data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Express the sum of fractions $\\frac{3}{8} + \\frac{5}{32}$ as a decimal. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.53125",
"style": "rule"
} | {
"index": "5800",
"split": "train"
} |
math | [
{
"content": "Consider a square flag with a red cross of uniform width and a blue triangular central region on a white background. The cross is symmetric with respect to each of the diagonals of the square. Let's say the entire cross, including the blue triangle, occupies 45% of the area of the flag. Calculate ... | {
"ground_truth": "1.08\\%",
"style": "rule"
} | {
"index": "5801",
"split": "train"
} |
math | [
{
"content": "What is the smallest positive integer $n$ for which $n^2$ is divisible by 18 and $n^3$ is divisible by 640? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "120",
"style": "rule"
} | {
"index": "5802",
"split": "train"
} |
math | [
{
"content": "The greatest common divisor of 15 and some number between 75 and 90 is 5. What is the number? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "85",
"style": "rule"
} | {
"index": "5803",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate system $xOy$, it is known that the distance from any point on curve $C$ to point $M(0, \\frac{1}{2})$ is equal to its distance to the line $y = -\\frac{1}{2}$.\n\n(Ⅰ) Find the equation of curve $C$; \n(Ⅱ) Let $A_1(x_1, 0)$ and $A_2(x_2, 0)$ be two points on the x-axis ... | {
"ground_truth": "\\frac{6}{7}",
"style": "rule"
} | {
"index": "5804",
"split": "train"
} |
math | [
{
"content": "How many integers $n \\geq 2$ are there such that whenever $z_1, z_2, \\dots, z_n$ are complex numbers such that\n\\[|z_1| = |z_2| = \\dots = |z_n| = 1 \\text{ and } z_1 + z_2 + \\dots + z_n = 0,\\]\nthen the numbers $z_1, z_2, \\dots, z_n$ are equally spaced on the unit circle in the comple... | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "5805",
"split": "train"
} |
math | [
{
"content": "Johnny has 7 different colored marbles in his bag. In how many ways can he choose three different marbles from his bag to play a game? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "35",
"style": "rule"
} | {
"index": "5806",
"split": "train"
} |
math | [
{
"content": "Triangle $ABC$ has vertices $A(0, 10)$, $B(3, 0)$, $C(9, 0)$. A horizontal line with equation $y=s$ intersects line segment $\\overline{AB}$ at $P$ and line segment $\\overline{AC}$ at $Q$, forming $\\triangle APQ$ with area 18. Compute $s$. Please output the final answer within \\boxed{}.",
"... | {
"ground_truth": "10 - 2\\sqrt{15}",
"style": "rule"
} | {
"index": "5807",
"split": "train"
} |
math | [
{
"content": "The graph shows the number of minutes studied by both Asha (black bar) and Sasha (grey bar) in one week. On the average, how many more minutes per day did Sasha study than Asha? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "5808",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, we have $\\angle A = 90^\\circ$, $BC = 20$, and $\\tan C = 3\\cos B$. What is $AB$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{40\\sqrt{2}}{3}",
"style": "rule"
} | {
"index": "5809",
"split": "train"
} |
math | [
{
"content": "A $10\\times10\\times10$ grid of points consists of all points in space of the form $(i,j,k)$, where $i$, $j$, and $k$ are integers between $1$ and $10$, inclusive. Find the number of different lines that contain exactly $8$ of these points. Please output the final answer within \\boxed{}.",
"... | {
"ground_truth": "168",
"style": "rule"
} | {
"index": "5810",
"split": "train"
} |
math | [
{
"content": "What is the sum of all integer solutions to the inequality $|n| < |n-3| < 10$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-20",
"style": "rule"
} | {
"index": "5811",
"split": "train"
} |
math | [
{
"content": "A point is randomly thrown onto the interval \\( [11; 18] \\) and let \\( k \\) be the resulting value. Find the probability that the roots of the equation \\((k^2 + 2k - 99)x^2 + (3k - 7)x + 2 = 0\\) satisfy the condition \\( x_1 \\leq 2x_2 \\). Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "5812",
"split": "train"
} |
