data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "A 10-cm-by-10-cm square is partitioned such that points $A$ and $B$ are on two opposite sides of the square at one-third and two-thirds the length of the sides, respectively. What is the area of the new shaded region formed by connecting points $A$, $B$, and their reflections a... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 29249,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the equation $\\frac{1}{7} + \\frac{7}{x} = \\frac{16}{x} + \\frac{1}{16}$, what is the value of $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "112"
} | {
"split": "train",
"index": 29250,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the number \\( 2016****02** \\), each of the 6 asterisks needs to be replaced with any of the digits \\( 0, 2, 4, 5, 7, 9 \\) (the digits may repeat) so that the resulting 12-digit number is divisible by 15. How many ways can this be done? Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "5184"
} | {
"split": "train",
"index": 29251,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Assume that $x$ is a positive real number. Which is equivalent to $\\sqrt[3]{x\\sqrt{x}}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "x^{\\frac{1}{2}}"
} | {
"split": "train",
"index": 29252,
"solution": "1. **Rewrite the expression inside the cube root:** \n We start by expressing the term inside the cube root in terms of powers of $x$:\n \\[\n x\\sqrt{x} = x \\cdot x^{1/2} = x^{1 + 1/2} = x^{3/2}\n \\]\n\n2. **Apply the cube root to the expression:** \n ... |
deepscaler | [
{
"role": "user",
"content": "A four-digit number $\\overline{a b c d}$ (where digits can repeat and are non-zero) is called a \"good number\" if it satisfies the conditions $\\overline{a b}<20$, $\\overline{b c}<21$, and $\\overline{c d}<22$. How many such \"good numbers\" are there? Let's think step by st... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 29253,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $(u_n)$ is a sequence of real numbers satisfying\n\\[u_{n+2}=2u_{n+1}+u_n\\]and that $u_3=9$ and $u_6=128$. What is $u_5$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "53"
} | {
"split": "train",
"index": 29254,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Antonette gets $70\\%$ on a 10-problem test, $80\\%$ on a 20-problem test and $90\\%$ on a 30-problem test. If the three tests are combined into one 60-problem test, which percent is her overall score, rounded to the nearest percent? Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "83\\%"
} | {
"split": "train",
"index": 29255,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCDEF$ be a regular hexagon. Let $G, H, I, J, K,$ and $L$ be the centers, respectively, of equilateral triangles with bases $\\overline{AB}, \\overline{BC}, \\overline{CD}, \\overline{DE}, \\overline{EF},$ and $\\overline{FA},$ each exterior to the hexagon. What is the ra... | math | {
"style": "rule",
"ground_truth": "\\frac{10}{3}"
} | {
"split": "train",
"index": 29256,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the least positive three-digit multiple of 7? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "105"
} | {
"split": "train",
"index": 29257,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of positive integer divisors of 12 ! that leave a remainder of 1 when divided by 3. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "66"
} | {
"split": "train",
"index": 29258,
"solution": "First we factor $12!=2^{10} 3^{5} 5^{2} 7^{1} 11^{1}$, and note that $2,5,11 \\equiv-1(\\bmod 3)$ while $7 \\equiv 1$ $(\\bmod 3)$. The desired divisors are precisely $2^{a} 5^{b} 7^{c} 11^{d}$ with $0 \\leq a \\leq 10,0 \\leq b \\leq 2,0 \\leq c \\leq 1,0 \\leq d ... |
deepscaler | [
{
"role": "user",
"content": "How many distinguishable rearrangements of the letters in \"BALANCE\" have all the vowels at the end. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 29259,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the value of $-1 + 2 + 3 + 4 - 5 - 6 - 7 - 8 - 9 + \\dots + 12100$, where the signs change after each perfect square. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1331000"
} | {
"split": "train",
"index": 29260,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $\\frac{(3150-3030)^2}{144}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 29261,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a line segment of length $6$ is divided into three line segments of lengths that are positive integers, calculate the probability that these three line segments can form a triangle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {1}{10}"
