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value | _fasttext_score float64 1 1 | reasoning stringlengths 1.58k 68.7k | deepseek_solution stringlengths 472 2.58k | __original_row_idx int64 8 31.6k | final_reasoning_trace stringlengths 2.07k 70.2k | conversations listlengths 2 2 |
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[
{
"content": "A cone is reshaped into a sphere such that the volume of the sphere is equal to the volume of the cone. If the height of the cone is equal to the diameter of the sphere, what is the ratio of the surface area of the cone to the surface area of the sphere? Express your answer as a common fraction.",... | A cone is reshaped into a sphere such that the volume of the sphere is equal to the volume of the cone. If the height of the cone is equal to the diameter of the sphere, what is the ratio of the surface area of the cone to the surface area of the sphere? Express your answer as a common fraction. | A cone is reshaped into a sphere such that the volume of the sphere is equal to the volume of the cone. If the height of the cone is equal to the diameter of the sphere, what is the ratio of the surface area of the cone to the surface area of the sphere? Express your answer as a common fraction. | ai2-adapt-dev/openmath-2-math | 0.999891 | Okay, so I need to find the ratio of the surface area of a cone to that of a sphere when the cone is reshaped into a sphere with the same volume. Also, the height of the cone is equal to the diameter of the sphere. Hmm, let me break this down step by step.
First, let's recall the formulas for the volumes of a cone and... | Given a cone reshaped into a sphere such that their volumes are equal, and the height of the cone is equal to the diameter of the sphere, we need to find the ratio of the surface area of the cone to the surface area of the sphere.
1. **Volume Equality:**
- Volume of the cone: \( V_{\text{cone}} = \frac{1}{3}\pi r^2... | 27,070 | <think>
Okay, so I need to find the ratio of the surface area of a cone to that of a sphere when the cone is reshaped into a sphere with the same volume. Also, the height of the cone is equal to the diameter of the sphere. Hmm, let me break this down step by step.
First, let's recall the formulas for the volumes of a ... | [
{
"from": "human",
"value": "A cone is reshaped into a sphere such that the volume of the sphere is equal to the volume of the cone. If the height of the cone is equal to the diameter of the sphere, what is the ratio of the surface area of the cone to the surface area of the sphere? Express your answer as a... |
[
{
"content": "Two right triangles are similar and the ratio of the sides is 3:5. If the hypotenuse of the smaller triangle is 6 inches, what is the ratio of the perimeters of the triangles?",
"role": "user"
},
{
"content": "Since the triangles are similar, the corresponding sides are in the same rat... | Two right triangles are similar and the ratio of the sides is 3:5. If the hypotenuse of the smaller triangle is 6 inches, what is the ratio of the perimeters of the triangles? | Two right triangles are similar and the ratio of the sides is 3:5. If the hypotenuse of the smaller triangle is 6 inches, what is the ratio of the perimeters of the triangles? | ai2-adapt-dev/openmath-2-math | 0.999756 | Okay, so I need to solve this problem about two similar right triangles where the ratio of their sides is 3:5. The hypotenuse of the smaller triangle is 6 inches, and I need to find the ratio of their perimeters. Let me start by recalling what I know about similar triangles.
First, similar triangles have their corresp... | Given two similar right triangles with a side ratio of 3:5, and the hypotenuse of the smaller triangle is 6 inches, we need to find the ratio of their perimeters.
1. **Similar Triangles and Side Ratios**:
- Since the triangles are similar, the ratio of their corresponding sides is 3:5.
- The hypotenuse of the sm... | 22,379 | <think>
Okay, so I need to solve this problem about two similar right triangles where the ratio of their sides is 3:5. The hypotenuse of the smaller triangle is 6 inches, and I need to find the ratio of their perimeters. Let me start by recalling what I know about similar triangles.
