut... | An elegant way to solve this problem is to estimate the total number of digits, $D(n)$, and the total number of zeros, $Z(n)$, in the decimal representations of the first $n$ natural numbers.
Let $k$ be the number of digits of $n$, so that $10^{k-1} \le n < 10^k$. Thus, $k \approx \log_{10} n$.
### 1. Estimating the ... | <think>
**Exploring Sequence Limits**
I'm currently analyzing the probability $p_n$ of drawing a '0' from the first $n$ natural numbers' digits. My focus is on how this probability behaves as $n$ grows infinitely large, aiming to determine the limiting value.
**Calculating Digit Distributions**
I'm focusing on the d... | true | gemini-3.5-flash | Google | gemini-3.5-flash | true |
openthoughts_000012 | How many plums will balance one pear, given that 3 apples and one pear weigh as much as 10 plums, and one apple and 6 plums balance one pear? Assume that fruits of the same kind have the same weight. | 7 | <|channel>thought
The problem asks us to find how many plums will balance one pear, given two weight balance conditions:
1. 3 apples and 1 pear weigh as much as 10 plums.
2. 1 apple and 6 plums balance 1 pear.
Let's define variables for the weights of the fruits:
Let $a$ be the weight of an apple.
Let $p$ be the weigh... | <think>
</think>
The problem asks us to find how many plums will balance one pear, given two weight balance conditions:
1. 3 apples and 1 pear weigh as much as 10 plums.
2. 1 apple and 6 plums balance 1 pear.
Let's define variables for the weights of the fruits:
Let $a$ be the weight of an apple.
Let $p$ be the weig... | true | gemini-3.5-flash | Google | gemini-3.5-flash | true | |
openthoughts_000013 | Determine the value of
$$
z=a \sqrt{a} \sqrt[4]{a} \sqrt[8]{a} \ldots \sqrt[2^{n}]{a} \ldots
$$
if \( n \) is infinitely large. | a^2 | ```bash
$ cat reasoning_trace.txt
The question asks for the value of the infinite product:
$$ z = a \sqrt{a} \sqrt[4]{a} \sqrt[8]{a} \ldots \sqrt[2^{n}]{a} \ldots $$
where $n$ is infinitely large.
Let's write the terms as powers of $a$:
$$ z = a^1 \cdot a^{1/2} \cdot a^{1/4} \cdot a^{1/8} \cdot \ldots \cdot a^{1/2^n} ... | <think>
</think>
The question asks for the value of the infinite product:
$$ z = a \sqrt{a} \sqrt[4]{a} \sqrt[8]{a} \ldots \sqrt[2^{n}]{a} \ldots $$
where $n$ is infinitely large.
Let's write the terms as powers of $a$:
$$ z = a^1 \cdot a^{1/2} \cdot a^{1/4} \cdot a^{1/8} \cdot \ldots \cdot a^{1/2^n} \cdot \ldots $$... | true | gemini-3.5-flash | Google | gemini-3.5-flash | true | |
openthoughts_000022 | Buratino calculated the time accurately and left Papa Carlo's house at 13:40 to reach the Field of Miracles and plant 4 coins exactly at sunset. If he had walked $25 \%$ faster, he would have arrived at the Field of Miracles 1.5 hours earlier and waited. At what time did Buratino arrive at the Field of Miracles? | 21:10 | An elegant way to solve this problem is by using the relationship between speed and time.
Let $v$ be Buratino's actual walking speed, and $t$ be the actual time (in hours) he spent walking.
If he had walked $25\%$ faster, his speed would have been:
$$v_{\text{new}} = 1.25v = \frac{5}{4}v$$
Since the distance to the ... | <think>
</think>
Let $v$ be Buratino's actual walking speed, and $t$ be the actual time (in hours) he spent walking.
