Datasets:

aochongoliverli
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math8k-sft-QwQ-32B-16k-reasoning-traces

problem stringlengths 17 4.31k	solution stringlengths 0 159	level int64 3 5	conversations listlengths 2 2
Let $a$ and $b$ be the two real values of $x$ for which\[\sqrt[3]{x} + \sqrt[3]{20 - x} = 2\]The smaller of the two values can be expressed as $p - \sqrt{q}$, where $p$ and $q$ are integers. Compute $p + q$.	118	5	[ { "content": "Let $a$ and $b$ be the two real values of $x$ for which\\[\\sqrt[3]{x} + \\sqrt[3]{20 - x} = 2\\]The smaller of the two values can be expressed as $p - \\sqrt{q}$, where $p$ and $q$ are integers. Compute $p + q$.", "role": "user" }, { "content": "<think>\nAlright, so I have this equati...
For how many integer values of $x$ is $5x^{2}+19x+16 > 20$ not satisfied?	5	5	[ { "content": "For how many integer values of $x$ is $5x^{2}+19x+16 > 20$ not satisfied?", "role": "user" }, { "content": "<think>\nOkay, so I need to figure out for how many integer values of x the inequality 5x² + 19x + 16 > 20 is not satisfied. Hmm, that means I need to find the integer x where th...
A car is averaging 50 miles per hour. If the car maintains this speed, how many minutes less would a 450-mile trip take than a 475-mile trip?	30	3	[ { "content": "A car is averaging 50 miles per hour. If the car maintains this speed, how many minutes less would a 450-mile trip take than a 475-mile trip?", "role": "user" }, { "content": "<think>\nOkay, so I need to figure out how many minutes less a 450-mile trip would take compared to a 475-mile...
Find the greatest common divisor of $10293$ and $29384$.	1	3	[ { "content": "Find the greatest common divisor of $10293$ and $29384$.", "role": "user" }, { "content": "<think>\nOkay, so I need to find the greatest common divisor (GCD) of 10293 and 29384. Hmm, GCD is the largest number that divides both numbers without leaving a remainder. I remember there's the...
How many ounces of pure water must be added to $30$ ounces of a $30\%$ solution of acid to yield a solution that is $20\%$ acid?	15	3	[ { "content": "How many ounces of pure water must be added to $30$ ounces of a $30\\%$ solution of acid to yield a solution that is $20\\%$ acid?", "role": "user" }, { "content": "<think>\nOkay, so I need to figure out how many ounces of pure water to add to 30 ounces of a 30% acid solution to make i...
An octahedron consists of two square-based pyramids glued together along their square bases to form a polyhedron with eight faces. Imagine an ant that begins at the top vertex and walks to one of the four adjacent vertices that he randomly selects and calls vertex A. From vertex A, he will then walk to one of the four ...	\frac{1}{4}	3	[ { "content": "An octahedron consists of two square-based pyramids glued together along their square bases to form a polyhedron with eight faces. Imagine an ant that begins at the top vertex and walks to one of the four adjacent vertices that he randomly selects and calls vertex A. From vertex A, he will then wa...
Suppose I have 6 shirts, 4 ties, and 3 pairs of pants. If an outfit requires a shirt and pants, and can either have a tie or not have a tie, how many outfits can I make?	90	4	[ { "content": "Suppose I have 6 shirts, 4 ties, and 3 pairs of pants. If an outfit requires a shirt and pants, and can either have a tie or not have a tie, how many outfits can I make?", "role": "user" }, { "content": "<think>\nOkay, so I need to figure out how many different outfits I can make with...
In right triangle $ABC$ with $\angle A = 90^\circ$, we have $AB =16$ and $BC = 24$. Find $\sin A$.	1	3	[ { "content": "In right triangle $ABC$ with $\\angle A = 90^\\circ$, we have $AB =16$ and $BC = 24$. Find $\\sin A$.", "role": "user" }, { "content": "<think>\nOkay, so I need to find sin A for the right triangle ABC where angle A is 90 degrees, AB is 16 units long, and BC is 24 units. Hmm, wait a s...
If $\sqrt{5 + x} + \sqrt{20 - x} = 7$, what is the value of $(5 + x)(20 - x)$?	144	4	[ { "content": "If $\\sqrt{5 + x} + \\sqrt{20 - x} = 7$, what is the value of $(5 + x)(20 - x)$?", "role": "user" }, { "content": "<think>\nOkay, so I have this equation here: √(5 + x) + √(20 - x) = 7, and I need to find the value of (5 + x)(20 - x). Hmm, let me think about how to approach this. \n\nF...