} | {
"split": "train",
"index": 29262,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $9s+5t=108$ and $s$ is two less than $t$, what is $t$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 29263,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the one-variable quadratic equation $x^{2}+2x+m+1=0$ has two distinct real roots with respect to $x$, determine the value of $m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 29264,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle is circumscribed about an equilateral triangle with side lengths of $12$ units each. Calculate the area of the circle, and then find the perimeter of the triangle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 29265,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mr. Jones has 6 children. Assuming that the gender of each child is determined independently and with equal likelihood of male and female, what is the probability that Mr. Jones has more sons than daughters or more daughters than sons? Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "\\dfrac{11}{16}"
} | {
"split": "train",
"index": 29266,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each face of a die is arranged so that the sum of the numbers on opposite faces is 7. In the arrangement shown with three dice, only seven faces are visible. What is the sum of the numbers on the faces that are not visible in the given image? Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "41"
} | {
"split": "train",
"index": 29267,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be an acute triangle with incenter $I$ ; ray $AI$ meets the circumcircle $\\Omega$ of $ABC$ at $M \\neq A$ . Suppose $T$ lies on line $BC$ such that $\\angle MIT=90^{\\circ}$ .\nLet $K$ be the foot of the altitude from $I$ to $\\overline{TM}$ . Giv... | math | {
"style": "rule",
"ground_truth": "128"
} | {
"split": "train",
"index": 29268,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For distinct positive integers $a, b<2012$, define $f(a, b)$ to be the number of integers $k$ with $1\\le k<2012$ such that the remainder when $ak$ divided by $2012$ is greater than that of $bk$ divided by $2012$. Let $S$ be the minimum value of $f(a, b)$, where $a$ and $b$ ran... | math | {
"style": "rule",
"ground_truth": "502"
} | {
"split": "train",
"index": 29269,
"solution": "\nTo solve for \\( S \\), the minimum value of \\( f(a, b) \\), where distinct positive integers \\( a, b < 2012 \\), we first need to analyze the function \\( f(a, b) \\). This function represents the number of integers \\( k \\) with \\( 1 \\leq k < 2012 \\) such... |
deepscaler | [
{
"role": "user",
"content": "Given the ranges $-6 \\leq x \\leq -2$ and $0 \\leq y \\leq 4$, determine the largest possible value of the expression $\\frac{x+y}{x}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 29270,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given triangle ABC, where sides $a$, $b$, and $c$ correspond to angles A, B, and C respectively, and $a=4$, $\\cos{B}=\\frac{4}{5}$.\n(1) If $b=6$, find the value of $\\sin{A}$;\n(2) If the area of triangle ABC, $S=12$, find the values of $b$ and $c$. Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{13}"
} | {
"split": "train",
"index": 29271,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the focus of the parabola $y=x^{2}$ is $F$, a line passing through point $F$ intersects the parabola at points $A$ and $B$. If $|AB|=4$, find the distance from the midpoint of chord $AB$ to the $x$-axis. Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{4}"
} | {
"split": "train",
"index": 29272,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 12 sprinters in an international track event final, including 5 Americans. Gold, silver, and bronze medals are awarded to the first, second, and third place finishers respectively. Determine the number of ways the medals can be awarded if no more than two Americans re... | math | {
"style": "rule",
"ground_truth": "1260"
} | {
"split": "train",
"index": 29273,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For the cubic function $f(x)=ax^3+bx^2+cx+d$ ($a\\neq 0$), define: Let $f''(x)$ be the derivative of the derivative of the function $y=f(x)$, that is, the second derivative of $f(x)$. If the equation $f''(x)=0$ has a real solution $x_0$, then the point $(x_0, f(x_0))$ is called... | math | {
"style": "rule",
"ground_truth": "2012"
} | {
"split": "train",
"index": 29274,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangle in a Cartesian coordinate plane has vertices (5, -2), (10, 5) and (5, 5). How many square units are in the area of the triangle? Express your answer as a decimal to the nearest tenth. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17.5"
