data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Anca and Bruce drove along a highway. Bruce drove at 50 km/h and Anca at 60 km/h, but stopped to rest. How long did Anca stop? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "40 \\text{ minutes}",
"style": "rule"
} | {
"index": "3500",
"split": "train"
} |
math | [
{
"content": "A line parallel to the side $AC$ of a triangle $ABC$ with $\\angle C = 90$ intersects side $AB$ at $M$ and side $BC$ at $N$ , so that $CN/BN = AC/BC = 2/1$ . The segments $CM$ and $AN$ meet at $O$ . Let $K$ be a point on the segment $ON$ such that $MO+OK = KN$ . The bisecto... | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "3501",
"split": "train"
} |
math | [
{
"content": "In how many ways can 9 people sit around a round table? (Two seatings are considered the same if one is a rotation of the other.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "40,\\!320",
"style": "rule"
} | {
"index": "3502",
"split": "train"
} |
math | [
{
"content": "Given that $| \\overrightarrow{a}|=12$, $| \\overrightarrow{b}|=9$, and $\\overrightarrow{a} \\cdot \\overrightarrow{b}=-54 \\sqrt {2}$, find the angle between $\\overrightarrow{a}$ and $\\overrightarrow{b}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3\\pi}{4}",
"style": "rule"
} | {
"index": "3503",
"split": "train"
} |
math | [
{
"content": "If the random variable $X$ follows a Bernoulli distribution with a success probability of $0.7$, and the random variable $Y$ follows a binomial distribution with $Y \\sim B(10, 0.8)$, then $EX$, $DX$, $EY$, $DY$ are respectively ........, ........, ........, ........ Please output the final answer... | {
"ground_truth": "1.6",
"style": "rule"
} | {
"index": "3504",
"split": "train"
} |
math | [
{
"content": "In the diagram below, $WXYZ$ is a trapezoid where $\\overline{WX}\\parallel \\overline{ZY}$ and $\\overline{WY}\\perp\\overline{ZY}$. Given $YZ = 15$, $\\tan Z = 2$, and $\\tan X = 2.5$, what is the length of $XY$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{261}",
"style": "rule"
} | {
"index": "3505",
"split": "train"
} |
math | [
{
"content": "Which of the following lines, when drawn together with the $x$-axis and the $y$-axis, encloses an isosceles triangle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "y=-x+4",
"style": "rule"
} | {
"index": "3506",
"split": "train"
} |
math | [
{
"content": "Given that $f(\\alpha) = \\cos\\alpha \\sqrt{\\frac{\\cot\\alpha - \\cos\\alpha}{\\cot\\alpha + \\cos\\alpha}} + \\sin\\alpha \\sqrt{\\frac{\\tan\\alpha - \\sin\\alpha}{\\tan\\alpha + \\sin\\alpha}}$, and $\\alpha$ is an angle in the second quadrant. \n(1) Simplify $f(\\alpha)$. \n(2) If $f(-\\a... | {
"ground_truth": "-\\frac{7}{12}",
"style": "rule"
} | {
"index": "3507",
"split": "train"
} |
math | [
{
"content": "A man travels $m$ feet due north at $2$ minutes per mile. He returns due south to his starting point at $2$ miles per minute. The average rate in miles per hour for the entire trip is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "48",
"style": "rule"
} | {
"index": "3508",
"split": "train"
} |
math | [
{
"content": "A $1.4 \\mathrm{~m}$ long rod has $3 \\mathrm{~kg}$ masses at both ends. Where should the rod be pivoted so that, when released from a horizontal position, the mass on the left side passes under the pivot with a speed of $1.6 \\mathrm{~m} /\\mathrm{s}$? Please output the final answer within \\boxe... | {
"ground_truth": "0.8",
"style": "rule"
} | {
"index": "3509",
"split": "train"
} |
math | [
{
"content": "Given that in triangle ABC, the lengths of the sides opposite to angles A, B, and C are a, b, and c respectively, and b = 3, c = 1, and A = 2B.\n(1) Find the value of a;\n(2) Find the value of sin(A + $\\frac{\\pi}{4}$). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{4 - \\sqrt{2}}{6}",
"style": "rule"
} | {
"index": "3510",
"split": "train"
} |
math | [
{
"content": "Three $1 \times 1 \times 1$ cubes are joined side by side. What is the surface area of the resulting prism? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "3511",
"split": "train"
} |