If he had walked $25\%$ faster, his speed would have been:
$$v_{\text{new}} = 1.25v = \frac{5}{4}v$$
Since the distance to the Field of Miracles remains the same, the time taken is inversely proporti... | true | gemini-3.5-flash | Google | gemini-3.5-flash | true | |
openthoughts_000025 | Find the variance of the discrete random variable $X$ that is distributed according to the Poisson distribution:
$$
\begin{array}{ccccccc}
X & 0 & 1 & 2 & \cdots & k & \cdots \\
P & \mathrm{e}^{-\lambda} & \lambda \mathrm{e}^{-\lambda} / 1! & \lambda^{2} \mathrm{e}^{-\lambda} / 2! & \cdots & \lambda^{k} \mathrm{e}^{-\... | \lambda | An elegant way to find the variance of a Poisson-distributed random variable $X$ is by using its moments.
The probability mass function of $X$ is given by:
$$P(X = k) = \frac{\lambda^k \mathrm{e}^{-\lambda}}{k!}, \quad k = 0, 1, 2, \dots$$
The variance of $X$ is defined as:
$$\text{Var}(X) = E[X^2] - (E[X])^2$$
###... | <think>
</think>
To find the variance of a Poisson random variable X with parameter \lambda:
1. Calculate the mean E[X] = \lambda.
2. Calculate the second factorial moment E[X(X-1)] = \lambda^2.
3. Express E[X^2] as E[X(X-1)] + E[X] = \lambda^2 + \lambda.
4. Compute Var(X) = E[X^2] - (E[X])^2 = \lambda^2 + \lambda - ... | true | gemini-3.5-flash | Google | gemini-3.5-flash | true | |
openthoughts_000037 | What is the largest result we can find when we divide a 2-digit number by the sum of its digits? | 10 | <|channel>thought
The question asks for the largest result we can find when we divide a 2-digit number by the sum of its digits.
Let the 2-digit number be represented as $10a + b$, where $a \in \{1, 2, \dots, 9\}$ and $b \in \{0, 1, \dots, 9\}$.
The sum of its digits is $a + b$.
We want to maximize the ratio:
$$ R = \... | <think>
</think>
Let the 2-digit number be represented as $10a + b$, where $a \in \{1, 2, \dots, 9\}$ and $b \in \{0, 1, \dots, 9\}$.
The sum of its digits is $a + b$.
We want to maximize the ratio:
$$ R = \frac{10a + b}{a + b} $$
We can rewrite the ratio as:
$$ R = \frac{10a + 10b - 9b}{a + b} = 10 - \frac{9b}{a + ... | true | gemini-3.5-flash | Google | gemini-3.5-flash | true | |
openthoughts_000070 | Professor Antônio discovered an interesting property related to the integer $x$ that represents his age. He told his students that $x^{2}=\overline{a b a c}$ and that $x=\overline{a b}+\overline{a c}$. What is the professor's age?
Note: We are using a bar to distinguish the decimal representation of the four-digit num... | 45 | An elegant way to solve this problem is by using the decimal representation of the numbers.
Let's write the four-digit number $x^2 = \overline{a b a c}$ as:
$$x^2 = 100 \cdot \overline{a b} + \overline{a c}$$
We are also given that:
$$x = \overline{a b} + \overline{a c} \implies \overline{a c} = x - \overline{a b}$$
... | <think>
</think>
An elegant way to solve this problem is by using the decimal representation of the numbers.
Let's write the four-digit number $x^2 = \overline{a b a c}$ as:
$$x^2 = 100 \cdot \overline{a b} + \overline{a c}$$
We are also given that:
$$x = \overline{a b} + \overline{a c} \implies \overline{a c} = x ... | true | gemini-3.5-flash | Google | gemini-3.5-flash | true | |
openthoughts_000071 | The faces of a cube are inscribed with six natural numbers (possibly repeating) such that the numbers on adjacent faces differ by more than 1. What is the smallest possible sum of these six numbers? | 18 | An elegant way to solve this problem is to use the properties of the cube's face adjacency.
### 1. Understanding the Adjacency of a Cube's Faces
A cube has 6 faces. Each face is adjacent to exactly 4 other faces and opposite to exactly 1 face.