math | [
{
"content": "Let $P$ and $Q$ be points on the circle $(x-0)^2+(y-6)^2=2$ and the ellipse $\\frac{x^2}{10}+y^2=1$, respectively. What is the maximum distance between $P$ and $Q$?\nA) $5\\sqrt{2}$\nB) $\\sqrt{46}+\\sqrt{2}$\nC) $7+\\sqrt{2}$\nD) $6\\sqrt{2}$ Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "6\\sqrt{2}",
"style": "rule"
} | {
"index": "5813",
"split": "train"
} |
math | [
{
"content": "In $\\triangle PQR$, points $X$ and $Y$ lie on $\\overline{QR}$ and $\\overline{PR}$, respectively. If $\\overline{PX}$ and $\\overline{QY}$ intersect at $Z$ such that $PZ/ZX = 2$ and $QZ/ZY = 5$, what is $RX/RY$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{4}",
"style": "rule"
} | {
"index": "5814",
"split": "train"
} |
math | [
{
"content": "Find $n$ if $n\\cdot n! + n! = 720$, where $n! = n\\cdot (n-1)\\cdot (n-2)\\cdots 2\\cdot 1$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "5815",
"split": "train"
} |
math | [
{
"content": "Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively in $\\triangle ABC$, and $(\\sin A + \\sin B)(a-b) = (\\sin C - \\sin B)c$.\n\n1. Find the measure of angle $A$.\n2. If $a=4$, find the maximum area of $\\triangle ABC$. Please output the final answer with... | {
"ground_truth": "4\\sqrt{3}",
"style": "rule"
} | {
"index": "5816",
"split": "train"
} |
math | [
{
"content": "Parallelogram $ABCD$ has $AB=CD=6$ and $BC=AD=10$ , where $\\angle ABC$ is obtuse. The circumcircle of $\\triangle ABD$ intersects $BC$ at $E$ such that $CE=4$ . Compute $BD$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4\\sqrt{6}",
"style": "rule"
} | {
"index": "5817",
"split": "train"
} |
math | [
{
"content": "The decimal $0.76$ is equal to the fraction $\\frac{4b+19}{6b+11}$, where $b$ is a positive integer. What is the value of $b$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "5818",
"split": "train"
} |
math | [
{
"content": "We label the sides of a dodecagon as \\(C_{1}, C_{2}, \\ldots, C_{12}\\). In how many ways can we color the sides of a dodecagon with four colors such that two adjacent sides are always colored differently? (Two colorings are considered different if any one side \\(C_{i}\\) is colored differently ... | {
"ground_truth": "531444",
"style": "rule"
} | {
"index": "5819",
"split": "train"
} |
math | [
{
"content": "The arithmetic mean of the nine numbers in the set $\\{9, 99, 999, 9999, \\ldots, 999999999\\}$ is a $9$-digit number $M$, all of whose digits are distinct. The number $M$ doesn't contain the digit Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule"
} | {
"index": "5820",
"split": "train"
} |
math | [
{
"content": "In the country of Anchuria, a day can either be sunny, with sunshine all day, or rainy, with rain all day. If today's weather is different from yesterday's, the Anchurians say that the weather has changed. Scientists have established that January 1st is always sunny, and each subsequent day in Jan... | {
"ground_truth": "2047",
"style": "rule"
} | {
"index": "5821",
"split": "train"
} |
math | [
{
"content": "How many ordered triples (x,y,z) of integers satisfy the system of equations below?\n\\begin{array}{l} x^2-3xy+2y^2-z^2=31 \\ -x^2+6yz+2z^2=44 \\ x^2+xy+8z^2=100\\ \\end{array} Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule"
} | {
"index": "5822",
"split": "train"
} |
math | [
{
"content": "A bus ticket is numbered with six digits: from 000000 to 999999. You buy one ticket. What is the probability that you will get a ticket whose digits are in ascending (or descending) order? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.00042",
"style": "rule"
} | {
"index": "5823",
"split": "train"
} |
math | [
{
"content": "Buses. At the stop near Absent-Minded Scientist's house, buses of two routes stop: No. 152 and No. 251. Both go to the metro station. The interval between buses No. 152 is exactly 5 minutes, and the interval between buses No. 251 is exactly 7 minutes. The intervals are strictly observed, but these... | {
"ground_truth": "5/14",
"style": "rule"
} | {
"index": "5824",
"split": "train"
} |
math | [
{