math | [
{
"content": "In a circuit with two independently controlled automatic switches connected in parallel, the probability that the circuit can operate normally is given that one of the switches can close, and the probabilities of the switches being able to close are $0.5$ and $0.7$ respectively, calculate this pro... | {
"ground_truth": "0.85",
"style": "rule"
} | {
"index": "3512",
"split": "train"
} |
math | [
{
"content": "Calculate the minimum of $\\sqrt{(x-1)^{2}+y^{2}}+\\sqrt{(x+1)^{2}+y^{2}}+|2-y|$ for $x, y \\in \\mathbb{R}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2 + \\sqrt{3}",
"style": "rule"
} | {
"index": "3513",
"split": "train"
} |
math | [
{
"content": "Let $T = TNFTPP$ . As $n$ ranges over the integers, the expression $n^4 - 898n^2 + T - 2160$ evaluates to just one prime number. Find this prime.\n\n[b]Note: This is part of the Ultimate Problem, where each question depended on the previous question. For those who w... | {
"ground_truth": "1801",
"style": "rule"
} | {
"index": "3514",
"split": "train"
} |
math | [
{
"content": "What is the average of all the integer values of $M$ such that $\\frac{M}{42}$ is strictly between $\\frac{5}{14}$ and $\\frac{1}{6}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "3515",
"split": "train"
} |
math | [
{
"content": "How many degrees are in each interior angle of a regular hexagon? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "120^\\circ",
"style": "rule"
} | {
"index": "3516",
"split": "train"
} |
math | [
{
"content": "In the figure below, $N$ congruent semicircles lie on the diameter of a large semicircle, with their diameters covering the diameter of the large semicircle with no overlap. Let $A$ be the combined area of the small semicircles and $B$ be the area of the region inside the large semicircle but outs... | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "3517",
"split": "train"
} |
math | [
{
"content": "Let the operation $x*y$ be defined as $x*y = (x+1)(y+1)$. The operation $x^{*2}$ is defined as $x^{*2} = x*x$. Calculate the value of the polynomial $3*(x^{*2}) - 2*x + 1$ when $x=2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "3518",
"split": "train"
} |
math | [
{
"content": "With four standard six-sided dice in play, Vivian rolls all four and can choose to reroll any subset of them. To win, Vivian needs the sum of the four dice after possibly rerolling some of them to be exactly 12. Vivian plays optimally to maximize her chances of winning. What is the probability tha... | {
"ground_truth": "\\frac{1}{8}",
"style": "rule"
} | {
"index": "3519",
"split": "train"
} |
math | [
{
"content": "A rectangular box has a total surface area of 94 square inches. The sum of the lengths of all its edges is 48 inches. What is the sum of the lengths in inches of all of its interior diagonals? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20\\sqrt{2}",
"style": "rule"
} | {
"index": "3520",
"split": "train"
} |
math | [
{
"content": "Calculate the definite integral:\n\n$$\n\\int_{0}^{2} \\frac{(4 \\sqrt{2-x}-\\sqrt{3 x+2}) d x}{(\\sqrt{3 x+2}+4 \\sqrt{2-x})(3 x+2)^{2}}\n$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{32} \\ln 5",
"style": "rule"
} | {
"index": "3521",
"split": "train"
} |
math | [
{
"content": "Calculate $[(12^{12} \\div 12^{11})^2 \\cdot 4^2] \\div 2^4$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "144",
"style": "rule"
} | {
"index": "3522",
"split": "train"
} |
math | [
{
"content": "When Carlos divides his marbles into groups of six, seven, or eight, he has one marble left over. Assuming Carlos has more than one marble, what is the smallest possible number of marbles in Carlos's collection? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "169",
"style": "rule"
} | {
"index": "3523",
"split": "train"
} |
math | [
{
"content": "Given the sequence of even counting numbers starting from $0$, find the sum of the first $1500$ terms. Then, given the sequence of odd counting numbers, find the sum of the first $1500$ terms, and calculate their difference. Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "1500",
