messages listlengths 2 2 | instruction_seed stringlengths 35 770 | response_seed stringlengths 35 770 | _source stringclasses 1
value | _fasttext_score float64 1 1 | reasoning stringlengths 1.55k 88.7k | deepseek_solution stringlengths 318 3.56k | __original_row_idx int64 24 31.6k | final_reasoning_trace stringlengths 2.13k 90k | conversations listlengths 2 2 |
|---|---|---|---|---|---|---|---|---|---|
[
{
"content": "Find the sum of the squares of all the integers from 1 to 49.",
"role": "user"
},
{
"content": "We can use the formula for the sum of squares:\n\\[ \\sum_{i=1}^{n} i^2 = \\frac{n(n+1)(2n+1)}{6} \\]\n\nSubstitute $n = 49$:\n\\[ \\sum_{i=1}^{49} i^2 = \\frac{49(49+1)(2 \\cdot 49+1)}{6} =... | Find the sum of the squares of all the integers from 1 to 49. | Find the sum of the squares of all the integers from 1 to 49. | ai2-adapt-dev/openmath-2-math | 0.999519 | Okay, so I need to find the sum of the squares of all the integers from 1 to 49. Hmm, let me think about how to approach this. I remember there's a formula for the sum of squares of the first n natural numbers. Wasn't it something like n(n + 1)(2n + 1)/6? Let me verify that. If I plug in n = 1, the sum should be 1² = 1... | To find the sum of the squares of all the integers from 1 to 49, we use the formula for the sum of the squares of the first \( n \) natural numbers:
\[
\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}
\]
For \( n = 49 \):
\[
\sum_{k=1}^{49} k^2 = \frac{49 \cdot 50 \cdot 99}{6}
\]
First, we compute the numerator:
\[
49... | 9,823 | <think>
Okay, so I need to find the sum of the squares of all the integers from 1 to 49. Hmm, let me think about how to approach this. I remember there's a formula for the sum of squares of the first n natural numbers. Wasn't it something like n(n + 1)(2n + 1)/6? Let me verify that. If I plug in n = 1, the sum should b... | [
{
"from": "human",
"value": "Find the sum of the squares of all the integers from 1 to 49."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the sum of the squares of all the integers from 1 to 49. Hmm, let me think about how to approach this. I remember there's a formula for the s... |
[
{
"content": "The sum of the digits of the sum of all the even positive integers between 20 and 200 is equal to 10 times the sum of the digits of their product. Determine the smallest possible value of the sum of the digits of their product.",
"role": "user"
},
{
"content": "To find the smallest pos... | The sum of the digits of the sum of all the even positive integers between 20 and 200 is equal to 10 times the sum of the digits of their product. Determine the smallest possible value of the sum of the digits of their product. | The sum of the digits of the sum of all the even positive integers between 20 and 200 is equal to 10 times the sum of the digits of their product. Determine the smallest possible value of the sum of the digits of their product. | ai2-adapt-dev/openmath-2-math | 0.999966 | Okay, let's see. I need to find the smallest possible value of the sum of the digits of the product of all even positive integers between 20 and 200. But there's a condition: the sum of the digits of the sum of all those even numbers should be equal to 10 times the sum of the digits of their product. Hmm, that sounds a... | First, we need to find the sum of all even positive integers between 20 and 200 inclusive. This forms an arithmetic sequence with the first term \(a_1 = 20\), the last term \(a_n = 200\), and the common difference \(d = 2\). The number of terms \(n\) is calculated as follows:
\[
n = \frac{200 - 20}{2} + 1 = \frac{180}... | 31,218 | <think>
Okay, let's see. I need to find the smallest possible value of the sum of the digits of the product of all even positive integers between 20 and 200. But there's a condition: the sum of the digits of the sum of all those even numbers should be equal to 10 times the sum of the digits of their product. Hmm, that ... | [
{
"from": "human",
"value": "The sum of the digits of the sum of all the even positive integers between 20 and 200 is equal to 10 times the sum of the digits of their product. Determine the smallest possible value of the sum of the digits of their product."