} | {
"split": "train",
"index": 29275,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A box contains tiles, numbered 1, 2, 3,..., 49, 50. Only tiles which are marked with a number congruent to $2 \\pmod{5}$ are blue. One tile is chosen randomly from the box. What is the probability that the tile is blue? Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"split": "train",
"index": 29276,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A train moves along a straight track, and from the moment it starts braking to the moment it stops, the distance $S$ in meters that the train travels in $t$ seconds after braking is given by $S=27t-0.45t^2$. Find the time in seconds after braking when the train stops, and the d... | math | {
"style": "rule",
"ground_truth": "405"
} | {
"split": "train",
"index": 29277,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The large cube shown is made up of $27$ identical sized smaller cubes. For each face of the large cube, the opposite face is shaded the same way. The total number of smaller cubes that must have at least one face shaded is Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 29278,
"solution": "To solve this problem, we need to determine the number of smaller cubes in the large cube that have at least one face shaded. We are given that the large cube is composed of $27$ smaller cubes and that the shading pattern on one face is mirrored exactly on the oppo... |
deepscaler | [
{
"role": "user",
"content": "Let x; y; z be real numbers, satisfying the relations\n$x \\ge 20$\n$y \u0015\\ge 40$\n$z \u0015\\ge 1675$\nx + y + z = 2015\nFind the greatest value of the product P = $x\u0001y z$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{721480000}{27}"
} | {
"split": "train",
"index": 29279,
"solution": "\nGiven the conditions:\n\n\\[\nx \\geq 20, \\quad y \\geq 40, \\quad z \\geq 1675\n\\]\n\nand the equation:\n\n\\[\nx + y + z = 2015\n\\]\n\nwe need to find the greatest value of the product \\( P = xyz \\).\n\n### Step 1: Analyze the Variables\n\nWe express \\( z... |
deepscaler | [
{
"role": "user",
"content": "The ratio of the areas of two squares is $\\frac{300}{147}$. Find the simplified form of the ratio of their side lengths, expressed as $\\frac{a\\sqrt{b}}{c}$ where $a$, $b$, and $c$ are integers. Additionally, if the perimeter of the larger square is 60 units, determine the si... | math | {
"style": "rule",
"ground_truth": "10.5"
} | {
"split": "train",
"index": 29280,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Koalas absorb only $25\\%$ of the fiber they eat. A koala absorbed 10.5 ounces of fiber in one day. How many ounces of fiber did he eat that day? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 29281,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bottle of cola costs 2 yuan, and two empty bottles can be exchanged for one more bottle of cola. With 30 yuan, what is the maximum number of bottles of cola that you can drink? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 29282,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$, $b$, $c$ be the sides of a triangle, and let $\\alpha$, $\\beta$, $\\gamma$ be the angles opposite them respectively. If $a^2+b^2=2c^2$, find the value of\n\\[\n\\frac{\\cot \\gamma}{\\cot \\alpha + \\cot \\beta}.\n\\] Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 29283,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a positive integer $n$, the factorial notation $n!$ represents the product of the integers from $n$ to $1$. What value of $N$ satisfies the following equation? $5!\\cdot 9!=12\\cdot N!$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 29284,
"solution": "1. **Understanding the given equation**: We start with the equation:\n \\[\n 5! \\cdot 9! = 12 \\cdot N!\n \\]\n We need to find the value of $N$ that satisfies this equation.\n\n2. **Expanding and simplifying the left-hand side**: We know that $5! = 120$ a... |
deepscaler | [
{
"role": "user",
"content": "Ten 6-sided dice are rolled. What is the probability that exactly three of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": ".155"
} | {
"split": "train",
"index": 29285,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of $x$ such that $\\sqrt{1 - 3x} = 7$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-16"
} | {
"split": "train",
"index": 29286,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the terminal side of angle $\\alpha$ passes through the point $(3a, 4a)$ ($a < 0$), then $\\sin\\alpha=$ ______, $\\tan(\\pi-2\\alpha)=$ ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{24}{7}"