"style": "rule"
} | {
"index": "3524",
"split": "train"
} |
math | [
{
"content": "Given that $D$ is a point on the side $AB$ of $\\triangle ABC$, and $\\overrightarrow{CD} = \\frac{1}{3}\\overrightarrow{AC} + \\lambda \\cdot \\overrightarrow{BC}$, determine the value of the real number $\\lambda$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{4}{3}",
"style": "rule"
} | {
"index": "3525",
"split": "train"
} |
math | [
{
"content": "Solve the equations: \n(1) $(x-3)^2+2x(x-3)=0$ \n(2) $x^2-4x+1=0$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2-\\sqrt{3}",
"style": "rule"
} | {
"index": "3526",
"split": "train"
} |
math | [
{
"content": "At the 2013 Winnebago County Fair a vendor is offering a \"fair special\" on sandals. If you buy one pair of sandals at the regular price of $50, you get a second pair at a 40% discount, and a third pair at half the regular price. Javier took advantage of the \"fair special\" to buy three pairs of... | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "3527",
"split": "train"
} |
math | [
{
"content": "The intercept on the x-axis, the intercept on the y-axis, and the slope of the line 4x-5y-20=0 are respectively. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{4}{5}",
"style": "rule"
} | {
"index": "3528",
"split": "train"
} |
math | [
{
"content": "The distance between the two intersections of $x=y^4$ and $x+y^2=1$ is $\\sqrt{u+v\\sqrt5}$. Find the ordered pair, $(u,v)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(-2,2)",
"style": "rule"
} | {
"index": "3529",
"split": "train"
} |
math | [
{
"content": "Assume a deck of 27 cards where each card features one of three symbols (star, circle, or square), each symbol painted in one of three colors (red, yellow, or blue), and each color applied in one of three intensities (light, medium, or dark). Each symbol-color-intensity combination is unique acros... | {
"ground_truth": "117",
"style": "rule"
} | {
"index": "3530",
"split": "train"
} |
math | [
{
"content": "How does the area of the quadrilateral formed by connecting the midpoints of the sides of a concave quadrilateral compare to the area of the original quadrilateral? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "3531",
"split": "train"
} |
math | [
{
"content": "If the system of equations \n\\[\n\\begin{align*}\n4x + y &= a, \\\\\n3x + 4y^2 &= 3a,\n\\end{align*}\n\\]\nhas a solution $(x,y)$ when $x=3$, compute $a$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9.75",
"style": "rule"
} | {
"index": "3532",
"split": "train"
} |
math | [
{
"content": "Given that $\\frac{x}{9}, \\frac{y}{15}, \\frac{z}{14}$ are all in their simplest forms and their product is $\\frac{1}{6}$, find the value of $x+y+z$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "3533",
"split": "train"
} |
math | [
{
"content": "Find the number of primes $p$ between $100$ and $200$ for which $x^{11}+y^{16}\\equiv 2013\\pmod p$ has a solution in integers $x$ and $y$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "3534",
"split": "train"
} |
math | [
{
"content": "Given one hundred numbers: \\(1, \\frac{1}{2}, \\frac{1}{3}, \\frac{1}{4}, \\ldots, \\frac{1}{100}\\). We compute 98 differences: \\(a_{1} = 1 - \\frac{1}{3}, a_{2} = \\frac{1}{2} - \\frac{1}{4}, \\ldots, a_{98} = \\frac{1}{98} - \\frac{1}{100}\\). What is the sum of all these differences? Please ... | {
"ground_truth": "\\frac{14651}{9900}",
"style": "rule"
} | {
"index": "3535",
"split": "train"
} |
math | [
{
"content": "The sequence \\( a_{0}, a_{1}, \\cdots, a_{n} \\) satisfies:\n\\[ \na_{0}=\\sqrt{3}, \\quad a_{n+1}=[a_{n}]+\\frac{1}{\\{a_{n}\\}} \n\\]\nwhere \\( [x] \\) denotes the greatest integer less than or equal to the real number \\( x \\), and \\( \\{x\\}=x-[x] \\). Find the value of \\( a_{2016} \\). P... | {
"ground_truth": "3024 + \\sqrt{3}",
"style": "rule"
} | {
"index": "3536",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it satisfies $\\sqrt{3}b\\cos A - a\\sin B = 0$.\n\n$(1)$ Find the measure of angle $A$; \n$(2)$ Given that $c=4$ and the area of $\\triangle ABC$ is $6\\sqrt{3}$, find the value... | {