"content": "Bob chooses a $4$ -digit binary string uniformly at random, and examines an infinite sequence of uniformly and independently random binary bits. If $N$ is the least number of bits Bob has to examine in order to find his chosen string, then find the expected value of $N$ . For example, if Bob’s... | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "5825",
"split": "train"
} |
math | [
{
"content": "Given an integer $k\\geq 2$, determine all functions $f$ from the positive integers into themselves such that $f(x_1)!+f(x_2)!+\\cdots f(x_k)!$ is divisibe by $x_1!+x_2!+\\cdots x_k!$ for all positive integers $x_1,x_2,\\cdots x_k$.\n\n$Albania$ Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "f(n) = n",
"style": "rule"
} | {
"index": "5826",
"split": "train"
} |
math | [
{
"content": "The minimum element of the set of positive integers $A_k$ is 1, the maximum element is 2007, and all elements can be arranged in ascending order to form an arithmetic sequence with a common difference of $k$. Then, the number of elements in the union $A_{17} \\cup A_{59}$ is ____. Please output th... | {
"ground_truth": "151",
"style": "rule"
} | {
"index": "5827",
"split": "train"
} |
math | [
{
"content": "The $GEB$ sequence $1,3,7,12, \\ldots$ is defined by the following properties: (i) the GEB sequence is increasing (that is, each term is larger than the previous term), (ii) the sequence formed using the differences between each pair of consecutive terms in the GEB sequence (namely, the sequence $... | {
"ground_truth": "5764",
"style": "rule"
} | {
"index": "5828",
"split": "train"
} |
math | [
{
"content": "A bag contains 4 red, 3 blue, and 6 yellow marbles. What is the probability, expressed as a decimal, of drawing one red marble followed by one blue marble without replacement? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.076923",
"style": "rule"
} | {
"index": "5829",
"split": "train"
} |
math | [
{
"content": "A freight train travels 1 mile in 1 minute 30 seconds. At this rate, how many miles will the train travel in 1 hour? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "5830",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)= \\frac {x}{1+x}$, then $f(1)+f(2)+f(3)+\\ldots+f(2017)+f( \\frac {1}{2})+f( \\frac {1}{3})+\\ldots+f( \\frac {1}{2017})=$ \\_\\_\\_\\_\\_\\_ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac {4033}{2}",
"style": "rule"
} | {
"index": "5831",
"split": "train"
} |
math | [
{
"content": "In the figure, if $A E=3, C E=1, B D=C D=2$, and $A B=5$, find $A G$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3\\sqrt{66} / 7",
"style": "rule"
} | {
"index": "5832",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x) = \\frac {a^{x}}{a^{x}+1}$ ($a>0$ and $a \\neq 1$).\n- (I) Find the range of $f(x)$.\n- (II) If the maximum value of $f(x)$ on the interval $[-1, 2]$ is $\\frac {3}{4}$, find the value of $a$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac {1}{3}",
"style": "rule"
} | {
"index": "5833",
"split": "train"
} |
math | [
{
"content": "The necessary and sufficient condition for the lines $ax+2y+1=0$ and $3x+(a-1)y+1=0$ to be parallel is \"$a=$ ______\". Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "5834",
"split": "train"
} |
math | [
{
"content": "In triangle $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, satisfying $\\left(a+b+c\\right)\\left(a+b-c\\right)=ab$. \n$(1)$ Find angle $C$; \n$(2)$ If the angle bisector of angle $C$ intersects $AB$ at point $D$ and $CD=2$, find the minimum ... | {
"ground_truth": "6+4\\sqrt{2}",
"style": "rule"
} | {
"index": "5835",
"split": "train"
} |
math | [
{
"content": "Six students are arranged into two rows, with 3 students per row, and the number of different arrangements must be calculated. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "720",
"style": "rule"
} | {
"index": "5836",
"split": "train"
} |
math | [
{
"content": "Given two arithmetic sequences $\\{a\\_n\\}$ and $\\{b\\_n\\}$, let $S\\_n$ and $T\\_n$ denote the sum of the first $n$ terms of each sequence, respectively. If $\\frac{S\\_n}{T\\_n} = \\frac{7n+2}{n+3}$, determine the value of $\\frac{a\\_5}{b\\_5}$. Please output the final answer within \\boxed{... | {
