data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Among the following functions, identify which pairs represent the same function.\n\n1. $f(x) = |x|, g(x) = \\sqrt{x^2}$;\n2. $f(x) = \\sqrt{x^2}, g(x) = (\\sqrt{x})^2$;\n3. $f(x) = \\frac{x^2 - 1}{x - 1}, g(x) = x + 1$;\n4. $f(x) = \\sqrt{x + 1} \\cdot \\sqrt{x - 1}, g(x) = \\sqrt{x^2 - 1}$. Pleas... | {
"ground_truth": "(1)",
"style": "rule"
} | {
"index": "34000",
"split": "train"
} |
math | [
{
"content": "\\[\\log_{10} x + \\log_{\\sqrt{10}} x + \\log_{\\sqrt[3]{10}} x + \\ldots + \\log_{\\sqrt[1]{10}} x = 5.5\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt[10]{10}",
"style": "rule"
} | {
"index": "34001",
"split": "train"
} |
math | [
{
"content": "[asy]\ndraw((0,1)--(4,1)--(4,2)--(0,2)--cycle);\ndraw((2,0)--(3,0)--(3,3)--(2,3)--cycle);\ndraw((1,1)--(1,2));\nlabel(\"1\",(0.5,1.5));\nlabel(\"2\",(1.5,1.5));\nlabel(\"32\",(2.5,1.5));\nlabel(\"16\",(3.5,1.5));\nlabel(\"8\",(2.5,0.5));\nlabel(\"6\",(2.5,2.5));\n[/asy]\nThe image above is a net o... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "34002",
"split": "train"
} |
math | [
{
"content": "Given that $x - \\frac{1}{x} = 4$, what is $x^3 - \\frac{1}{x^3}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "76",
"style": "rule"
} | {
"index": "34003",
"split": "train"
} |
math | [
{
"content": "Calculate the infinite sum:\n\\[\n\\sum_{n=1}^\\infty \\frac{n^3 - n}{(n+3)!}\n\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{6}",
"style": "rule"
} | {
"index": "34004",
"split": "train"
} |
math | [
{
"content": "Let $A B C D E$ be a convex pentagon such that $\\angle A B C=\\angle A C D=\\angle A D E=90^{\\circ}$ and $A B=B C=C D=D E=1$. Compute $A E$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "34005",
"split": "train"
} |
math | [
{
"content": "If $\\left(a + \\frac{1}{a}\\right)^2 = 3$, then $a^3 + \\frac{1}{a^3}$ equals: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule"
} | {
"index": "34006",
"split": "train"
} |
math | [
{
"content": "There exist constants $a_1, a_2, a_3, a_4, a_5, a_6, a_7$ such that\n\\[\n\\cos^7 \\theta = a_1 \\cos \\theta + a_2 \\cos 2 \\theta + a_3 \\cos 3 \\theta + a_4 \\cos 4 \\theta + a_5 \\cos 5 \\theta + a_6 \\cos 6 \\theta + a_7 \\cos 7 \\theta\n\\]\nfor all angles $\\theta.$ Find $a_1^2 + a_2^2 + a_... | {
"ground_truth": "\\frac{1716}{4096}",
"style": "rule"
} | {
"index": "34007",
"split": "train"
} |
math | [
{
"content": "Given that the sum of the first $n$ terms of the sequence $\\{a_n\\}$ is $S_n=\\ln (1+ \\frac {1}{n})$, find the value of $e^{a_7+a_8+a_9}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac {20}{21}",
"style": "rule"
} | {
"index": "34008",
"split": "train"
} |
math | [
{
"content": "In the quadratic equation $3x^{2}-6x-7=0$, the coefficient of the quadratic term is ____ and the constant term is ____. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-7",
"style": "rule"
} | {
"index": "34009",
"split": "train"
} |
math | [
{
"content": "Find $n$ such that $2^6 \\cdot 3^3 \\cdot n = 10!$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "350",
"style": "rule"
} | {
"index": "34010",
"split": "train"
} |
math | [
{
"content": "Calculate the limit of the function:\n\n\\[ \\lim _{x \\rightarrow \\frac{1}{2}} \\frac{\\sqrt[3]{\\frac{x}{4}}-\\frac{1}{2}}{\\sqrt{\\frac{1}{2}+x}-\\sqrt{2x}} \\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{2}{3}",
"style": "rule"
} | {
"index": "34011",
"split": "train"
} |
math | [
{
"content": "A person has a probability of $\\frac{1}{2}$ to hit the target in each shot. What is the probability of hitting the target 3 times out of 6 shots, with exactly 2 consecutive hits? (Answer with a numerical value) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{16}",
"style": "rule"
} | {
"index": "34012",
"split": "train"
} |
math | [
{
"content": "In $\\triangle PQR$, we have $PQ = QR = 34$ and $PR = 32$. Point $M$ is the midpoint of $\\overline{QR}$. Find $PM$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3\\sqrt{89}",
"style": "rule"
} | {
"index": "34013",
"split": "train"
} |
math | [
{
"content": "Let $a$ and $b$ be angles such that $\\cos a + \\cos b = \\frac{1}{2}$ and $\\sin a + \\sin b = \\frac{3}{11}.$ Find\n\\[\\tan \\left( \\frac{a + b}{2} \\right).\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{6}{11}",