"ground_truth": "2\\sqrt{7}",
"style": "rule"
} | {
"index": "3537",
"split": "train"
} |
math | [
{
"content": "What is the least positive integer with exactly $10$ positive factors? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "48",
"style": "rule"
} | {
"index": "3538",
"split": "train"
} |
math | [
{
"content": "Find the remainder when $5^{2021}$ is divided by $17$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "3539",
"split": "train"
} |
math | [
{
"content": "Little kids were eating candies. Each ate 11 candies less than the rest combined but still more than one candy. How many candies were eaten in total? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "33",
"style": "rule"
} | {
"index": "3540",
"split": "train"
} |
math | [
{
"content": "The room numbers of a hotel are all three-digit numbers. The first digit represents the floor and the last two digits represent the room number. The hotel has rooms on five floors, numbered 1 to 5. It has 35 rooms on each floor, numbered $\\mathrm{n}01$ to $\\mathrm{n}35$ where $\\mathrm{n}$ is th... | {
"ground_truth": "105",
"style": "rule"
} | {
"index": "3541",
"split": "train"
} |
math | [
{
"content": "A positive integer divisor of $10!$ is chosen at random. The probability that the divisor chosen is a perfect square can be expressed as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "3542",
"split": "train"
} |
math | [
{
"content": "If $\\sin x = 3 \\cos x,$ then what is $\\sin x \\cos x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{10}",
"style": "rule"
} | {
"index": "3543",
"split": "train"
} |
math | [
{
"content": "Given non-negative real numbers \\( a, b, c, x, y, z \\) that satisfy \\( a + b + c = x + y + z = 1 \\), find the minimum value of \\( \\left(a - x^{2}\\right)\\left(b - y^{2}\\right)\\left(c - z^{2}\\right) \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1/4",
"style": "rule"
} | {
"index": "3544",
"split": "train"
} |
math | [
{
"content": "Let $\\overline{MN}$ be a diameter of a circle with diameter 1. Let $A$ and $B$ be points on one of the semicircular arcs determined by $\\overline{MN}$ such that $A$ is the midpoint of the semicircle and $MB=\\frac{3}5$. Point $C$ lies on the other semicircular arc. Let $d$ be the length of the l... | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "3545",
"split": "train"
} |
math | [
{
"content": "Complex numbers $a,$ $b,$ $c$ form an equilateral triangle with side length 18 in the complex plane. If $|a + b + c| = 36,$ find $|ab + ac + bc|.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "432",
"style": "rule"
} | {
"index": "3546",
"split": "train"
} |
math | [
{
"content": "What is the sum of all of the positive even factors of $504$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1456",
"style": "rule"
} | {
"index": "3547",
"split": "train"
} |
math | [
{
"content": "Let $ a,b,c,d$ be rational numbers with $ a>0$ . If for every integer $ n\\ge 0$ , the number $ an^{3} \\plus{}bn^{2} \\plus{}cn\\plus{}d$ is also integer, then the minimal value of $ a$ will be Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$\\frac{1}{6}$",
"style": "rule"
} | {
"index": "3548",
"split": "train"
} |
math | [
{
"content": "Given two non-collinear vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ satisfying $|\\overrightarrow{a}|=|\\overrightarrow{b}|$, and $\\overrightarrow{a}\\perp(\\overrightarrow{a}-2\\overrightarrow{b})$, the angle between $\\overrightarrow{a}$ and $\\overrightarrow{b}$ is ______. Please o... | {
"ground_truth": "\\frac{\\pi}{3}",
"style": "rule"
} | {
"index": "3549",
"split": "train"
} |
math | [
{
"content": "Let \\(r(x)\\) have a domain of \\(\\{-2,-1,0,1\\}\\) and a range of \\(\\{-1,0,2,3\\}\\). Let \\(t(x)\\) have a domain of \\(\\{-1,0,1,2,3\\}\\) and be defined as \\(t(x) = 2x + 1\\). Furthermore, \\(s(x)\\) is defined on the domain \\(\\{1, 2, 3, 4, 5, 6\\}\\) by \\(s(x) = x + 2\\). What is the ... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "3550",
"split": "train"
} |
math | [
{