"ground_truth": "\\frac{65}{12}",
"style": "rule"
} | {
"index": "5837",
"split": "train"
} |
math | [
{
"content": "Given a cube of unit side. Let $A$ and $B$ be two opposite vertex. Determine the radius of the sphere, with center inside the cube, tangent to the three faces of the cube with common point $A$ and tangent to the three sides with common point $B$ . Please output the final answer within \\box... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "5838",
"split": "train"
} |
math | [
{
"content": "Given a circle $C: (x-3)^{2}+y^{2}=25$ and a line $l: (m+1)x+(m-1)y-2=0$ (where $m$ is a parameter), the minimum length of the chord intercepted by the circle $C$ and the line $l$ is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4\\sqrt{5}",
"style": "rule"
} | {
"index": "5839",
"split": "train"
} |
math | [
{
"content": "In the three-dimensional Cartesian coordinate system, given points $A(2,a,-1)$, $B(-2,3,b)$, $C(1,2,-2)$.<br/>$(1)$ If points $A$, $B$, and $C$ are collinear, find the values of $a$ and $b$;<br/>$(2)$ Given $b=-3$, $D(-1,3,-3)$, and points $A$, $B$, $C$, and $D$ are coplanar, find the value of $a$... | {
"ground_truth": "a=1",
"style": "rule"
} | {
"index": "5840",
"split": "train"
} |
math | [
{
"content": "Two cards are chosen at random from a standard 52-card deck. What is the probability that both cards are numbers (2 through 10) and total to 15? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{8}{221}",
"style": "rule"
} | {
"index": "5841",
"split": "train"
} |
math | [
{
"content": "Given the complex numbers \\( z_{1} \\) and \\( z_{2} \\) such that \\( \\left| z_{2} \\right| = 4 \\) and \\( 4z_{1}^{2} - 2z_{1}z_{2} + z_{2}^{2} = 0 \\), find the maximum value of \\( \\left| \\left( z_{1} + 1 \\right)^{2} \\left( z_{1} - 2 \\right) \\right| \\). Please output the final answer ... | {
"ground_truth": "6\\sqrt{6}",
"style": "rule"
} | {
"index": "5842",
"split": "train"
} |
math | [
{
"content": "Three different numbers are chosen from the set $\\{-4, -3, -1, 3, 5, 8\\}$ and multiplied. Find the largest possible positive product. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "120",
"style": "rule"
} | {
"index": "5843",
"split": "train"
} |
math | [
{
"content": "Let \\(ABCD\\) be a trapezoid such that \\(AB \\parallel CD, \\angle BAC=25^{\\circ}, \\angle ABC=125^{\\circ}\\), and \\(AB+AD=CD\\). Compute \\(\\angle ADC\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "70^{\\circ}",
"style": "rule"
} | {
"index": "5844",
"split": "train"
} |
math | [
{
"content": "David drives from his home to the airport to catch a flight. He drives $35$ miles in the first hour, but realizes that he will be $1$ hour late if he continues at this speed. He increases his speed by $15$ miles per hour for the rest of the way to the airport and arrives $30$ minutes early. How ma... | {
"ground_truth": "210",
"style": "rule"
} | {
"index": "5845",
"split": "train"
} |
math | [
{
"content": "Write the four numbers \\(1, 9, 8, 8\\) in a row and perform the following operation: for each pair of adjacent numbers, subtract the left number from the right number, then write the difference between these two numbers in between them. This completes one operation. Then, perform the same operati... | {
"ground_truth": "726",
"style": "rule"
} | {
"index": "5846",
"split": "train"
} |
math | [
{
"content": "In a 6 x 4 grid (6 rows, 4 columns), 12 of the 24 squares are to be shaded so that there are two shaded squares in each row and three shaded squares in each column. Let $N$ be the number of shadings with this property. Find the remainder when $N$ is divided by 1000. Please output the final answer ... | {
"ground_truth": "860",
"style": "rule"
} | {
"index": "5847",
"split": "train"
} |
math | [
{
"content": "Given that line $l$ intersects the hyperbola $\\frac{x^2}{9} - \\frac{y^2}{4} = 1$ at points $A$ and $B$, and the midpoint of segment $AB$ has coordinates $(6, 2)$, determine the slope of line $l$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{4}{3}",
"style": "rule"
} | {
"index": "5848",
"split": "train"
} |
math | [
{
