instruction_seed stringlengths 21 1.44k | response_seed stringlengths 0 1.44k ⌀ | reasoning stringlengths 1.35k 80.8k | deepseek_solution stringlengths 237 3.8k | __original_row_idx int64 36 40k ⌀ | source stringclasses 4
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$(*)$ Let $ABC$ be a triangle with $\angle ABC$ obtuse. The $A$-excircle is a circle in the exterior of $\triangle ABC$ that is tangent to side $\overline{BC}$ of the triangle and tangent to the extensions of the other two sides. Let $E$, $F$ be the feet of the altitudes from $B$ and $C$ to lines $AC$ and $AB$, respect... | null | Okay, so I need to figure out if the line EF can be tangent to the A-excircle in an obtuse triangle ABC where angle ABC is obtuse. Let me start by recalling some definitions and properties to make sure I have the right setup.
First, the A-excircle. Ex circles are also called escribed circles. The A-excircle is tangent... | Given an obtuse triangle \(ABC\) with \(\angle ABC\) obtuse, we need to determine if the line \(EF\) formed by the feet of the altitudes from \(B\) and \(C\) to \(AC\) and \(AB\) respectively can be tangent to the \(A\)-excircle.
1. **Understanding the A-excircle**: The \(A\)-excircle is tangent to \(BC\) and the exte... | null | OpenMathInstruct-AIME | <think>
Okay, so I need to figure out if the line EF can be tangent to the A-excircle in an obtuse triangle ABC where angle ABC is obtuse. Let me start by recalling some definitions and properties to make sure I have the right setup.
First, the A-excircle. Ex circles are also called escribed circles. The A-excircle is... | [
{
"from": "human",
"value": "$(*)$ Let $ABC$ be a triangle with $\\angle ABC$ obtuse. The $A$-excircle is a circle in the exterior of $\\triangle ABC$ that is tangent to side $\\overline{BC}$ of the triangle and tangent to the extensions of the other two sides. Let $E$, $F$ be the feet of the altitudes from... |
15. Let the complex numbers $z_{1}, z_{2}$ satisfy $\left|z_{1}\right|=\left|z_{1}+z_{2}\right|=3,\left|z_{1}-z_{2}\right|=3 \sqrt{3}$, find the value of $\log _{3}\left|\left(z_{1} \overline{z_{2}}\right)^{2000}+\left(\overline{z_{1}} z_{2}\right)^{2000}\right|$. | null | Okay, so I need to solve this problem involving complex numbers. Let me read it again carefully.
We have two complex numbers, z₁ and z₂. They satisfy |z₁| = |z₁ + z₂| = 3, and |z₁ - z₂| = 3√3. We need to find the value of log base 3 of the absolute value of (z₁ times the conjugate of z₂) raised to the 2000th power plu... | Given the complex numbers \( z_1 \) and \( z_2 \) satisfying \( |z_1| = |z_1 + z_2| = 3 \) and \( |z_1 - z_2| = 3\sqrt{3} \), we need to find the value of \( \log_3 \left| \left( z_1 \overline{z_2} \right)^{2000} + \left( \overline{z_1} z_2 \right)^{2000} \right| \).
First, we use the given magnitudes to set up equati... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to solve this problem involving complex numbers. Let me read it again carefully.
We have two complex numbers, z₁ and z₂. They satisfy |z₁| = |z₁ + z₂| = 3, and |z₁ - z₂| = 3√3. We need to find the value of log base 3 of the absolute value of (z₁ times the conjugate of z₂) raised to the 2000th p... | [
{
"from": "human",
"value": "15. Let the complex numbers $z_{1}, z_{2}$ satisfy $\\left|z_{1}\\right|=\\left|z_{1}+z_{2}\\right|=3,\\left|z_{1}-z_{2}\\right|=3 \\sqrt{3}$, find the value of $\\log _{3}\\left|\\left(z_{1} \\overline{z_{2}}\\right)^{2000}+\\left(\\overline{z_{1}} z_{2}\\right)^{2000}\\right|$... |
"Each corner cube is removed from this $3\\text{ cm}\\times 3\\text{ cm}\\times 3\\text{ cm}$ cube. (...TRUNCATED) | null | "Okay, let me try to figure out this problem step by step. So, we have a 3 cm × 3 cm × 3 cm cube, (...TRUNCATED) | "The original 3 cm × 3 cm × 3 cm cube has a surface area of 6 faces × 9 sq.cm each = 54 sq.cm. Wh(...TRUNCATED) | null | OpenMathInstruct-AIME | "<think>\nOkay, let me try to figure out this problem step by step. So, we have a 3 cm × 3 cm × 3 (...TRUNCATED) | [{"from":"human","value":"Each corner cube is removed from this $3\\text{ cm}\\times 3\\text{ cm}\\t(...TRUNCATED) |