} | {
"split": "train",
"index": 29287,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $B$ and $C$ lie on $\\overline{AD}$. The length of $\\overline{AB}$ is 4 times the length of $\\overline{BD}$, and the length of $\\overline{AC}$ is 9 times the length of $\\overline{CD}$. The length of $\\overline{BC}$ is what fraction of the length of $\\overline{AD}$?... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{10}"
} | {
"split": "train",
"index": 29288,
"solution": "1. **Assign Variables:**\n Let the length of $\\overline{CD}$ be $x$. Therefore, the length of $\\overline{AC}$ is $9x$.\n\n2. **Express $\\overline{AB}$ and $\\overline{BD}$:**\n Since $\\overline{AB}$ is $4$ times the length of $\\overline{BD}$, let $\\overli... |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system, the graphs of the functions $y=\\frac{3}{x}$ and $y=x+1$ intersect at the point $\\left(m,n\\right)$. Evaluate the algebraic expression $\\left(m-n\\right)^{2}\\cdot \\left(\\frac{1}{n}-\\frac{1}{m}\\right)$. Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{3}"
} | {
"split": "train",
"index": 29289,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the shortest distance from the origin to the circle defined by \\( x^2 - 30x + y^2 - 8y + 325 = 0 \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{241} - 2\\sqrt{21}"
} | {
"split": "train",
"index": 29290,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}=(2\\cos \\alpha,\\sin ^{2}\\alpha)$, $\\overrightarrow{b}=(2\\sin \\alpha,t)$, where $\\alpha\\in(0, \\frac {\\pi}{2})$, and $t$ is a real number. \n$(1)$ If $\\overrightarrow{a}- \\overrightarrow{b}=( \\frac {2}{5},0)$, find the value of $t$... | math | {
"style": "rule",
"ground_truth": "\\frac {23}{7}"
} | {
"split": "train",
"index": 29291,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "John now has 15 marbles of different colors, including two red, two green, and one blue marble. In how many ways can he choose 5 marbles, if exactly two of the chosen marbles are red and one is green? Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "110"
} | {
"split": "train",
"index": 29292,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the line $15x + 6y = 90$ which forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?\nA) $21$\nB) $35$\nC) $41$\nD) $21 + 10\\sqrt{\\frac{1}{29}}$ Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "21 + 10\\sqrt{\\frac{1}{29}}"
} | {
"split": "train",
"index": 29293,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Using the numbers $2, 4, 12, 40$ each exactly once, you can perform operations to obtain 24. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40 \\div 4 + 12 + 2"
} | {
"split": "train",
"index": 29294,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\operatorname{tg} \\theta$ and $\\operatorname{ctg} \\theta$ are the real roots of the equation $2x^{2} - 2kx = 3 - k^{2}$, and $\\alpha < \\theta < \\frac{5 \\pi}{4}$, find the value of $\\cos \\theta - \\sin \\theta$. Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "-\\sqrt{\\frac{5 - 2\\sqrt{5}}{5}}"
} | {
"split": "train",
"index": 29295,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The annual incomes of $1,000$ families range from $8200$ dollars to $98,000$ dollars. In error, the largest income was entered on the computer as $980,000$ dollars. The difference between the mean of the incorrect data and the mean of the actual data is Let's think step by st... | math | {
"style": "rule",
"ground_truth": "882"
} | {
"split": "train",
"index": 29296,
"solution": "1. **Define the variables:**\n Let $S$ be the sum of the incomes of all families except the one with the highest income, which is $98,000$.\n\n2. **Calculate the mean of the actual data:**\n The mean of the actual data is calculated by adding the highest income... |
deepscaler | [
{
"role": "user",
"content": "John has 6 green marbles and 4 purple marbles. He chooses a marble at random, writes down its color, and then puts the marble back. He performs this process 5 times. What is the probability that he chooses exactly two green marbles? Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "\\frac{144}{625}"
} | {
"split": "train",
"index": 29297,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(a_n\\) be the sequence defined by \\(a_1 = 3\\) and \\(a_{n+1} = 3^{k}\\), where \\(k = a_n\\). Let \\(b_n\\) be the remainder when \\(a_n\\) is divided by 100. Which values \\(b_n\\) occur for infinitely many \\(n\\)? Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "87"
} | {
"split": "train",