"content": "Given a geometric sequence $\\{a_{n}\\}$, the sum of the first n terms is $S_{n}$, $S_{2}=7$ and $S_{6}=91$. Calculate $S_{4}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "3551",
"split": "train"
} |
math | [
{
"content": "How many equilateral hexagons of side length $\\sqrt{13}$ have one vertex at $(0,0)$ and the other five vertices at lattice points? (A lattice point is a point whose Cartesian coordinates are both integers. A hexagon may be concave but not self-intersecting.) Please output the final answer within ... | {
"ground_truth": "216",
"style": "rule"
} | {
"index": "3552",
"split": "train"
} |
math | [
{
"content": "An 8-by-8 square is divided into 64 unit squares in the usual way. Each unit square is colored black or white. The number of black unit squares is even. We can take two adjacent unit squares (forming a 1-by-2 or 2-by-1 rectangle), and flip their colors: black becomes white and white becomes bla... | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "3553",
"split": "train"
} |
math | [
{
"content": "Given that $a, b, c$ are the sides opposite to angles $A, B, C$ respectively in $\\triangle ABC$, and $\\cos A \\sin B \\sin C + \\cos B \\sin A \\sin C = 2 \\cos C \\sin A \\sin B$, find the maximum value of $C$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\pi}{3}",
"style": "rule"
} | {
"index": "3554",
"split": "train"
} |
math | [
{
"content": "The average (arithmetic mean) age of a group consisting of doctors and lawyers in 40. If the doctors average 35 and the lawyers 50 years old, then the ratio of the numbers of doctors to the number of lawyers is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2: 1",
"style": "rule"
} | {
"index": "3555",
"split": "train"
} |
math | [
{
"content": "Compute $$\\lim _{A \\rightarrow+\\infty} \\frac{1}{A} \\int_{1}^{A} A^{\\frac{1}{x}} \\mathrm{~d} x$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "3556",
"split": "train"
} |
math | [
{
"content": "The acronym XYZ is displayed on a rectangular grid similarly spaced 1 unit apart. The acronym starts X having length 2 units at the top and the bottom and slanted sides making triangles on both ends. Y is made with a vertical line of 3 units and two slanted lines extending from the midpoint down t... | {
"ground_truth": "13 + 4\\sqrt{2} + \\sqrt{10}",
"style": "rule"
} | {
"index": "3557",
"split": "train"
} |
math | [
{
"content": "The graph of $y = f(x)$ is shown below.\n\n[asy]\nunitsize(0.5 cm);\n\nreal func(real x) {\n real y;\n if (x >= -3 && x <= 0) {y = -2 - x;}\n if (x >= 0 && x <= 2) {y = sqrt(4 - (x - 2)^2) - 2;}\n if (x >= 2 && x <= 3) {y = 2*(x - 2);}\n return(y);\n}\n\nint i, n;\n\nfor (i = -5; i <= 5; ++i)... | {
"ground_truth": "\\text{C}",
"style": "rule"
} | {
"index": "3558",
"split": "train"
} |
math | [
{
"content": "In a regular quadrilateral prism $ABCDA'A'B'C'D'$ with vertices on the same sphere, $AB = 1$ and $AA' = \\sqrt{2}$, the spherical distance between points $A$ and $C$ is ( ). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$\\frac{\\pi}{2}$",
"style": "rule"
} | {
"index": "3559",
"split": "train"
} |
math | [
{
"content": "If $cos2α=-\\frac{{\\sqrt{10}}}{{10}}$, $sin({α-β})=\\frac{{\\sqrt{5}}}{5}$, and $α∈({\\frac{π}{4},\\frac{π}{2}})$, $β∈({-π,-\\frac{π}{2}})$, then $\\alpha +\\beta =$____. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{\\pi}{4}",
"style": "rule"
} | {
"index": "3560",
"split": "train"
} |
math | [
{
"content": "Right triangle $ABC$ is inscribed in circle $W$ . $\\angle{CAB}=65$ degrees, and $\\angle{CBA}=25$ degrees. The median from $C$ to $AB$ intersects $W$ and line $D$ . Line $l_1$ is drawn tangent to $W$ at $A$ . Line $l_2$ is drawn tangent to $W$ at $D$ . The lines $l_1$ an... | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "3561",
"split": "train"
} |
math | [
{
"content": "Given a device purchased at $40 less a 15% discount, and a goal of achieving a 25% selling profit based on the cost after allowing a 10% discount on the final sales price, determine the marked price of the device in dollars. Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "47.22",
"style": "rule"
} | {