"style": "rule"
} | {
"index": "34014",
"split": "train"
} |
math | [
{
"content": "The left and right foci of a hyperbola are $F_{1}$ and $F_{2}$, respectively. A line passing through $F_{2}$ intersects the right branch of the hyperbola at points $A$ and $B$. If $\\triangle F_{1} A B$ is an equilateral triangle, what is the eccentricity of the hyperbola? Please output the final ... | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "34015",
"split": "train"
} |
math | [
{
"content": "How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\\times 5$ square array of dots? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "34016",
"split": "train"
} |
math | [
{
"content": "Let $a,$ $b,$ and $t$ be real numbers such that $a + b = t.$ Find, in terms of $t,$ the minimum value of $a^2 + b^2.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{t^2}{2}",
"style": "rule"
} | {
"index": "34017",
"split": "train"
} |
math | [
{
"content": "What is the largest base-4 number that has four digits? Express your answer in base 10. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "255",
"style": "rule"
} | {
"index": "34018",
"split": "train"
} |
math | [
{
"content": "Given $m=(\\sqrt{3}\\sin \\omega x,\\cos \\omega x)$, $n=(\\cos \\omega x,-\\cos \\omega x)$ ($\\omega > 0$, $x\\in\\mathbb{R}$), $f(x)=m\\cdot n-\\frac{1}{2}$ and the distance between two adjacent axes of symmetry on the graph of $f(x)$ is $\\frac{\\pi}{2}$.\n\n$(1)$ Find the intervals of monoton... | {
"ground_truth": "\\frac{3\\sqrt{3}}{4}",
"style": "rule"
} | {
"index": "34019",
"split": "train"
} |
math | [
{
"content": "Find the sum of all positive integers $a=2^n3^m$ where $n$ and $m$ are non-negative integers, for which $a^6$ is not a divisor of $6^a$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "42",
"style": "rule"
} | {
"index": "34020",
"split": "train"
} |
math | [
{
"content": "Given that $F$ is the right focus of the ellipse $C:\\frac{x^2}{4}+\\frac{y^2}{3}=1$, $P$ is a point on the ellipse $C$, and $A(1,2\\sqrt{2})$, find the maximum value of $|PA|+|PF|$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4 + 2\\sqrt{3}",
"style": "rule"
} | {
"index": "34021",
"split": "train"
} |
math | [
{
"content": "Given vectors $\\overrightarrow{m}=( \\sqrt {3}\\sin x-\\cos x,1)$ and $\\overrightarrow{n}=(\\cos x, \\frac {1}{2})$, and the function $f(x)= \\overrightarrow{m}\\cdot \\overrightarrow{n}$,\n(1) Find the interval(s) where the function $f(x)$ is monotonically increasing;\n(2) If $a$, $b$, $c$ are ... | {
"ground_truth": "2 \\sqrt {3}",
"style": "rule"
} | {
"index": "34022",
"split": "train"
} |
math | [
{
"content": "Square $ABCD$ has center $O,\\ AB=900,\\ E$ and $F$ are on $AB$ with $AE<BF$ and $E$ between $A$ and $F, m\\angle EOF =45^\\circ,$ and $EF=400.$ Given that $BF=p+q\\sqrt{r},$ where $p,q,$ and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p+q+r.$ Please output... | {
"ground_truth": "307",
"style": "rule"
} | {
"index": "34023",
"split": "train"
} |
math | [
{
"content": "Consider a 4-by-4 grid where each of the unit squares can be colored either purple or green. Each color choice is equally likely independent of the others. Compute the probability that the grid does not contain a 3-by-3 grid of squares all colored purple. Express your result in the form $\\frac{m}... | {
"ground_truth": "255",
"style": "rule"
} | {
"index": "34024",
"split": "train"
} |
math | [
{
"content": "$a,b,c$ - are sides of triangle $T$ . It is known, that if we increase any one side by $1$ , we get new\na) triangle\nb)acute triangle\n\nFind minimal possible area of triangle $T$ in case of a) and in case b) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{3}}{4}",
"style": "rule"
} | {
"index": "34025",
"split": "train"
} |
math | [
{
"content": "A circle centered at $O$ is circumscribed about $\\triangle ABC$ as follows: [asy]\npair pA, pB, pC, pO;\npO = (0, 0);\npA = pO + dir(-20);\npB = pO + dir(90);\npC = pO + dir(190);\ndraw(pA--pB--pC--pA);\ndraw(pO--pA);\ndraw(pO--pB);\ndraw(pO--pC);\nlabel(\"$O$\", pO, S);\nlabel(\"$110^\\circ$\", ... | {
"ground_truth": "50^\\circ",
"style": "rule"
} | {
"index": "34026",
"split": "train"
} |
math | [
{