"index": 29298,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $P$ and $Q$ are on a circle with radius $7$ and $PQ = 8$. Point $R$ is the midpoint of the minor arc $PQ$. Calculate the length of line segment $PR$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{32}"
} | {
"split": "train",
"index": 29299,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jeremy's friend Steven will pay him $\\dfrac{11}{2}$ dollars for every room he cleans. Yesterday, Jeremy cleaned $\\dfrac{7}{3}$ rooms. How many dollars does Steven owe Jeremy? Leave your answer in fraction form. Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "\\frac{77}{6}"
} | {
"split": "train",
"index": 29300,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer $k$ such that $1^2 + 2^2 + 3^2 + \\ldots + k^2$ is a multiple of $150$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 29301,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The kindergarten received flashcards for reading: some say \"MA\", and the others say \"NYA\". Each child took three cards to form words. It turned out that 20 children can form the word \"MAMA\" from their cards, 30 children can form the word \"NYANYA\", and 40 children can fo... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 29302,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The national student loan is a credit loan subsidized by the finance department, aimed at helping college students from families with financial difficulties to pay for tuition, accommodation, and living expenses during their study period in college. The total amount applied for... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 29303,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of the rational function $\\frac{q(x)}{2x^5+x^4-7x^2+1}$ has a horizontal asymptote. What is the largest possible degree of $q(x)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 29304,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $x$, $y$, and $z$ are all positive real numbers, and $x^{2}+y^{2}+z^{2}=1$, then the minimum value of $\\frac{(z+1)^{2}}{2xyz}$ is $\\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3+2 \\sqrt{2}"
} | {
"split": "train",
"index": 29305,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( A \\) be the set of any 20 points on the circumference of a circle. Joining any two points in \\( A \\) produces one chord of this circle. Suppose every three such chords are not concurrent. Find the number of regions within the circle which are divided by all these cho... | math | {
"style": "rule",
"ground_truth": "5036"
} | {
"split": "train",
"index": 29306,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The circle centered at $(2,-1)$ and with radius $4$ intersects the circle centered at $(2,5)$ and with radius $\\sqrt{10}$ at two points $A$ and $B$. Find $(AB)^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 29307,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Squares $A B C D$ and $D E F G$ have side lengths 1 and $\\frac{1}{3}$, respectively, where $E$ is on $\\overline{C D}$ and points $A, D, G$ lie on a line in that order. Line $C F$ meets line $A G$ at $X$. The length $A X$ can be written as $\\frac{m}{n}$, where $m, n$ are posi... | math | {
"style": "rule",
"ground_truth": "302"
} | {
"split": "train",
"index": 29308,
"solution": "There are a variety of solutions involving similar triangles. One fast way to solve the problem without hunting for many geometric relationships is to notice that, if one continues to add squares inscribed between $\\overline{A X}$ and $\\overline{X C}$ as shown in... |
deepscaler | [
{
"role": "user",
"content": "Let $S_{0}$ be a unit square in the Cartesian plane with horizontal and vertical sides. For any $n>0$, the shape $S_{n}$ is formed by adjoining 9 copies of $S_{n-1}$ in a $3 \\times 3$ grid, and then removing the center copy. Let $a_{n}$ be the expected value of $\\left|x-x^{\\... | math | {
"style": "rule",
"ground_truth": "1217"
} | {
"split": "train",
"index": 29309,
"solution": "By symmetry, we only need to consider the $x$-distance, then we can multiply our answer by 2. Let this quantity be $g(n)=a_{n} / 2$. Divide the $n$th iteration fractal into three meta-columns of equal width. Then the probability that a random point is in the first,... |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the roots of the equation $(x - 5)^2 = 9$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 29310,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate $\\displaystyle \\sum_{n=1}^\\infty \\ln \\left(1+\\frac{1}{n}\\right) \\ln\\left( 1+\\frac{1}{2n}\\right)\\ln\\left( 1+\\frac{1}{2n+1}\\right)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\ln^3(2)}{3}"
} | {
"split": "train",