"index": "3562",
"split": "train"
} |
math | [
{
"content": "A conical glass is in the form of a right circular cone. The slant height is $21$ and the radius of the top rim of the glass is $14$ . An ant at the mid point of a slant line on the outside wall of the glass sees a honey drop diametrically opposite to it on the inside wall of the glass. If $d$... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "3563",
"split": "train"
} |
math | [
{
"content": "Knot is on an epic quest to save the land of Hyruler from the evil Gammadorf. To do this, he must collect the two pieces of the Lineforce, then go to the Temple of Lime. As shown on the figure, Knot starts on point $K$, and must travel to point $T$, where $O K=2$ and $O T=4$. However, he must firs... | {
"ground_truth": "2 \\sqrt{5}",
"style": "rule"
} | {
"index": "3564",
"split": "train"
} |
math | [
{
"content": "The table below lists the air distances in kilometers between several cities. If two different cities from the table are chosen at random, what is the probability that the distance between them is less than 9000 kilometers? Express your answer as a common fraction.\n\n\\begin{tabular}{|c|c|c|c|c|}... | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "3565",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate system $xOy$, the sum of distances from point $P$ to the centers of two circles $C_1$ and $C_2$ is equal to 4, where $C_1: x^2+y^2-2\\sqrt{3}y+2=0$, $C_2: x^2+y^2+2\\sqrt{3}y-3=0$. Let the trajectory of point $P$ be $C$.\n(1) Find the equation of $C$;\n(2) Suppose the l... | {
"ground_truth": "\\frac{4\\sqrt{65}}{17}",
"style": "rule"
} | {
"index": "3566",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, let $AB=6$, $BC=7$, $AC=4$, and $O$ be the incenter of $\\triangle ABC$. If $\\overrightarrow{AO} = p\\overrightarrow{AB} + q\\overrightarrow{AC}$, determine the value of $\\frac{p}{q}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "3567",
"split": "train"
} |
math | [
{
"content": "$x$ is a base- $10$ number such that when the digits of $x$ are interpreted as a base- $20$ number, the resulting number is twice the value as when they are interpreted as a base- $13$ number. Find the sum of all possible values of $x$ . Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "198",
"style": "rule"
} | {
"index": "3568",
"split": "train"
} |
math | [
{
"content": "Given triangle $\\triangle ABC$ with $\\cos C = \\frac{2}{3}$, $AC = 4$, and $BC = 3$, calculate the value of $\\tan B$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4\\sqrt{5}",
"style": "rule"
} | {
"index": "3569",
"split": "train"
} |
math | [
{
"content": "Determine the number of digits in the value of $2^{15} \\times 5^{11}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "3570",
"split": "train"
} |
math | [
{
"content": "If 600 were expressed as a sum of at least two distinct powers of 2, what would be the least possible sum of the exponents of these powers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "3571",
"split": "train"
} |
math | [
{
"content": "How many four-digit whole numbers are there such that the leftmost digit is an odd prime, the second digit is a multiple of 3, and all four digits are different? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "616",
"style": "rule"
} | {
"index": "3572",
"split": "train"
} |
math | [
{
"content": "An event occurs many years ago. It occurs periodically in $x$ consecutive years, then there is a break of $y$ consecutive years. We know that the event occured in $1964$ , $1986$ , $1996$ , $2008$ and it didn't occur in $1976$ , $1993$ , $2006$ , $2013$ . What is the first year in tha... | {
"ground_truth": "2018",
"style": "rule"
} | {
"index": "3573",
"split": "train"
} |
math | [
{
"content": "What is the greatest common divisor of $2^{1998}-1$ and $2^{1989}-1$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "511",
"style": "rule"
} | {
"index": "3574",
"split": "train"
} |
math | [
{