Find the sum of the coefficients in the polynomial $3(3x^{7} + 8x^4 - 7) + 7(x^5 - 7x^2 + 5)$ when it is fully simplified.	5	4	[ { "content": "Find the sum of the coefficients in the polynomial $3(3x^{7} + 8x^4 - 7) + 7(x^5 - 7x^2 + 5)$ when it is fully simplified.", "role": "user" }, { "content": "<think>\nOkay, so I need to find the sum of the coefficients in this polynomial: 3(3x⁷ + 8x⁴ - 7) + 7(x⁵ - 7x² + 5). Hmm, let me ...
A mother purchases 5 blue plates, 2 red plates, 2 green plates, and 1 orange plate. How many ways are there for her to arrange these plates for dinner around her circular table if she doesn't want the 2 green plates to be adjacent?	588	5	[ { "content": "A mother purchases 5 blue plates, 2 red plates, 2 green plates, and 1 orange plate. How many ways are there for her to arrange these plates for dinner around her circular table if she doesn't want the 2 green plates to be adjacent?", "role": "user" }, { "content": "<think>\nOkay, so th...
How many positive integers, not exceeding 100, are multiples of 2 or 3 but not 4?	42	5	[ { "content": "How many positive integers, not exceeding 100, are multiples of 2 or 3 but not 4?", "role": "user" }, { "content": "<think>\nOkay, so I need to figure out how many numbers from 1 to 100 are multiples of 2 or 3 but not 4. Hmm, let me think step by step. \n\nFirst, maybe I should start b...
A function $f$ is defined on the complex numbers by $f(z)=(a+bi)z,$ where $a$ and $b$ are positive numbers. This function has the property that for each complex number $z$, $f(z)$ is equidistant from both $z$ and the origin. Given that $\|a+bi\|=8$, find $b^2.$	\frac{255}{4}	5	[ { "content": "A function $f$ is defined on the complex numbers by $f(z)=(a+bi)z,$ where $a$ and $b$ are positive numbers. This function has the property that for each complex number $z$, $f(z)$ is equidistant from both $z$ and the origin. Given that $\|a+bi\|=8$, find $b^2.$", "role": "user" }, { "con...
The equation $y = \frac{x + A}{Bx + C}$, where $A,B,$ and $C$ are integers, is shown below. What is $A + B + C$? [asy] import graph; size(8.14cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-2.52,xmax=5.62,ymin=-4.28,ymax=3.32; pen cqcqcq=rgb(0.75,0.75,0.75); /grid/...	-3	5	[ { "content": "The equation $y = \\frac{x + A}{Bx + C}$, where $A,B,$ and $C$ are integers, is shown below. What is $A + B + C$?\n\n[asy]\nimport graph; size(8.14cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-2.52,xmax=5.62,ymin=-4.28,ymax=3.32;\n\npen cqcqcq=rg...
What is the least integer value of $x$ such that $\lvert2x+ 7\rvert\le 16$?	-11	4	[ { "content": "What is the least integer value of $x$ such that $\\lvert2x+ 7\\rvert\\le 16$?", "role": "user" }, { "content": "<think>\nOkay, so I need to find the least integer value of x that satisfies the inequality \|2x + 7\| ≤ 16. Hmm, absolute value inequalities can sometimes be tricky, but let...
Let $A = (8,0,0),$ $B = (0,-4,0),$ $C = (0,0,6),$ and $D = (0,0,0).$ Find the point $P$ such that \[AP = BP = CP = DP.\]	(4,-2,3)	3	[ { "content": "Let $A = (8,0,0),$ $B = (0,-4,0),$ $C = (0,0,6),$ and $D = (0,0,0).$ Find the point $P$ such that\n\\[AP = BP = CP = DP.\\]", "role": "user" }, { "content": "<think>\nOkay, so I need to find a point P that is equidistant from points A, B, C, and D. All these points are in 3D space. Le...
A rectangle has a perimeter of 30 units and its dimensions are whole numbers. What is the maximum possible area of the rectangle in square units?	56	4	[ { "content": "A rectangle has a perimeter of 30 units and its dimensions are whole numbers. What is the maximum possible area of the rectangle in square units?", "role": "user" }, { "content": "<think>\nOkay, so I have this problem here: A rectangle has a perimeter of 30 units, and its dimensions ar...
The number $5\,41G\,507\,2H6$ is divisible by $72.$ If $G$ and $H$ each represent a single digit, what is the sum of all distinct possible values of the product $GH?$ (Count each possible value of $GH$ only once, even if it results from multiple $G,$ $H$ pairs.)	59	5	[ { "content": "The number $5\\,41G\\,507\\,2H6$ is divisible by $72.$ If $G$ and $H$ each represent a single digit, what is the sum of all distinct possible values of the product $GH?$ (Count each possible value of $GH$ only once, even if it results from multiple $G,$ $H$ pairs.)", "role": "user" }, { ...