"content": "Let $a$ and $b$ be distinct real numbers for which\n\\[\\frac{a}{b} + \\frac{a+10b}{b+10a} = 2.\\]Find $\\frac{a}{b}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.8",
"style": "rule"
} | {
"index": "5849",
"split": "train"
} |
math | [
{
"content": "Given that $$\\sqrt {0.1587}$$≈0.3984, $$\\sqrt {1.587}$$≈1.260, $$\\sqrt[3]{0.1587}$$≈0.5414, $$\\sqrt[3]{1.587}$$≈1.166, can you find without using a calculator:\n(1) $$\\sqrt {15.87}$$≈\\_\\_\\_\\_\\_\\_;\n(2) - $$\\sqrt {0.001587}$$≈\\_\\_\\_\\_\\_\\_;\n(3) $$\\sqrt[3]{1.587\\times 10^{-4}}$$≈... | {
"ground_truth": "0.05414",
"style": "rule"
} | {
"index": "5850",
"split": "train"
} |
math | [
{
"content": "Express $\\frac{\\sin 10+\\sin 20+\\sin 30+\\sin 40+\\sin 50+\\sin 60+\\sin 70+\\sin 80}{\\cos 5 \\cos 10 \\cos 20}$ without using trigonometric functions. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4 \\sqrt{2}",
"style": "rule"
} | {
"index": "5851",
"split": "train"
} |
math | [
{
"content": "What is left when $\\frac{1}{8}$ is subtracted from $\\frac{3}{4}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{8}",
"style": "rule"
} | {
"index": "5852",
"split": "train"
} |
math | [
{
"content": "Find the smallest positive integer $n$ such that there exists a complex number $z$, with positive real and imaginary part, satisfying $z^{n}=(\\bar{z})^{n}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "5853",
"split": "train"
} |
math | [
{
"content": "Calculate the volume in cubic centimeters of a truncated cone formed by cutting a smaller cone from a larger cone. The larger cone has a diameter of 8 cm at the base and a height of 10 cm. The smaller cone, which is cut from the top, has a diameter of 4 cm and a height of 4 cm. Express your answer... | {
"ground_truth": "48\\pi",
"style": "rule"
} | {
"index": "5854",
"split": "train"
} |
math | [
{
"content": "A square carpet of side length 9 feet is designed with one large shaded square and eight smaller, congruent shaded squares, as shown. [asy]\n\ndraw((0,0)--(9,0)--(9,9)--(0,9)--(0,0));\n\nfill((1,1)--(2,1)--(2,2)--(1,2)--cycle,gray(.8));\n\nfill((4,1)--(5,1)--(5,2)--(4,2)--cycle,gray(.8));\n\nfill(... | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "5855",
"split": "train"
} |
math | [
{
"content": "Let \\((x_1, y_1), (x_2, y_2),\\) and \\((x_3, y_3)\\) be three different real solutions to the system of equations:\n\\[\n\\begin{cases}\nx^3 - 5xy^2 = 21 \\\\\ny^3 - 5x^2y = 28\n\\end{cases}\n\\]\nFind the value of \\(\\left(11 - \\frac{x_1}{y_1}\\right)\\left(11 - \\frac{x_2}{y_2}\\right)\\left... | {
"ground_truth": "1729",
"style": "rule"
} | {
"index": "5856",
"split": "train"
} |
math | [
{
"content": "What is the $x$-coordinate of the point on the $x$-axis that is equidistant from points $A( -2, 0)$ and $B(0,4)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "5857",
"split": "train"
} |
math | [
{
"content": "Given $\\cos \\left(\\alpha- \\frac{\\pi}{6}\\right) + \\sin \\alpha = \\frac{4}{5} \\sqrt{3}$, find the value of $\\sin \\left(\\alpha+ \\frac{7\\pi}{6}\\right)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{4}{5}",
"style": "rule"
} | {
"index": "5858",
"split": "train"
} |
math | [
{
"content": "Samantha lives 2 blocks west and 1 block south of the southwest corner of City Park. Her school is 2 blocks east and 2 blocks north of the northeast corner of City Park. On school days she bikes on streets to the southwest corner of City Park, then takes a diagonal path through the park to the nor... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "5859",
"split": "train"
} |
math | [
{
"content": "\\begin{align*}\n4a + 2b + 5c + 8d &= 67 \\\\\n4(d+c) &= b \\\\\n2b + 3c &= a \\\\\nc + 1 &= d \\\\\n\\end{align*}\nGiven the above system of equations, find \\(a \\cdot b \\cdot c \\cdot d\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1201 \\times 572 \\times 19 \\times 124}{105^4}",
"style": "rule"
} | {
"index": "5860",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=2\\sqrt{2}\\cos x\\sin\\left(x+\\frac{\\pi}{4}\\right)-1$. \n(I) Find the value of $f\\left(\\frac{\\pi}{4}\\right)$; \n(II) Find the maximum and minimum values of $f(x)$ in the interval $\\left[0, \\frac{\\pi}{2}\\right]$. Please output the final answer within \\boxed{}... | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "5861",