},
{
"from": "gpt",
"value": "<think>\... |
[{"content":"In the triangle $ABC$ the points $D$, $E$ and $F$ are midpoints of the sides $BC$, $CA$(...TRUNCATED) | "In the triangle $ABC$ the points $D$, $E$ and $F$ are midpoints of the sides $BC$, $CA$ and $AB$ re(...TRUNCATED) | "In the triangle $ABC$ the points $D$, $E$ and $F$ are midpoints of the sides $BC$, $CA$ and $AB$ re(...TRUNCATED) | ai2-adapt-dev/openmath-2-math | 0.999777 | "Okay, so I have this problem here: In triangle ABC, the points D, E, and F are midpoints of the sid(...TRUNCATED) | "Given triangle \\(ABC\\) with \\(D\\), \\(E\\), and \\(F\\) as the midpoints of sides \\(BC\\), \\((...TRUNCATED) | 16,690 | "<think>\nOkay, so I have this problem here: In triangle ABC, the points D, E, and F are midpoints o(...TRUNCATED) | [{"from":"human","value":"In the triangle $ABC$ the points $D$, $E$ and $F$ are midpoints of the sid(...TRUNCATED) |
[{"content":"Let $\\{a_n\\}$ be an arithmetic sequence such that $a_1$ and $a_3$ are the roots of th(...TRUNCATED) | "Let $\\{a_n\\}$ be an arithmetic sequence such that $a_1$ and $a_3$ are the roots of the equation $(...TRUNCATED) | "Let $\\{a_n\\}$ be an arithmetic sequence such that $a_1$ and $a_3$ are the roots of the equation $(...TRUNCATED) | ai2-adapt-dev/openmath-2-math | 0.999899 | "Okay, so I have this problem here: there's an arithmetic sequence {a_n}, and the first term a₁ an(...TRUNCATED) | "Given that \\(\\{a_n\\}\\) is an arithmetic sequence where \\(a_1\\) and \\(a_3\\) are the roots of(...TRUNCATED) | 25,289 | "<think>\nOkay, so I have this problem here: there's an arithmetic sequence {a_n}, and the first ter(...TRUNCATED) | [{"from":"human","value":"Let $\\{a_n\\}$ be an arithmetic sequence such that $a_1$ and $a_3$ are th(...TRUNCATED) |
[{"content":"If the ratio of the height of a rectangle to its width is 2, and its area is $12\\sqrt{(...TRUNCATED) | "If the ratio of the height of a rectangle to its width is 2, and its area is $12\\sqrt{3}$, what is(...TRUNCATED) | "If the ratio of the height of a rectangle to its width is 2, and its area is $12\\sqrt{3}$, what is(...TRUNCATED) | ai2-adapt-dev/openmath-2-math | 0.999401 | "Okay, let's see. The problem says that the ratio of the height of a rectangle to its width is 2, an(...TRUNCATED) | "Given the ratio of the height to the width of a rectangle is 2, we denote the width as \\( w \\) an(...TRUNCATED) | 20,972 | "<think>\nOkay, let's see. The problem says that the ratio of the height of a rectangle to its width(...TRUNCATED) | [{"from":"human","value":"If the ratio of the height of a rectangle to its width is 2, and its area (...TRUNCATED) |
[{"content":"The endpoints of a line segment are (2, 3) and (8, 15). What is the sum of the coordina(...TRUNCATED) | "The endpoints of a line segment are (2, 3) and (8, 15). What is the sum of the coordinates of the m(...TRUNCATED) | "The endpoints of a line segment are (2, 3) and (8, 15). What is the sum of the coordinates of the m(...TRUNCATED) | ai2-adapt-dev/openmath-2-math | 0.999583 | "Okay, so I have this problem here: The endpoints of a line segment are (2, 3) and (8, 15). What is (...TRUNCATED) | "To find the sum of the coordinates of the midpoint of the line segment with endpoints (2, 3) and (8(...TRUNCATED) | 5,591 | "<think>\nOkay, so I have this problem here: The endpoints of a line segment are (2, 3) and (8, 15).(...TRUNCATED) | [{"from":"human","value":"The endpoints of a line segment are (2, 3) and (8, 15). What is the sum of(...TRUNCATED) |