"content": "A regular hexagon `LMNOPQ` has sides of length 4. Find the area of triangle `LNP`. Express your answer in simplest radical form. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8\\sqrt{3}",
"style": "rule"
} | {
"index": "34027",
"split": "train"
} |
math | [
{
"content": "Find $t$ such that $(t,5)$ lies on the line through $(0,3)$ and $(-8,0)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{16}{3}",
"style": "rule"
} | {
"index": "34028",
"split": "train"
} |
math | [
{
"content": "To calculate $41^2$, David mentally figures the value $40^2$ and adds 81. David subtracts a number from $40^2$ to calculate $39^2$. What number does he subtract? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "79",
"style": "rule"
} | {
"index": "34029",
"split": "train"
} |
math | [
{
"content": "Given a function $f(x)$ $(x \\in \\mathbb{R})$ that satisfies the equation $f(-x) = 8 - f(4 + x)$, and another function $g(x) = \\frac{4x + 3}{x - 2}$. If the graph of $f(x)$ has 168 intersection points with the graph of $g(x)$, denoted as $P_i(x_i, y_i)$ $(i = 1,2, \\dots, 168)$, calculate the va... | {
"ground_truth": "1008",
"style": "rule"
} | {
"index": "34030",
"split": "train"
} |
math | [
{
"content": "What is the greatest integer less than 150 for which the greatest common divisor of that integer and 18 is 6? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "144",
"style": "rule"
} | {
"index": "34031",
"split": "train"
} |
math | [
{
"content": "The largest divisor of a natural number \\( N \\), smaller than \\( N \\), was added to \\( N \\), producing a power of ten. Find all such \\( N \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "75",
"style": "rule"
} | {
"index": "34032",
"split": "train"
} |
math | [
{
"content": "For each positive integer $n$ let $a_n$ be the least positive integer multiple of $23$ such that $a_n \\equiv 1 \\pmod{2^n}.$ Find the number of positive integers $n$ less than or equal to $1000$ that satisfy $a_n = a_{n+1}.$ Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "363",
"style": "rule"
} | {
"index": "34033",
"split": "train"
} |
math | [
{
"content": "Determine all such pairs pf positive integers $(a, b)$ such that $a + b + (gcd (a, b))^ 2 = lcm (a, b) = 2 \\cdot lcm(a -1, b)$, where $lcm (a, b)$ denotes the smallest common multiple, and $gcd (a, b)$ denotes the greatest common divisor of numbers $a, b$. Please output the final answer within... | {
"ground_truth": "(2, 3) \\text{ and } (6, 15)",
"style": "rule"
} | {
"index": "34034",
"split": "train"
} |
math | [
{
"content": "A five-digit number is called a \"hill\" if its first three digits are in ascending order and its last three digits are in descending order. For example, 13760 and 28932 are hills, whereas 78821 and 86521 are not hills. How many hills exist that are greater than the number 77777? Please output the... | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "34035",
"split": "train"
} |
math | [
{
"content": "Six people are arranged in a row. In how many ways can the three people A, B, and C be arranged such that they are not adjacent to each other? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "144",
"style": "rule"
} | {
"index": "34036",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, $\\overrightarrow {AD}=3 \\overrightarrow {DC}$, $\\overrightarrow {BP}=2 \\overrightarrow {PD}$, if $\\overrightarrow {AP}=λ \\overrightarrow {BA}+μ \\overrightarrow {BC}$, then $λ+μ=\\_\\_\\_\\_\\_\\_$. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "- \\frac {1}{3}",
"style": "rule"
} | {
"index": "34037",
"split": "train"
} |
math | [
{
"content": "The first term of a sequence is 934. Each subsequent term is equal to the sum of the digits of the previous term multiplied by 13. Find the 2019th term of the sequence. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "130",
"style": "rule"
} | {
"index": "34038",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $c = a \\cos B + 2b \\sin^2 \\frac{A}{2}$.\n(1) Find angle $A$.\n(2) If $b=4$ and the length of median drawn to side $AC$ is $\\sqrt{7}$, find $a$. Please output the final answer with... | {
"ground_truth": "\\sqrt{13}",
"style": "rule"
} | {
"index": "34039",
"split": "train"
} |
math | [
{
"content": "Let $M = 36 \\cdot 36 \\cdot 77 \\cdot 330$. Find the ratio of the sum of the odd divisors of $M$ to the sum of the even divisors of $M$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1 : 62",