"index": 29311,
"solution": "\nThe problem requires evaluating the infinite series:\n\n\\[\n\\sum_{n=1}^\\infty \\ln \\left(1+\\frac{1}{n}\\right) \\ln\\left( 1+\\frac{1}{2n}\\right)\\ln\\left( 1+\\frac{1}{2n+1}\\right).\n\\]\n\nFirstly, observe the behavior of the logarithmic terms for large ... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=|\\log_{4}x|$, and real numbers $m$, $n$ satisfy $0 < m < n$ and $f(m)=f(n)$. If the maximum value of $f(x)$ in the interval $[m^{2},n]$ is $2$, then $\\frac{n}{m}=$ ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 29312,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the distance between the points $(2,2)$ and $(-1,-1)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{2}"
} | {
"split": "train",
"index": 29313,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle ABC, BR = RC, CS = 3SA, and (AT)/(TB) = p/q. If the area of △RST is twice the area of △TBR, determine the value of p/q. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{3}"
} | {
"split": "train",
"index": 29314,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If real numbers \\(x\\) and \\(y\\) satisfy \\(x^2 + y^2 = 20\\), what is the maximum value of \\(xy + 8x + y\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 29315,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the minimum number of straight cuts needed to cut a cake in 100 pieces? The pieces do not need to be the same size or shape but cannot be rearranged between cuts. You may assume that the cake is a large cube and may be cut from any direction. Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 29316,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the positive numbers \\( x \\) and \\( y \\) satisfy \\( x^{3} + y^{3} = x - y \\). Find the maximum value of the real number \\( \\lambda \\) such that \\( x^{2} + \\lambda y^{2} \\leq 1 \\) always holds. Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "2 + 2\\sqrt{2}"
} | {
"split": "train",
"index": 29317,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Distinct lines $\\ell$ and $m$ lie in the $xy$-plane. They intersect at the origin. Point $P(-1, 4)$ is reflected about line $\\ell$ to point $P'$, and then $P'$ is reflected about line $m$ to point $P''$. The equation of line $\\ell$ is $5x - y = 0$, and the coordinates of $P'... | math | {
"style": "rule",
"ground_truth": "2x-3y=0"
} | {
"split": "train",
"index": 29318,
"solution": "1. **Identify the given information:**\n - Line $\\ell$ has the equation $5x - y = 0$.\n - Point $P$ has coordinates $(-1, 4)$.\n - Point $P''$ has coordinates $(4, 1)$.\n - Lines $\\ell$ and $m$ intersect at the origin $O$.\n\n2. **Understand the geometric... |
deepscaler | [
{
"role": "user",
"content": "Select two distinct diagonals at random from a regular octagon. What is the probability that the two diagonals intersect at a point strictly within the octagon? Express your answer as $a + b$ , where the probability is $\\tfrac{a}{b}$ and $a$ and $b$ are relatively prime... | math | {
"style": "rule",
"ground_truth": "7 + 19"
} | {
"split": "train",
"index": 29319,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the plane vectors $\\overrightarrow{a}=(1,0)$ and $\\overrightarrow{b}=\\left( -\\frac{1}{2}, \\frac{\\sqrt{3}}{2} \\right)$, find the angle between $\\overrightarrow{a}$ and $\\overrightarrow{a}+ \\overrightarrow{b}$. Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"split": "train",
"index": 29320,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Out of sixteen Easter eggs, three are red. Ten eggs were placed in a larger box and six in a smaller box at random. What is the probability that both boxes contain at least one red egg? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3/4"
} | {
"split": "train",
"index": 29321,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the numbers 0, 1, 2, 3, 4, 5, 6, determine the total number of 3-digit numbers that can be formed from these digits without repetition and divided by 5. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "55"
} | {
"split": "train",
"index": 29322,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The segment connecting the centers of two intersecting circles is divided by their common chord into segments of 4 and 1. Find the length of the common chord, given that the radii of the circles are in the ratio $3:2$. Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{11}"
} | {
"split": "train",
"index": 29323,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a tetrahedron \\( S-ABC \\), point \\( A_1 \\) is the centroid of \\( \\triangle SBC \\). Point \\( G \\) is on the segment \\( AA_1 \\) such that \\(\\frac{|AG|}{|GA_1|}=3\\). Line \\( SG \\) intersects the plane of \\( \\triangle ABC \\) at point \\( M \\). Determine \\... | math | {