"Simplify\n\\[\\frac{\\sin{10^\\circ}+\\sin{20^\\circ}}{\\cos{10^\\circ}+\\cos{20^\\circ}}.\\]Enter (...TRUNCATED) | "Simplify\n\\[\\frac{\\sin{10^\\circ}+\\sin{20^\\circ}}{\\cos{10^\\circ}+\\cos{20^\\circ}}.\\]Enter (...TRUNCATED) | "Okay, let's see. I need to simplify this trigonometric expression: (sin10° + sin20°)/(cos10° + c(...TRUNCATED) | "To simplify the expression \\(\\frac{\\sin{10^\\circ}+\\sin{20^\\circ}}{\\cos{10^\\circ}+\\cos{20^\(...TRUNCATED) | 33,695 | ai2-adapt-dev/openmath-2-math | "<think>\nOkay, let's see. I need to simplify this trigonometric expression: (sin10° + sin20°)/(co(...TRUNCATED) | [{"from":"human","value":"Simplify\n\\[\\frac{\\sin{10^\\circ}+\\sin{20^\\circ}}{\\cos{10^\\circ}+\\(...TRUNCATED) |
"Prove that any knot or link in a $3$-highly twisted $2m$-plat diagram with $m \\geq 2$ and at least(...TRUNCATED) | "Okay, so I need to prove that any knot or link in a 3-highly twisted 2m-plat diagram with m ≥ 2 a(...TRUNCATED) | "To prove that any knot or link in a 3-highly twisted 2m-plat diagram with \\( m \\geq 2 \\) and at (...TRUNCATED) | 18,039 | mlfoundations-dev/gair_mathpile | "<think>\nOkay, so I need to prove that any knot or link in a 3-highly twisted 2m-plat diagram with (...TRUNCATED) | [{"from":"human","value":"Prove that any knot or link in a $3$-highly twisted $2m$-plat diagram with(...TRUNCATED) | |
"A bookshelf has 5 shelves, and each shelf can hold at most 8 books of the same size. If the bookshe(...TRUNCATED) | "A bookshelf has 5 shelves, and each shelf can hold at most 8 books of the same size. If the bookshe(...TRUNCATED) | "Alright, so I need to figure out the least number of shelves required to store 24 books of differen(...TRUNCATED) | "The problem requires determining the least number of shelves needed to store 24 books of different (...TRUNCATED) | 3,003 | ai2-adapt-dev/openmath-2-math | "<think>\nAlright, so I need to figure out the least number of shelves required to store 24 books of(...TRUNCATED) | [{"from":"human","value":"A bookshelf has 5 shelves, and each shelf can hold at most 8 books of the (...TRUNCATED) |
"On June 1, a group of students is standing in rows, with 15 students in each row. On June 2, the sa(...TRUNCATED) | null | "Okay, let me try to figure out this problem step by step. So, the question is about finding the sma(...TRUNCATED) | "To determine the smallest number of students \\( N \\) such that the group can be arranged in diffe(...TRUNCATED) | null | OpenMathInstruct-AIME | "<think>\nOkay, let me try to figure out this problem step by step. So, the question is about findin(...TRUNCATED) | [{"from":"human","value":"On June 1, a group of students is standing in rows, with 15 students in ea(...TRUNCATED) |
"Given the curve $\\frac{x^2}{12-m}+\\frac{y^2}{m-4}=1$ represents an ellipse, which of the followin(...TRUNCATED) | null | "Okay, let's see. I have this problem about an ellipse given by the equation (x²)/(12 - m) + (y²)/(...TRUNCATED) | "Given the curve \\(\\frac{x^2}{12-m} + \\frac{y^2}{m-4} = 1\\) represents an ellipse, we need to de(...TRUNCATED) | null | AI-MO/NuminaMath-1.5 | "<think>\nOkay, let's see. I have this problem about an ellipse given by the equation (x²)/(12 - m)(...TRUNCATED) | [{"from":"human","value":"Given the curve $\\frac{x^2}{12-m}+\\frac{y^2}{m-4}=1$ represents an ellip(...TRUNCATED) |
"Consider the damped hyperbolic Allen-Cahn equation given by\n\n\\[\n\\tau u_{tt} + g(u) u_t = \\Del(...TRUNCATED) | "Okay, so I need to figure out the motion of the interface in the damped hyperbolic Allen-Cahn equat(...TRUNCATED) | "To analyze the motion of the interface in the damped hyperbolic Allen-Cahn equation as \\(\\varepsi(...TRUNCATED) | 4,303 | mlfoundations-dev/gair_mathpile | "<think>\nOkay, so I need to figure out the motion of the interface in the damped hyperbolic Allen-C(...TRUNCATED) | [{"from":"human","value":"Consider the damped hyperbolic Allen-Cahn equation given by\n\n\\[\n\\tau (...TRUNCATED) | |
"From time $t=0$ to time $t=1$ a population increased by $i\\%$, and from time $t=1$ to time $t=2$ t(...TRUNCATED) | null | "Okay, so I need to figure out the total percentage increase in population from time t=0 to t=2, giv(...TRUNCATED) | "Given a population that increases by \\(i\\%\\) from time \\(t=0\\) to \\(t=1\\) and then by \\(j\\(...TRUNCATED) | null | OpenMathInstruct-AIME | "<think>\nOkay, so I need to figure out the total percentage increase in population from time t=0 to(...TRUNCATED) | [{"from":"human","value":"From time $t=0$ to time $t=1$ a population increased by $i\\%$, and from t(...TRUNCATED) |
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