Let the three pairs of opposite faces be:
- Pair 1: $\{A, A'\}$
- Pair 2:... | <think>
</think>
The problem asks for the smallest possible sum of six natural numbers inscribed on the faces of a cube, such that the numbers on adjacent faces differ by more than 1.
Let the six faces of the cube be F_1, F_2, F_3, F_4, F_5, F_6.
The numbers on these faces are natural numbers (positive integers: 1, 2... | true | gemini-3.5-flash | Google | gemini-3.5-flash | true | |
openthoughts_000084 | It is known that all 'krakozyabrs' have either horns or wings (or both). From the results of a global census of 'krakozyabrs,' it was found that 20% of the 'krakozyabrs' with horns also have wings, and 25% of the 'krakozyabrs' with wings also have horns. How many 'krakozyabrs' are left in the world if it is known that ... | 32 | **Analyzing Krakozyabr Numbers**
I'm currently working through the problem, focusing on the percentages related to krakozyabrs possessing horns and wings. I've noted that all krakozyabrs have at least one of these features, and I'm processing the conditional probabilities provided.
**Calculating Krakozyabr Totals**
... | An elegant way to solve this problem is by using set theory and the properties of percentages.
Let:
* $H$ be the set of 'krakozyabrs' with horns.
* $W$ be the set of 'krakozyabrs' with wings.
* $N$ be the total number of 'krakozyabrs'.
Since every 'krakozyabrs' has either horns, wings, or both, the total number of 'k... | <think>
**Analyzing Krakozyabr Numbers**
I'm currently working through the problem, focusing on the percentages related to krakozyabrs possessing horns and wings. I've noted that all krakozyabrs have at least one of these features, and I'm processing the conditional probabilities provided.
**Calculating Krakozyabr To... | true | gemini-3.5-flash | Google | gemini-3.5-flash | true |
openthoughts_000098 | Find the integer part of the expression
\[a=\sqrt{1981+\sqrt{1981+\sqrt{1981+\ldots+\sqrt{1981+\sqrt{1981}}}}},\]
if the number 1981 appears $n$ times ($n \geq 2$). | 45 | An elegant way to find the integer part of the expression is to analyze the bounds of the sequence of nested radicals.
Let us define the sequence $a_k$ for $k \geq 1$ as:
\[a_1 = \sqrt{1981}\]
\[a_k = \sqrt{1981 + a_{k-1}} \quad \text{for } k \geq 2\]
We want to find the integer part of $a_n$ for $n \geq 2$.
First, ... | <think>
</think>
The question asks for the integer part of the expression
\[a=\sqrt{1981+\sqrt{1981+\sqrt{1981+\ldots+\sqrt{1981+\sqrt{1981}}}}},\]
where the number 1981 appears $n$ times ($n \geq 2$).
Let's denote the sequence of nested radicals as:
$a_1 = \sqrt{1981}$
$a_2 = \sqrt{1981 + \sqrt{1981}}$
...
$a_n = \... | true | gemini-3.5-flash | Google | gemini-3.5-flash | true | |
openthoughts_000125 | Given a positive integer \( n (n \geq 2) \), find the maximum real number \( \lambda \) such that the inequality \( a_{n}^{2} \geq \lambda \left(a_{1}+a_{2}+\cdots+a_{n-1}\right) + 2 a_{n} \) holds for any positive integers \( a_{1}, a_{2}, \cdots, a_{n} \) satisfying \( a_{1} < a_{2} < \cdots < a_{n} \). | \frac{2(n-2)}{n-1} | An elegant way to solve this problem is to analyze the constraints on the sum of the first $n-1$ terms of the sequence and then find the optimal value for $\lambda$.
### Step 1: Upper bound for the sum of the first $n-1$ terms
Since $a_1, a_2, \cdots, a_n$ are strictly increasing positive integers, we have:
$$a_{n-1} ... | <think>
</think>
The problem asks for the maximum real number $\lambda$ such that the inequality
$$ a_{n}^{2} \geq \lambda \left(a_{1}+a_{2}+\cdots+a_{n-1}\right) + 2 a_{n} $$
holds for any positive integers $a_{1}, a_{2}, \cdots, a_{n}$ satisfying $a_{1} < a_{2} < \cdots < a_{n}$.