"split": "train"
} |
math | [
{
"content": "The increasing sequence $3, 15, 24, 48, \\ldots$ consists of those positive multiples of 3 that are one less than a perfect square. What is the remainder when the 1994th term of the sequence is divided by 1000? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "935",
"style": "rule"
} | {
"index": "5862",
"split": "train"
} |
math | [
{
"content": "Miki's father is saving money in a piggy bank for the family's vacation, adding to it once a week. Miki counts and notes how much money has accumulated every week and looks for patterns in the growth. Let $P_{n}$ denote the amount in the piggy bank in the $n$-th week (in forints). Here are a few o... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "5863",
"split": "train"
} |
math | [
{
"content": "The quadratic function \\( f(x) = x^2 + mx + n \\) has real roots. The inequality \\( s \\leq (m-1)^2 + (n-1)^2 + (m-n)^2 \\) holds for any quadratic function satisfying the above conditions. What is the maximum value of \\( s \\)? Please output the final answer within \\boxed{}.",
"role": "us... | {
"ground_truth": "9/8",
"style": "rule"
} | {
"index": "5864",
"split": "train"
} |
math | [
{
"content": "Express the sum as a common fraction: $.2 + .04 + .006 + .0008 + .00001.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{24681}{100000}",
"style": "rule"
} | {
"index": "5865",
"split": "train"
} |
math | [
{
"content": "What is the smallest four-digit number that is divisible by $35$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1200",
"style": "rule"
} | {
"index": "5866",
"split": "train"
} |
math | [
{
"content": "Given a sequence $\\{a_n\\}$ whose general term formula is $a_n = -n^2 + 12n - 32$, and the sum of the first $n$ terms is $S_n$, then for any $n > m$ (where $m, n \\in \\mathbb{N}^*$), the maximum value of $S_n - S_m$ is __. Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "5867",
"split": "train"
} |
math | [
{
"content": "In triangle $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, $c$, with $b=6$, $c=10$, and $\\cos C=-\\frac{2}{3}$. \n$(1)$ Find $\\cos B$; \n$(2)$ Find the height on side $AB$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{20 - 4\\sqrt{5}}{5}",
"style": "rule"
} | {
"index": "5868",
"split": "train"
} |
math | [
{
"content": "Allison, Brian and Noah each have a 6-sided cube. All of the faces on Allison's cube have a 5. The faces on Brian's cube are numbered 1, 2, 3, 4, 5 and 6. Three of the faces on Noah's cube have a 2 and three of the faces have a 6. All three cubes are rolled. What is the probability that Allison's ... | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "5869",
"split": "train"
} |
math | [
{
"content": "Find the maximum number of real roots to a polynomial of the form\n\\[x^n + x^{n - 1} + \\dots + x + 1 = 0,\\]where $n$ is a positive integer. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "5870",
"split": "train"
} |
math | [
{
"content": "The length of the shortest trip from $A$ to $B$ along the edges of the cube shown is the length of 3 edges. How many different 3-edge trips are there from $A$ to $B$?\n\n[asy]\n\nsize(4cm,4cm);\n\npair a1, b1, c1, d1;\n\na1=(1,1);\nb1=(0,1);\nc1=(1.6,1.4);\nd1=(1,0);\n\npair e1, f1, g1, h1;\n\ne1=... | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "5871",
"split": "train"
} |
math | [
{
"content": "Let $S_n$ be the sum of the first $n$ terms of the sequence $\\{a_n\\}$. Given that $3S_n=a_{n+1}-2$, and $a_2=1$, find $a_6$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "256",
"style": "rule"
} | {
"index": "5872",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x) = |x^2 + bx|$ ($b \\in \\mathbb{R}$), when $x \\in [0, 1]$, find the minimum value of the maximum value of $f(x)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3-2\\sqrt{2}",
"style": "rule"
} | {
"index": "5873",
"split": "train"
} |
math | [
{