"style": "rule",
"ground_truth": "1/3"
} | {
"split": "train",
"index": 29324,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, $a_6$, $a_7$ are distinct positive integers whose sum equals 159, find the maximum value of the smallest number $a_1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 29325,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence $\\{a_n\\}$ satisfies $a_n+a_{n+1}=n^2+(-1)^n$. Find the value of $a_{101}-a_1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5150"
} | {
"split": "train",
"index": 29326,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\overrightarrow {e_{1}}$ and $\\overrightarrow {e_{2}}$ are unit vectors with an angle of $\\frac {2π}{3}$ between them, and $\\overrightarrow {a}$ = 3 $\\overrightarrow {e_{1}}$ + 2 $\\overrightarrow {e_{2}}$, $\\overrightarrow {b}$ = 3 $\\overrightarrow {e_{2}}$,... | math | {
"style": "rule",
"ground_truth": "\\frac {1}{2}"
} | {
"split": "train",
"index": 29327,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $f(x)$ is a function, then we define the function $f^{(n)}(x)$ to be the result of $n$ applications of $f$ to $x$, where $n$ is a positive integer. For example, $f^{(3)}(x)=f(f(f(x)))$.\n\nWe define the $\\textit{order}$ of an input $x$ with respect to $f$ to be the smallest... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 29328,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alice wants to compare the percentage increase in area when her pizza size increases first from an 8-inch pizza to a 10-inch pizza, and then from the 10-inch pizza to a 14-inch pizza. Calculate the percent increase in area for both size changes. Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "96\\%"
} | {
"split": "train",
"index": 29329,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Harold, Tanya, and Ulysses paint a very long picket fence.\nHarold starts with the first picket and paints every $h$ th picket;\nTanya starts with the second picket and paints every $t$ th picket; and\nUlysses starts with the third picket and paints every $u$ th picket.\nCall t... | math | {
"style": "rule",
"ground_truth": "757"
} | {
"split": "train",
"index": 29330,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system xOy, curve $C_1: x^2+y^2=1$ is given. Taking the origin O of the Cartesian coordinate system xOy as the pole and the positive half-axis of x as the polar axis, a polar coordinate system is established with the same unit length. It is known tha... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{5}"
} | {
"split": "train",
"index": 29331,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Fifty ping-pong-balls are numbered 1, 2, 3, ..., 49, 50. A ball is chosen at random. What is the probability that the number on the selected ball is a multiple of 5 or 7 or both? Express your answer as a common fraction. Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{25}"
} | {
"split": "train",
"index": 29332,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Complex numbers are often used when dealing with alternating current (AC) circuits. In the equation $V = IZ$, $V$ is voltage, $I$ is current, and $Z$ is a value known as impedance. If $V = 1-i$ and $Z=1+3i$, find $I$. Express your answer as a complex number in the form $a+bi$,... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{5} - \\frac{2}{5}i"
} | {
"split": "train",
"index": 29333,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In rectangle $PQRS$, $PQ = 150$. Let $T$ be the midpoint of $\\overline{PS}$. Given that line $PT$ and line $QT$ are perpendicular, find the greatest integer less than $PS$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "212"
} | {
"split": "train",
"index": 29334,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $f(x)=3x^{2}+2x+1$, if $\\int_{-1}^{1}f(x)\\,dx=2f(a)$, then $a=$ ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 29335,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all values of $a$ for which the points $(0,0,0),$ $(1,a,0),$ $(0,1,a),$ and $(a,0,1)$ are coplanar. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 29336,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the graph of \\( y = g(x) \\), with \\( 1 \\) unit between grid lines, where \\( g(x) = \\frac{(x-4)(x-2)(x)(x+2)(x+4)(x+6)}{720} - 2.5 \\), defined only on the shown domain. \n\nDetermine the sum of all integers \\( c \\) for which the equation \\( g(x) = c \\) has ex... | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"split": "train",