"content": "Given the circle \\(\\Gamma: x^{2} + y^{2} = 1\\) with two intersection points with the \\(x\\)-axis as \\(A\\) and \\(B\\) (from left to right), \\(P\\) is a moving point on circle \\(\\Gamma\\). Line \\(l\\) passes through point \\(P\\) and is tangent to circle \\(\\Gamma\\). A line perpendicula... | {
"ground_truth": "2\\sqrt{5} + 2",
"style": "rule"
} | {
"index": "3575",
"split": "train"
} |
math | [
{
"content": "Given $\\tan \\left(α- \\frac {π}{4}\\right)=2$, find the value of $\\sin \\left(2α- \\frac {π}{4}\\right)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac {\\sqrt {2}}{10}",
"style": "rule"
} | {
"index": "3576",
"split": "train"
} |
math | [
{
"content": "Given the sequence $a_1, a_2, \\ldots, a_n, \\ldots$, insert 3 numbers between each pair of consecutive terms to form a new sequence. Determine the term number of the 69th term of the new sequence. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "3577",
"split": "train"
} |
math | [
{
"content": "Three squares \\( GQOP, HJNO \\), and \\( RKMN \\) have vertices which sit on the sides of triangle \\( FIL \\) as shown. The squares have areas of 10, 90, and 40 respectively. What is the area of triangle \\( FIL \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "220.5",
"style": "rule"
} | {
"index": "3578",
"split": "train"
} |
math | [
{
"content": "Determine the absolute value of the difference between single-digit integers $C$ and $D$ such that:\n$$ \\begin{array}{c@{}c@{\\;}c@{}c@{}c@{}c}\n & & & D& D & C_6\\\\\n & & & \\mathbf{5} & \\mathbf{2} & D_6\\\\\n & & + & C & \\mathbf{2} & \\mathbf{4_6}\\\\\n \\cline{2-6}\n & & D & \\mathbf{2} & \... | {
"ground_truth": "1_6",
"style": "rule"
} | {
"index": "3579",
"split": "train"
} |
math | [
{
"content": "Let \\(x\\), \\(y\\), and \\(z\\) be complex numbers such that:\n\\[\nxy + 3y = -9, \\\\\nyz + 3z = -9, \\\\\nzx + 3x = -9.\n\\]\n\nFind all possible values of \\(xyz\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "3580",
"split": "train"
} |
math | [
{
"content": "A \\(3 \\times 3 \\times 3\\) cube composed of 27 unit cubes rests on a horizontal plane. Determine the number of ways of selecting two distinct unit cubes (order is irrelevant) from a \\(3 \\times 3 \\times 1\\) block with the property that the line joining the centers of the two cubes makes a \\... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "3581",
"split": "train"
} |
math | [
{
"content": "In $\\triangle PQR$, $\\angle PQR = 150^\\circ$, $PQ = 4$ and $QR = 6$. If perpendiculars are constructed to $\\overline{PQ}$ at $P$ and to $\\overline{QR}$ at $R$, and they meet at point $S$, calculate the length of $RS$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{24}{\\sqrt{52 + 24\\sqrt{3}}}",
"style": "rule"
} | {
"index": "3582",
"split": "train"
} |
math | [
{
"content": "Find the volume of the region in space defined by\n\\[|x + y + 2z| + |x + y - 2z| \\le 12\\]\nand $x, y, z \\ge 0$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "54",
"style": "rule"
} | {
"index": "3583",
"split": "train"
} |
math | [
{
"content": "Consider the statements:\n(1) p and q are both true\n(2) p is true and q is false\n(3) p is false and q is true\n(4) p is false and q is false.\nHow many of these imply the negative of the statement \"p and q are both true?\" Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "3584",
"split": "train"
} |
math | [
{
"content": "If three angles \\(x, y, z\\) form an arithmetic sequence with a common difference of \\(\\frac{\\pi}{2}\\), then \\(\\tan x \\tan y + \\tan y \\tan z + \\tan z \\tan x = \\) ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "3585",
"split": "train"
} |
math | [
{
"content": "Below is a portion of the graph of a function, $y=p(x)$:\n\n[asy]\nimport graph; size(6cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-3.5,xmax=3.5,ymin=-4.25,ymax=6.25;\n\npen cqcqcq=rgb(0.75,0.75,0.75);\n/*grid*/ pen gs=linewidth(0.7)+cqcqcq+line... | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "3586",
"split": "train"
} |
math | [
{
"content": "Given the point \\( A(0,1) \\) and the curve \\( C: y = \\log_a x \\) which always passes through point \\( B \\), if \\( P \\) is a moving point on the curve \\( C \\) and the minimum value of \\( \\overrightarrow{AB} \\cdot \\overrightarrow{AP} \\) is 2, then the real number \\( a = \\) _______.... | {
