instruction_seed stringlengths 16 5.67k | response_seed stringlengths 0 5.67k ⌀ | reasoning stringlengths 526 104k | deepseek_solution stringlengths 7 5.23k | __original_row_idx int64 0 40k ⌀ | source stringclasses 4
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In triangle $ABC$, $AB = 10$ and $AC = 17$. Let $D$ be the foot of the perpendicular from $A$ to $BC$. If $BD:CD = 2:5$, then find $AD$. | In triangle $ABC$, $AB = 10$ and $AC = 17$. Let $D$ be the foot of the perpendicular from $A$ to $BC$. If $BD:CD = 2:5$, then find $AD$. | Okay, so we have triangle ABC with AB = 10 and AC = 17. D is the foot of the perpendicular from A to BC, so AD is perpendicular to BC. We're told the ratio BD:CD is 2:5, and we need to find the length of AD. Hmm, let's see.
First, since D is the foot of the perpendicular from A to BC, triangle ABD and triangle ACD ar... | Given triangle \(ABC\) with \(AB = 10\) and \(AC = 17\). \(D\) is the foot of the perpendicular from \(A\) to \(BC\) such that \(BD:CD = 2:5\). We need to find the length of \(AD\).
First, denote \(BD = 2k\) and \(CD = 5k\). Therefore, \(BC = BD + CD = 7k\).
Using the Pythagorean theorem in triangles \(ABD\) and \(AC... | 22,660 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so we have triangle ABC with AB = 10 and AC = 17. D is the foot of the perpendicular from A to BC, so AD is perpendicular to BC. We're told the ratio BD:CD is 2:5, and we need to find the length of AD. Hmm, let's see.
First, since D is the foot of the perpendicular from A to BC, triangle ABD and triangl... | [
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"from": "human",
"value": "In triangle $ABC$, $AB = 10$ and $AC = 17$. Let $D$ be the foot of the perpendicular from $A$ to $BC$. If $BD:CD = 2:5$, then find $AD$."
},
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"from": "gpt",
"value": "<think>\nOkay, so we have triangle ABC with AB = 10 and AC = 17. D is the foot of the perpendicul... |
Given a homogeneous simplicial affine semigroup \( S \) with \( \operatorname{deg} K[S] = \alpha^{d-1} \), where \( \alpha \) is greater than or equal to 3 and \( d \) is 3 or more, how does the presence of at least \( \left( \frac{3}{4} + \frac{1}{4d} \right) \alpha + 2 \) integer points on any edge of the associated ... | Okay, let's try to tackle this problem step by step. First, I need to make sure I understand all the terms and what's being asked here.
We have a homogeneous simplicial affine semigroup \( S \). The degree of the associated semigroup ring \( K[S] \) is given as \( \alpha^{d-1} \), where \( \alpha \geq 3 \) and \( d \... | Given a homogeneous simplicial affine semigroup \( S \) with \( \operatorname{deg} K[S] = \alpha^{d-1} \), where \( \alpha \geq 3 \) and \( d \geq 3 \), we need to determine how the presence of at least \( \left( \frac{3}{4} + \frac{1}{4d} \right) \alpha + 2 \) integer points on any edge of the associated convex polyto... | 24,903 | mlfoundations-dev/gair_mathpile | <think>
Okay, let's try to tackle this problem step by step. First, I need to make sure I understand all the terms and what's being asked here.
We have a homogeneous simplicial affine semigroup \( S \). The degree of the associated semigroup ring \( K[S] \) is given as \( \alpha^{d-1} \), where \( \alpha \geq 3 \) an... | [
{
"from": "human",
"value": "Given a homogeneous simplicial affine semigroup \\( S \\) with \\( \\operatorname{deg} K[S] = \\alpha^{d-1} \\), where \\( \\alpha \\) is greater than or equal to 3 and \\( d \\) is 3 or more, how does the presence of at least \\( \\left( \\frac{3}{4} + \\frac{1}{4d} \\right) \\... | |
"Consider a signed graph where the adjacency matrix \\( B \\) has the following structure:\n\n\\[\nB(...TRUNCATED) | "Okay, let me try to work through this problem step by step. The problem is about a signed graph wit(...TRUNCATED) | "**Answer**\n\n1. **Convergence with Non-Negative Initial Opinions** \n The system converges to t(...TRUNCATED) | 32,617 | mlfoundations-dev/gair_mathpile | "<think>\nOkay, let me try to work through this problem step by step. The problem is about a signed (...TRUNCATED) | [{"from":"human","value":"Consider a signed graph where the adjacency matrix \\( B \\) has the follo(...TRUNCATED) | |