[{"content":"In a triangle, the length of the hypotenuse and one leg are relatively prime, while the(...TRUNCATED) | "In a triangle, the length of the hypotenuse and one leg are relatively prime, while the length of t(...TRUNCATED) | "In a triangle, the length of the hypotenuse and one leg are relatively prime, while the length of t(...TRUNCATED) | ai2-adapt-dev/openmath-2-math | 0.999806 | "Okay, let me try to figure out this triangle problem. So, we have a right-angled triangle, I assume(...TRUNCATED) | "Given a right-angled triangle with hypotenuse 65, one leg coprime with the hypotenuse, and the othe(...TRUNCATED) | 7,008 | "<think>\nOkay, let me try to figure out this triangle problem. So, we have a right-angled triangle,(...TRUNCATED) | [{"from":"human","value":"In a triangle, the length of the hypotenuse and one leg are relatively pri(...TRUNCATED) |
[{"content":"The roots of the equation\n$$x^2-12x+36=0$$\nare such that the difference of their cube(...TRUNCATED) | "The roots of the equation\n$$x^2-12x+36=0$$\nare such that the difference of their cubes is equal t(...TRUNCATED) | "The roots of the equation\n$$x^2-12x+36=0$$\nare such that the difference of their cubes is equal t(...TRUNCATED) | ai2-adapt-dev/openmath-2-math | 0.999684 | "Okay, so I need to figure out the value of k in the equation x² - 12x + 36 = 0, where the differen(...TRUNCATED) | "Given the equation \\(x^2 - 12x + 36 = 0\\), we need to determine the value of \\(k\\) such that th(...TRUNCATED) | 11,245 | "<think>\nOkay, so I need to figure out the value of k in the equation x² - 12x + 36 = 0, where the(...TRUNCATED) | [{"from":"human","value":"The roots of the equation\n$$x^2-12x+36=0$$\nare such that the difference (...TRUNCATED) |
[{"content":"The sides of a right triangle have lengths $a, a+2$, and $2a$. Find the ratio of the le(...TRUNCATED) | "The sides of a right triangle have lengths $a, a+2$, and $2a$. Find the ratio of the lengths of the(...TRUNCATED) | "The sides of a right triangle have lengths $a, a+2$, and $2a$. Find the ratio of the lengths of the(...TRUNCATED) | ai2-adapt-dev/openmath-2-math | 0.999443 | "Okay, so I need to find the ratio of the sides of a right triangle with lengths a, a+2, and 2a. Hmm(...TRUNCATED) | "Given a right triangle with sides \\(a\\), \\(a+2\\), and \\(2a\\), we need to determine the ratio (...TRUNCATED) | 16,987 | "<think>\nOkay, so I need to find the ratio of the sides of a right triangle with lengths a, a+2, an(...TRUNCATED) | [{"from":"human","value":"The sides of a right triangle have lengths $a, a+2$, and $2a$. Find the ra(...TRUNCATED) |
[{"content":"In a right triangle, the length of the hypotenuse is 20 cm and the length of one of the(...TRUNCATED) | "In a right triangle, the length of the hypotenuse is 20 cm and the length of one of the legs is 16 (...TRUNCATED) | "In a right triangle, the length of the hypotenuse is 20 cm and the length of one of the legs is 16 (...TRUNCATED) | ai2-adapt-dev/openmath-2-math | 0.999562 | "Okay, so I need to find the area of the circle inscribed in a right triangle where the hypotenuse i(...TRUNCATED) | "Given a right triangle with a hypotenuse of 20 cm and one leg of 16 cm, we need to determine the ar(...TRUNCATED) | 10,495 | "<think>\nOkay, so I need to find the area of the circle inscribed in a right triangle where the hyp(...TRUNCATED) | [{"from":"human","value":"In a right triangle, the length of the hypotenuse is 20 cm and the length (...TRUNCATED) |
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