"ground_truth": "e",
"style": "rule"
} | {
"index": "3587",
"split": "train"
} |
math | [
{
"content": "22. In the polar coordinate system, the polar equation of circle $C$ is $\\rho =4\\cos \\theta$. Taking the pole as the origin and the direction of the polar axis as the positive direction of the $x$-axis, a Cartesian coordinate system is established using the same unit length. The parametric equa... | {
"ground_truth": "\\dfrac{\\sqrt{46}}{2}",
"style": "rule"
} | {
"index": "3588",
"split": "train"
} |
math | [
{
"content": "Find the fraction that equals $0.\\overline{4}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{4}{9}",
"style": "rule"
} | {
"index": "3589",
"split": "train"
} |
math | [
{
"content": "Let \\( A B C D \\) be a quadrilateral and \\( P \\) the intersection of \\( (A C) \\) and \\( (B D) \\). Assume that \\( \\widehat{C A D} = 50^\\circ \\), \\( \\widehat{B A C} = 70^\\circ \\), \\( \\widehat{D C A} = 40^\\circ \\), and \\( \\widehat{A C B} = 20^\\circ \\). Calculate the angle \\( ... | {
"ground_truth": "70",
"style": "rule"
} | {
"index": "3590",
"split": "train"
} |
math | [
{
"content": "Simplify first, then evaluate: $\\left(\\frac{{a}^{2}-1}{a-3}-a-1\\right) \\div \\frac{a+1}{{a}^{2}-6a+9}$, where $a=3-\\sqrt{2}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2\\sqrt{2}",
"style": "rule"
} | {
"index": "3591",
"split": "train"
} |
math | [
{
"content": "A Gareth sequence is a sequence of numbers where each number after the second is the non-negative difference between the two previous numbers. For example, if a Gareth sequence begins 15, 12, then:\n\n- The third number in the sequence is \\(15 - 12 = 3\\),\n- The fourth number is \\(12 - 3 = 9\\)... | {
"ground_truth": "64",
"style": "rule"
} | {
"index": "3592",
"split": "train"
} |
math | [
{
"content": "In right triangle $XYZ$, we have $\\angle X = \\angle Z$ and $XZ = 8\\sqrt{2}$. What is the area of $\\triangle XYZ$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "3593",
"split": "train"
} |
math | [
{
"content": "Given that the two lines $ax+2y+6=0$ and $x+(a-1)y+(a^{2}-1)=0$ are parallel, determine the set of possible values for $a$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\{-1\\}",
"style": "rule"
} | {
"index": "3594",
"split": "train"
} |
math | [
{
"content": "How many perfect squares less than 5000 have a ones digit of 4, 5, or 6? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "3595",
"split": "train"
} |
math | [
{
"content": "Objects $A$ and $B$ move simultaneously in the coordinate plane via a sequence of steps, each of length one. Object $A$ starts at $(0,0)$ and each of its steps is either right or up, both equally likely. Object $B$ starts at $(5,7)$ and each of its steps is either left or down, both equally likely... | {
"ground_truth": "0.20",
"style": "rule"
} | {
"index": "3596",
"split": "train"
} |
math | [
{
"content": "In three-dimensional space, the volume of the geometric body formed by points whose distance to line segment $A B$ is no greater than three units is $216 \\pi$. What is the length of the line segment $A B$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "3597",
"split": "train"
} |
math | [
{
"content": "A block of wood has the shape of a right circular cylinder with radius $6$ and height $8$, and its entire surface has been painted blue. Points $A$ and $B$ are chosen on the edge of one of the circular faces of the cylinder so that $\\overarc{AB}$ on that face measures $120^\\text{o}$. The block i... | {
"ground_truth": "53",
"style": "rule"
} | {
"index": "3598",
"split": "train"
} |
math | [
{
"content": "Problems 8, 9 and 10 use the data found in the accompanying paragraph and figures\nFour friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown.\n$\\circ$ Art's cookies are trapezoids: \n\n$\\circ$ Roger's cookies are ... | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "3599",
"split": "train"
} |
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