First, similar triangles have their... | [
{
"from": "human",
"value": "Two right triangles are similar and the ratio of the sides is 3:5. If the hypotenuse of the smaller triangle is 6 inches, what is the ratio of the perimeters of the triangles?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to solve this problem about two sim... |
[
{
"content": "Determine the last two digits of the number $2^{12345}$.",
"role": "user"
},
{
"content": "The last two digits of $2^n$ follow a cycle for different values of $n$. Let's calculate the last two digits of the first few powers of 2 to find the cycle:\n \\begin{align*}\n 2^1 = 02, \\qu... | Determine the last two digits of the number $2^{12345}$. | Determine the last two digits of the number $2^{12345}$. | ai2-adapt-dev/openmath-2-math | 0.999855 | Okay, so I need to find the last two digits of 2 raised to the power of 12345. That means I need to compute 2^12345 mod 100, right? Because the last two digits of any number are equivalent to the number modulo 100.
Hmm, modulus operations with exponents. I remember something about Euler's theorem here. Euler's theore... | To determine the last two digits of \(2^{12345}\), we need to compute \(2^{12345} \mod 100\). We can use the Chinese Remainder Theorem by breaking it down into modulo 4 and modulo 25.
1. **Modulo 4 Calculation:**
\[
2^{12345} \mod 4
\]
Since \(2^2 \equiv 0 \mod 4\), any higher power of 2 will also be 0 mod... | 22,394 | <think>
Okay, so I need to find the last two digits of 2 raised to the power of 12345. That means I need to compute 2^12345 mod 100, right? Because the last two digits of any number are equivalent to the number modulo 100.
Hmm, modulus operations with exponents. I remember something about Euler's theorem here. Euler'... | [
{
"from": "human",
"value": "Determine the last two digits of the number $2^{12345}$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the last two digits of 2 raised to the power of 12345. That means I need to compute 2^12345 mod 100, right? Because the last two digits of any num... |
[{"content":"Determine the largest integer $n$ such that $\\frac{2^{n}}{5^{n}}$ is a factor of $\\fr(...TRUNCATED) | "Determine the largest integer $n$ such that $\\frac{2^{n}}{5^{n}}$ is a factor of $\\frac{2400}{3^{(...TRUNCATED) | "Determine the largest integer $n$ such that $\\frac{2^{n}}{5^{n}}$ is a factor of $\\frac{2400}{3^{(...TRUNCATED) | ai2-adapt-dev/openmath-2-math | 0.999525 | "Okay, so I need to figure out the largest integer n such that (2^n)/(5^n) is a factor of 2400 divid(...TRUNCATED) | "To determine the largest integer \\( n \\) such that \\( \\frac{2^n}{5^n} \\) is a factor of \\( \\(...TRUNCATED) | 17,884 | "<think>\nOkay, so I need to figure out the largest integer n such that (2^n)/(5^n) is a factor of 2(...TRUNCATED) | [{"from":"human","value":"Determine the largest integer $n$ such that $\\frac{2^{n}}{5^{n}}$ is a fa(...TRUNCATED) |
[{"content":"Find the interval of all $x$ such that the sum of $2x$ and $3x$ is in the interval $(1,(...TRUNCATED) | "Find the interval of all $x$ such that the sum of $2x$ and $3x$ is in the interval $(1,2)$, and the(...TRUNCATED) | "Find the interval of all $x$ such that the sum of $2x$ and $3x$ is in the interval $(1,2)$, and the(...TRUNCATED) | ai2-adapt-dev/openmath-2-math | 0.999762 | "Okay, let's see. I need to find all x such that two conditions are satisfied: first, the sum of 2x (...TRUNCATED) | "To find the interval of all \\( x \\) such that the sum of \\( 2x \\) and \\( 3x \\) is in the inte(...TRUNCATED) | 24,260 | "<think>\nOkay, let's see. I need to find all x such that two conditions are satisfied: first, the s(...TRUNCATED) | [{"from":"human","value":"Find the interval of all $x$ such that the sum of $2x$ and $3x$ is in the (...TRUNCATED) |