Let's analyze the inequality.
We c... | true | gemini-3.5-flash | Google | gemini-3.5-flash | true | |
openthoughts_000127 | In an isosceles triangle with a lateral side of 100 and a base of 60, a circle is inscribed. Find the distance between the points of tangency located on the lateral sides. | 42 | <|channel>thought
The problem asks for the distance between the points of tangency located on the lateral sides of an isosceles triangle.
Let the isosceles triangle be $ABC$ with lateral sides $AB = AC = 100$ and base $BC = 60$.
Let the incircle of $\triangle ABC$ touch the sides $AB$, $AC$, and $BC$ at points $D$, $E$... | <think>
</think>
The problem asks for the distance between the points of tangency located on the lateral sides of an isosceles triangle.
Let the isosceles triangle be $ABC$ with lateral sides $AB = AC = 100$ and base $BC = 60$.
Let the incircle of $\triangle ABC$ touch the sides $AB$, $AC$, and $BC$ at points $D$, $E... | true | gemini-3.5-flash | Google | gemini-3.5-flash | true | |
openthoughts_000142 | Given the polynomial \( g(x) = x^5 + x^4 + x^3 + x^2 + x + 1 \), find the remainder when the polynomial \( g(x^{12}) \) is divided by \( g(x) \).
(A) 6
(B) \( 5 - x \)
(C) \( 4 - x + x^2 \)
(D) \( 3 - x + x^2 - x^3 \)
(E) \( 2 - x + x^2 - x^3 + x^4 \) | A | **Analyzing the Polynomial Roots**
I'm currently focused on understanding the roots of $g(x) = x^5 + x^4 + x^3 + x^2 + x + 1$. My current approach is to leverage the relationship $g(x) = \frac{x^6 - 1}{x - 1}$, which will be key in determining the remainder of $g(x^{12})$ divided by $g(x)$.
**Confirming the Remainder... | An elegant way to find the remainder is to work modulo the polynomial $g(x)$.
First, we observe that:
\[ g(x) = x^5 + x^4 + x^3 + x^2 + x + 1 \]
We can multiply $g(x)$ by $(x - 1)$ to get:
\[ (x - 1)g(x) = x^6 - 1 \]
This implies that:
\[ x^6 - 1 \equiv 0 \pmod{g(x)} \implies x^6 \equiv 1 \pmod{g(x)} \]
Now, we wan... | <think>
**Analyzing the Polynomial Roots**
I'm currently focused on understanding the roots of $g(x) = x^5 + x^4 + x^3 + x^2 + x + 1$. My current approach is to leverage the relationship $g(x) = \frac{x^6 - 1}{x - 1}$, which will be key in determining the remainder of $g(x^{12})$ divided by $g(x)$.
**Confirming the R... | true | gemini-3.5-flash | Google | gemini-3.5-flash | true |
openthoughts_000151 | There is a special calculator. When a number is input, the calculator will multiply the number by 2, then reverse the digits of the result. Finally, it will add 2 and display the final result. If you input a two-digit number and the final displayed result is 27, what was the initial input? | 26 | **Exploring the Calculator's Functionality**
I'm currently delving into the calculator's initial operations. My focus is on understanding how it processes a two-digit input, which I'm representing as 'x'. The first step involves a multiplication, and I'm working to decipher its specific parameters and effect on 'x'.
... | <|channel>thought
The problem asks us to find the initial two-digit number input into a special calculator.
Let the input number be $x$, which is a two-digit number.