"content": "Points $P$, $Q$, $R$, and $S$ are in space such that each of $\\overline{SP}$, $\\overline{SQ}$, and $\\overline{SR}$ is perpendicular to the other two. If $SP = SQ = 12$ and $SR = 7$, then what is the volume of pyramid $SPQR$? Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "168",
"style": "rule"
} | {
"index": "5874",
"split": "train"
} |
math | [
{
"content": "Given the arithmetic sequence $\\{a_n\\}$, find the maximum number of different arithmetic sequences that can be formed by choosing any 3 distinct numbers from the first 20 terms. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "180",
"style": "rule"
} | {
"index": "5875",
"split": "train"
} |
math | [
{
"content": "An airline company is planning to introduce a network of connections between the ten different airports of Sawubonia. The airports are ranked by priority from first to last (with no ties). We call such a network *feasible* if it satisfies the following conditions:\n\n\n\n- All connections operate ... | {
"ground_truth": "512",
"style": "rule"
} | {
"index": "5876",
"split": "train"
} |
math | [
{
"content": "Calculate the roundness of 1,728,000. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "5877",
"split": "train"
} |
math | [
{
"content": "A new window design consists of a rectangle topped with a semi-circle at both ends. The ratio of the length AD of the rectangle to its width AB is 4:3. If AB is 36 inches, calculate the ratio of the area of the rectangle to the combined area of the semicircles. Please output the final answer withi... | {
"ground_truth": "\\frac{16}{3\\pi}",
"style": "rule"
} | {
"index": "5878",
"split": "train"
} |
math | [
{
"content": "For $ n \\in \\mathbb{N}$ , let $ f(n)\\equal{}1^n\\plus{}2^{n\\minus{}1}\\plus{}3^{n\\minus{}2}\\plus{}...\\plus{}n^1$ . Determine the minimum value of: $ \\frac{f(n\\plus{}1)}{f(n)}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8/3",
"style": "rule"
} | {
"index": "5879",
"split": "train"
} |
math | [
{
"content": "Calculate and simplify $\\sqrt[4]{3^5 \\cdot 5^3}$ to the form $c\\sqrt[4]{d}$, where $c$ and $d$ are positive integers. What is $c+d$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "378",
"style": "rule"
} | {
"index": "5880",
"split": "train"
} |
math | [
{
"content": "Two diagonals of a regular decagon (a 10-sided polygon) are chosen. What is the probability that their intersection lies inside the decagon and forms a convex quadrilateral? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{42}{119}",
"style": "rule"
} | {
"index": "5881",
"split": "train"
} |
math | [
{
"content": "Ivan wanted to buy nails. In one store, where 100 grams of nails cost 180 rubles, he couldn't buy the required amount because he was short 1430 rubles. Then he went to another store where 100 grams cost 120 rubles. He bought the required amount and received 490 rubles in change. How many kilograms... | {
"ground_truth": "3.2",
"style": "rule"
} | {
"index": "5882",
"split": "train"
} |
math | [
{
"content": "If\n\\[ 1 \\cdot 1995 + 2 \\cdot 1994 + 3 \\cdot 1993 + \\dots + 1994 \\cdot 2 + 1995 \\cdot 1 = 1995 \\cdot 997 \\cdot y, \\]\ncompute the integer \\( y \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "665",
"style": "rule"
} | {
"index": "5883",
"split": "train"
} |
math | [
{
"content": "Given that the terminal side of the angle $α+ \\frac {π}{6}$ passes through point P($-1$, $-2\\sqrt {2}$), find the value of $\\sinα$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1-2\\sqrt {6}}{6}",
"style": "rule"
} | {
"index": "5884",
"split": "train"
} |
math | [
{
"content": "During the manufacture of a steel cable, it was found that the cable has the same length as the curve given by the system of equations:\n\n$$\n\\left\\{\\begin{array}{l}\nx+y+z=8 \\\\\nx y+y z+x z=14\n\\end{array}\\right.\n$$\n\nFind the length of the cable. Please output the final answer within \... | {
"ground_truth": "4\\pi \\sqrt{\\frac{11}{3}}",
"style": "rule"
} | {
"index": "5885",
"split": "train"
} |
math | [
{
"content": "The only prime factors of an integer $n$ are 2 and 3. If the sum of the divisors of $n$ (including itself) is $1815$ , find $n$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "648",
"style": "rule"
} | {
"index": "5886",
"split": "train"