[{"content":"Given a set of circular cones, each having the same radius and axis, find the ratio of (...TRUNCATED) | "Given a set of circular cones, each having the same radius and axis, find the ratio of the lateral (...TRUNCATED) | "Given a set of circular cones, each having the same radius and axis, find the ratio of the lateral (...TRUNCATED) | ai2-adapt-dev/openmath-2-math | 0.999591 | "Okay, so I have this problem about circular cones. All the cones have the same radius and the same (...TRUNCATED) | "Given a set of circular cones, each having the same radius and axis, we need to find the ratio of t(...TRUNCATED) | 1,730 | "<think>\nOkay, so I have this problem about circular cones. All the cones have the same radius and (...TRUNCATED) | [{"from":"human","value":"Given a set of circular cones, each having the same radius and axis, find (...TRUNCATED) |
[{"content":"Let $z$ be a complex number such that $\\ds z^3+z^{-3}=2i$. Determine the value of $z^6(...TRUNCATED) | Let $z$ be a complex number such that $\ds z^3+z^{-3}=2i$. Determine the value of $z^6+z^{-6}$. | Let $z$ be a complex number such that $\ds z^3+z^{-3}=2i$. Determine the value of $z^6+z^{-6}$. | ai2-adapt-dev/openmath-2-math | 0.99979 | "Okay, so I need to find the value of \\( z^6 + z^{-6} \\) given that \\( z^3 + z^{-3} = 2i \\). Hmm(...TRUNCATED) | "Given the equation \\( z^3 + z^{-3} = 2i \\), we need to determine the value of \\( z^6 + z^{-6} \\(...TRUNCATED) | 28,625 | "<think>\nOkay, so I need to find the value of \\( z^6 + z^{-6} \\) given that \\( z^3 + z^{-3} = 2i(...TRUNCATED) | [{"from":"human","value":"Let $z$ be a complex number such that $\\ds z^3+z^{-3}=2i$. Determine the (...TRUNCATED) |
[{"content":"A square box with a volume of 64 $\\text{cm}^3$ is inscribed inside a cylinder such tha(...TRUNCATED) | "A square box with a volume of 64 $\\text{cm}^3$ is inscribed inside a cylinder such that the height(...TRUNCATED) | "A square box with a volume of 64 $\\text{cm}^3$ is inscribed inside a cylinder such that the height(...TRUNCATED) | ai2-adapt-dev/openmath-2-math | 0.999492 | "Okay, so I need to find the surface area of a cylinder that has a square box inscribed inside it. T(...TRUNCATED) | "Given a square box with a volume of 64 cm³ inscribed inside a cylinder, where the height of the bo(...TRUNCATED) | 12,696 | "<think>\nOkay, so I need to find the surface area of a cylinder that has a square box inscribed ins(...TRUNCATED) | [{"from":"human","value":"A square box with a volume of 64 $\\text{cm}^3$ is inscribed inside a cyli(...TRUNCATED) |
[{"content":"In the diagram, $AB$ and $AC$ are two chords of a circle that intersect at $X$. $M$ is (...TRUNCATED) | "In the diagram, $AB$ and $AC$ are two chords of a circle that intersect at $X$. $M$ is the midpoint(...TRUNCATED) | "In the diagram, $AB$ and $AC$ are two chords of a circle that intersect at $X$. $M$ is the midpoint(...TRUNCATED) | ai2-adapt-dev/openmath-2-math | 0.999593 | "Okay, so I need to find the ratio of the measure of angle MXC to angle NXB in this circle where AB (...TRUNCATED) | "Given a circle with chords \\(AB\\) and \\(AC\\) intersecting at point \\(X\\). \\(M\\) is the midp(...TRUNCATED) | 7,778 | "<think>\nOkay, so I need to find the ratio of the measure of angle MXC to angle NXB in this circle (...TRUNCATED) | [{"from":"human","value":"In the diagram, $AB$ and $AC$ are two chords of a circle that intersect at(...TRUNCATED) |
[{"content":"In a right circular cone, the volume of the frustum obtained by cutting off the top is (...TRUNCATED) | "In a right circular cone, the volume of the frustum obtained by cutting off the top is one eighth t(...TRUNCATED) | "In a right circular cone, the volume of the frustum obtained by cutting off the top is one eighth t(...TRUNCATED) | ai2-adapt-dev/openmath-2-math | 0.999797 | "Okay, let's see. I need to find the ratio of the height of the removed top to the height of the ori(...TRUNCATED) | "Given a right circular cone, the volume of the frustum obtained by cutting off the top is one eight(...TRUNCATED) | 13,044 | "<think>\nOkay, let's see. I need to find the ratio of the height of the removed top to the height o(...TRUNCATED) | [{"from":"human","value":"In a right circular cone, the volume of the frustum obtained by cutting of(...TRUNCATED) |
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