The calculator performs the following operations:
1. Multiplies the number by 2: $2x$.
2. Reverses the digits of the result. Let's denote the digit revers... | <think>
**Exploring the Calculator's Functionality**
I'm currently delving into the calculator's initial operations. My focus is on understanding how it processes a two-digit input, which I'm representing as 'x'. The first step involves a multiplication, and I'm working to decipher its specific parameters and effect o... | true | gemini-3.5-flash | Google | gemini-3.5-flash | true |
openthoughts_000167 | Let the set \( T = \left\{ x_{1}, x_{2}, \cdots, x_{10} \right\} \) contain five-element subsets such that any two elements of \( T \) appear together in at most two of these subsets. What is the maximum number of such subsets? | 8 | **Exploring Subset Bounds**
I'm currently analyzing the constraints on a family of 5-element subsets from a 10-element set, specifically focusing on how frequently any pair of elements can co-occur. This involves determining the maximum possible size of such a family under the given condition of at most two co-occurre... | An elegant way to solve this problem is by using design theory and double counting.
### 1. Upper Bound on the Number of Subsets
Let $v = 10$ be the number of elements in the set $T$, $k = 5$ be the size of each subset, and $\lambda = 2$ be the maximum number of times any pair of elements can appear together. Let $b$ b... | <think>
**Exploring Subset Bounds**
I'm currently analyzing the constraints on a family of 5-element subsets from a 10-element set, specifically focusing on how frequently any pair of elements can co-occur. This involves determining the maximum possible size of such a family under the given condition of at most two co... | true | gemini-3.5-flash | Google | gemini-3.5-flash | true |
openthoughts_000182 | Let $[x]$ denote the greatest integer less than or equal to the real number $x$. Determine the number of elements in the set \(\left\{ n \, \middle| \, n=\left[ \frac{k^{2}}{2005} \right], \, 1 \leq k \leq 2004, \, k \in \mathbf{N} \right\} \). | 1503 | An elegant way to solve this problem is to analyze the behavior of the function $f(k) = \left[ \frac{k^2}{2005} \right]$ for $1 \leq k \leq 2004$.
Let $g(k) = \frac{k^2}{2005}$. The difference between consecutive terms is:
\[ g(k+1) - g(k) = \frac{(k+1)^2 - k^2}{2005} = \frac{2k+1}{2005} \]
### Case 1: When the diffe... | <think>
</think>
An elegant way to solve this problem is to analyze the behavior of the function $f(k) = \left[ \frac{k^2}{2005} \right]$ for $1 \leq k \leq 2004$.
Let $g(k) = \frac{k^2}{2005}$. The difference between consecutive terms is:
\[ g(k+1) - g(k) = \frac{(k+1)^2 - k^2}{2005} = \frac{2k+1}{2005} \]
### Cas... | true | gemini-3.5-flash | Google | gemini-3.5-flash | true | |
openthoughts_000186 | A trapezoid has parallel sides $a$ and $c$. What is the length of a segment that is parallel to the given sides of the trapezoid and bisects its area? | \sqrt{\frac{a^2 + c^2}{2}} | An elegant way to solve this problem is by using the properties of similar triangles or coordinate geometry.
### Derivation:
Let the trapezoid have parallel sides of lengths $a$ and $c$. We can place the trapezoid in a coordinate system such that the side of length $c$ lies on the line $y = 0$ (the x-axis) and the sid... | <think>
</think>
The question asks for the length of a segment that is parallel to the parallel sides $a$ and $c$ of a trapezoid and bisects its area.
Let the parallel sides of the trapezoid be $a$ and $c$.
Let the height of the trapezoid be $h$.
The area of the trapezoid is $S = \frac{a+c}{2} h$.
We want to find the... | true | gemini-3.5-flash | Google | gemini-3.5-flash | true |
End of preview. Expand in Data Studio
openthoughts-rep-victim-gemini-3.5-flash-correct
Rejection-sampled subset of the gemini-3.5-flash REP-exposed reasoning traces (Chia-Mu-Lab/openthoughts-rep-victim-gemini-3.5-flash-8k source harvest): only rows whose leaked CoT's \boxed answer matches the OpenThoughts gold (4707 rows). Variant = correct. Same pure-CoT target as the parent dataset (theft wrapper stripped). Built for the distillation MATH500 push.
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