"index": 29337,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, pentagon \\( PQRST \\) has \\( PQ = 13 \\), \\( QR = 18 \\), \\( ST = 30 \\), and a perimeter of 82. Also, \\( \\angle QRS = \\angle RST = \\angle STP = 90^\\circ \\). The area of the pentagon \\( PQRST \\) is: Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "270"
} | {
"split": "train",
"index": 29338,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two mathematicians, Kelly and Jason, play a cooperative game. The computer selects some secret positive integer $n<60$ (both Kelly and Jason know that $n<60$, but that they don't know what the value of $n$ is). The computer tells Kelly the unit digit of $n$, and it tells Jason ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 29339,
"solution": "The only way in which Kelly can know that $n$ is divisible by at least two different primes is if she is given 0 as the unit digit of $n$, since if she received anything else, then there is some number with that unit digit and not divisible by two primes (i.e., $1,... |
deepscaler | [
{
"role": "user",
"content": "Hiram's algebra notes are $50$ pages long and are printed on $25$ sheets of paper; the first sheet contains pages $1$ and $2$, the second sheet contains pages $3$ and $4$, and so on. One day he leaves his notes on the table before leaving for lunch, and his roommate decides to ... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 29340,
"solution": "1. **Define Variables:**\n Let $c$ be the number of consecutive sheets Hiram’s roommate borrows, and let $b$ be the number of sheets preceding the $c$ borrowed sheets. For example, if the roommate borrows sheets $3$, $4$, and $5$, then $c=3$ and $b=2$.\n\n2. **Ca... |
deepscaler | [
{
"role": "user",
"content": "Piravena must make a trip from $A$ to $B$, then from $B$ to $C$, then from $C$ to $A$. Each of these three parts of the trip is made entirely by bus or entirely by airplane. The cities form a right-angled triangle as shown, with $C$ a distance of 3000 km from $A$ and with $B$... | math | {
"style": "rule",
"ground_truth": "7500"
} | {
"split": "train",
"index": 29341,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a labor and technical competition among five students: A, B, C, D, and E, the rankings from first to fifth place were determined. When A and B asked about their results, the respondent told A, \"Unfortunately, both you and B did not win the championship\"; and told B, \"You ... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 29342,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two noncongruent integer-sided isosceles triangles have the same perimeter and the same area. The ratio of the lengths of the bases of the two triangles is $8: 7$. Find the minimum possible value of their common perimeter. Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "676"
} | {
"split": "train",
"index": 29343,
"solution": "Let the first triangle have sides $16n,a,a$, so the second has sides $14n,a+n,a+n$. The height of the first triangle is $\\frac{7}{8}$ the height of the second triangle. Therefore, we have \\[a^2-64n^2=\\frac{49}{64}((a+n)^2-49n^2).\\] Multiplying this, we get \\[6... |
deepscaler | [
{
"role": "user",
"content": "Given a circle $C: (x-3)^{2}+(y-4)^{2}=1$, and points $A(-1,0)$, $B(1,0)$, let $P$ be a moving point on the circle, then the maximum and minimum values of $d=|PA|^{2}+|PB|^{2}$ are \\_\\_\\_\\_\\_\\_ and \\_\\_\\_\\_\\_\\_ respectively. Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 29344,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the total surface area of a cube if the distance between the non-intersecting diagonals of two adjacent faces of this cube is 8. If the answer is not an integer, round it to the nearest whole number. Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "1152"
} | {
"split": "train",
"index": 29345,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all positive integers \\( n \\) such that \\( n \\) is equal to 100 times the number of positive divisors of \\( n \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2000"
} | {
"split": "train",
"index": 29346,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $-a-b^3+ab$ if $a=-3$ and $b=2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-11"
} | {
"split": "train",
"index": 29347,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Harry, Ron and Neville are having a race on their broomsticks. If there are no ties, in how many different possible orders can they finish? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 29348,
"solution": ""
} |
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