} |
math | [
{
"content": "Find the maximum and minimum values of the function $f(x)=x^{3}-12x$ on the interval $[-3,5]$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-16",
"style": "rule"
} | {
"index": "5887",
"split": "train"
} |
math | [
{
"content": "There are two arithmetic sequences $\\\\{a_{n}\\\\}$ and $\\\\{b_{n}\\\\}$, with respective sums of the first $n$ terms denoted by $S_{n}$ and $T_{n}$. Given that $\\dfrac{S_{n}}{T_{n}} = \\dfrac{3n}{2n+1}$, find the value of $\\dfrac{a_{1}+a_{2}+a_{14}+a_{19}}{b_{1}+b_{3}+b_{17}+b_{19}}$.\nA) $\\... | {
"ground_truth": "\\dfrac{17}{13}",
"style": "rule"
} | {
"index": "5888",
"split": "train"
} |
math | [
{
"content": "Find the number of ordered $64$-tuples $(x_0,x_1,\\dots,x_{63})$ such that $x_0,x_1,\\dots,x_{63}$ are distinct elements of $\\{1,2,\\dots,2017\\}$ and \\[ x_0 + x_1 + 2x_2 + 3x_3 + \\cdots + 63 x_{63} \\] is divisible by 2017. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "$\\frac{2016!}{1953!}- 63! \\cdot 2016$",
"style": "rule"
} | {
"index": "5889",
"split": "train"
} |
math | [
{
"content": "Find \\( k \\) such that for all real numbers \\( a, b, \\) and \\( c \\):\n\n\\[ \n(a+b)(b+c)(c+a) = (a+b+c)(ab + bc + ca) + k \\, abc \n\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "5890",
"split": "train"
} |
math | [
{
"content": "Given the parabola $C$: $y^{2}=4x$ with focus $F$, a line $l$ passing through $F$ intersects $C$ at points $A$ and $B$. Let $M$ be the midpoint of segment $AB$, and $O$ be the origin. The extensions of $AO$ and $BO$ intersect the line $x=-4$ at points $P$ and $Q$ respectively. \n(Ⅰ) Find the equa... | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "5891",
"split": "train"
} |
math | [
{
"content": "The angles of a pentagon are in arithmetic progression. One of the angles in degrees, must be: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "108",
"style": "rule"
} | {
"index": "5892",
"split": "train"
} |
math | [
{
"content": "What is $\\frac{1}{(-7^{3})^{3}}\\cdot(-7)^{10}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-7",
"style": "rule"
} | {
"index": "5893",
"split": "train"
} |
math | [
{
"content": "It is known that the complex number \\( z \\) satisfies \\( |z| = 1 \\). Find the maximum value of \\( u = \\left| z^3 - 3z + 2 \\right| \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3\\sqrt{3}",
"style": "rule"
} | {
"index": "5894",
"split": "train"
} |
math | [
{
"content": "Let \\( z \\) be a complex number such that \\( |z| = 1 \\). If the equation \\( z x^{2} + 2 \\bar{z} x + 2 = 0 \\) in terms of \\( x \\) has a real root, find the sum of all such complex numbers \\( z \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{3}{2}",
"style": "rule"
} | {
"index": "5895",
"split": "train"
} |
math | [
{
"content": "Select three digits from 1, 3, 5, 7, 9, and two digits from 0, 2, 4, 6, 8 to form a five-digit number without any repeating digits. How many such numbers can be formed? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11040",
"style": "rule"
} | {
"index": "5896",
"split": "train"
} |
math | [
{
"content": "Points $(1,-5)$ and $(11,7)$ are the opposite vertices of a parallelogram. What are the coordinates of the point where the diagonals of the parallelogram intersect? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(6,1)",
"style": "rule"
} | {
"index": "5897",
"split": "train"
} |
math | [
{
"content": "To celebrate her birthday, Ana is going to prepare pear and apple pies. In the market, an apple weighs $300 \\text{ g}$ and a pear weighs $200 \\text{ g}$. Ana's bag can hold a maximum weight of $7 \\text{ kg}$. What is the maximum number of fruits she can buy to make pies with both types of fruit... | {
"ground_truth": "34",
"style": "rule"
} | {
"index": "5898",
"split": "train"
} |
math | [
{
"content": "Petya cut an 8x8 square along the borders of the cells into parts of equal perimeter. It turned out that not all parts are equal. What is the maximum possible number of parts he could get? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "5899",
"split": "train"
} |
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