"The perimeter of a rectangle is 48 units, and its area is 135 square units. The length of one side (...TRUNCATED) | "The perimeter of a rectangle is 48 units, and its area is 135 square units. The length of one side (...TRUNCATED) | "Okay, so I need to find the length of the diagonal of a rectangle. The problem says the perimeter i(...TRUNCATED) | "Given the problem, we need to find the length of the diagonal of a rectangle with a perimeter of 48(...TRUNCATED) | 9,311 | ai2-adapt-dev/openmath-2-math | "<think>\nOkay, so I need to find the length of the diagonal of a rectangle. The problem says the pe(...TRUNCATED) | [{"from":"human","value":"The perimeter of a rectangle is 48 units, and its area is 135 square units(...TRUNCATED) |
"Given a positive integer \\( k \\) and a dominant integral weight \\( \\lambda \\) for the affine L(...TRUNCATED) | "Okay, so I have this problem about affine Lie algebras, specifically dealing with modules and highe(...TRUNCATED) | "Given a positive integer \\( k \\) and a dominant integral weight \\( \\lambda \\) for the affine L(...TRUNCATED) | 342 | mlfoundations-dev/gair_mathpile | "<think>\nOkay, so I have this problem about affine Lie algebras, specifically dealing with modules (...TRUNCATED) | [{"from":"human","value":"Given a positive integer \\( k \\) and a dominant integral weight \\( \\la(...TRUNCATED) | |
"A bookstore owner wants to display 5 bestsellers in a row on a shelf, but no two books can be writt(...TRUNCATED) | "A bookstore owner wants to display 5 bestsellers in a row on a shelf, but no two books can be writt(...TRUNCATED) | "Okay, so there's this problem where a bookstore owner wants to display 5 bestsellers in a row, and (...TRUNCATED) | "The bookstore owner wants to display 5 bestsellers in a row, with each book by a different author. (...TRUNCATED) | 34,876 | ai2-adapt-dev/openmath-2-math | "<think>\nOkay, so there's this problem where a bookstore owner wants to display 5 bestsellers in a (...TRUNCATED) | [{"from":"human","value":"A bookstore owner wants to display 5 bestsellers in a row on a shelf, but (...TRUNCATED) |
"A cone with a base radius of X cm and a height of 9 cm is full of sand. The sand is poured into a c(...TRUNCATED) | "To solve this problem, we need to compare the volume of the sand in the cone to the volume of the s(...TRUNCATED) | "Okay, let's see here. There's a cone filled with sand, and then the sand is poured into a cylinder.(...TRUNCATED) | "To find the value of \\( X \\), we start by equating the volume of the cone to the volume of the sa(...TRUNCATED) | 27,300 | MetaMath-AIME | "<think>\nOkay, let's see here. There's a cone filled with sand, and then the sand is poured into a (...TRUNCATED) | [{"from":"human","value":"A cone with a base radius of X cm and a height of 9 cm is full of sand. Th(...TRUNCATED) |
"What is the smallest positive integer that is divisible by both 4 and 9 and has exactly 4 distinct (...TRUNCATED) | "What is the smallest positive integer that is divisible by both 4 and 9 and has exactly 4 distinct (...TRUNCATED) | "Alright, let me try to figure out this problem. The question is asking for the smallest positive in(...TRUNCATED) | "To find the smallest positive integer that is divisible by both 4 and 9 and has exactly 4 distinct (...TRUNCATED) | 32,996 | ai2-adapt-dev/openmath-2-math | "<think>\nAlright, let me try to figure out this problem. The question is asking for the smallest po(...TRUNCATED) | [{"from":"human","value":"What is the smallest positive integer that is divisible by both 4 and 9 an(...TRUNCATED) |
"From July 1 to July 5, Max the monkey ate X coconuts. Each day, he ate 10 more coconuts than he did(...TRUNCATED) | "To solve for X, which represents the number of coconuts Max ate from July 1 to July 5, we need to f(...TRUNCATED) | "Okay, so we have this problem where Max the monkey ate X coconuts from July 1 to July 5. Each day, (...TRUNCATED) | "Max the monkey ate coconuts from July 1 to July 5, with each day's consumption forming an arithmeti(...TRUNCATED) | 25,640 | MetaMath-AIME | "<think>\nOkay, so we have this problem where Max the monkey ate X coconuts from July 1 to July 5. E(...TRUNCATED) | [{"from":"human","value":"From July 1 to July 5, Max the monkey ate X coconuts. Each day, he ate 10 (...TRUNCATED) |
G6.4 The average of $x, y, z$ is 10 . The average of $x, y, z, t$ is 12 . Find $t$. | null | "Okay, let's see. The problem says that the average of x, y, z is 10. Then the average of x, y, z, t(...TRUNCATED) | "The average of \\( x, y, z \\) is 10. This can be written as:\n\\[\n\\frac{x + y + z}{3} = 10\n\\]\(...TRUNCATED) | null | AI-MO/NuminaMath-1.5 | "<think>\nOkay, let's see. The problem says that the average of x, y, z is 10. Then the average of x(...TRUNCATED) | [{"from":"human","value":"G6.4 The average of $x, y, z$ is 10 . The average of $x, y, z, t$ is 12 . (...TRUNCATED) |
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