"style": "rule"
} | {
"index": "34040",
"split": "train"
} |
math | [
{
"content": "Find the maximum value of the function\n\n$$\nf(x)=\\sin (x+\\sin x)+\\sin (x-\\sin x)+\\left(\\frac{\\pi}{2}-2\\right) \\sin (\\sin x)\n$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\pi - 2}{\\sqrt{2}}",
"style": "rule"
} | {
"index": "34041",
"split": "train"
} |
math | [
{
"content": "The expression $\\left(1+\\frac{1}{2}\\right)\\left(1+\\frac{1}{3}\\right)\\left(1+\\frac{1}{4}\\right)\\left(1+\\frac{1}{5}\\right)\\left(1+\\frac{1}{6}\\right)\\left(1+\\frac{1}{7}\\right)\\left(1+\\frac{1}{8}\\right)\\left(1+\\frac{1}{9}\\right)$ is equal to what? Please output the final answer... | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "34042",
"split": "train"
} |
math | [
{
"content": "Let $m$ and $n$ be positive integers with $m\\le 2000$ and $k=3-\\frac{m}{n}$. Find the smallest positive value of $k$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$\\boxed{ \\frac{1}{667}} .$",
"style": "rule"
} | {
"index": "34043",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, $D$ is a point on side $\\overline{AC}$ such that $BD=DC$ and $\\angle BCD$ measures $70^\\circ$. What is the degree measure of $\\angle ADB$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "140",
"style": "rule"
} | {
"index": "34044",
"split": "train"
} |
math | [
{
"content": "What is the second number in the row of Pascal's triangle that has 43 numbers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "42",
"style": "rule"
} | {
"index": "34045",
"split": "train"
} |
math | [
{
"content": "Given positive numbers $m$ and $n$ that satisfy $m^2 + n^2 = 100$, find the maximum or minimum value of $m + n$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10\\sqrt{2}",
"style": "rule"
} | {
"index": "34046",
"split": "train"
} |
math | [
{
"content": "The measures of the interior angles of a convex polygon of $n$ sides are in arithmetic progression. If the common difference is $5^{\\circ}$ and the largest angle is $160^{\\circ}$, then $n$ equals: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "34047",
"split": "train"
} |
math | [
{
"content": "Sixteen wooden Cs are placed in a 4-by-4 grid, all with the same orientation, and each is to be colored either red or blue. A quadrant operation on the grid consists of choosing one of the four two-by-two subgrids of Cs found at the corners of the grid and moving each C in the subgrid to the adjac... | {
"ground_truth": "1296",
"style": "rule"
} | {
"index": "34048",
"split": "train"
} |
math | [
{
"content": "A cube, all of whose surfaces are painted, is cut into $1000$ smaller cubes of the same size. Find the expected value $E(X)$, where $X$ denotes the number of painted faces of a small cube randomly selected. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{5}",
"style": "rule"
} | {
"index": "34049",
"split": "train"
} |
math | [
{
"content": "Compute $\\frac{x^8+12x^4+36}{x^4+6}$ when $x=5$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "631",
"style": "rule"
} | {
"index": "34050",
"split": "train"
} |
math | [
{
"content": "A certain electronic device contains three components, with probabilities of failure for each component being $0.1, 0.2, 0.3$, respectively. If the probabilities of the device failing when one, two, or three components fail are $0.25, 0.6, 0.9$, respectively, find the probability that the device f... | {
"ground_truth": "0.1601",
"style": "rule"
} | {
"index": "34051",
"split": "train"
} |
math | [
{
"content": "Let $ABC$ be a right triangle where $\\measuredangle A = 90^\\circ$ and $M\\in (AB)$ such that $\\frac{AM}{MB}=3\\sqrt{3}-4$ . It is known that the symmetric point of $M$ with respect to the line $GI$ lies on $AC$ . Find the measure of $\\measuredangle B$ . Please output the final answ... | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "34052",
"split": "train"
} |
math | [
{
"content": "Maurice travels to work either by his own car (and then due to traffic jams, he is late in half the cases) or by subway (and then he is late only one out of four times). If on a given day Maurice arrives at work on time, he always uses the same mode of transportation the next day as he did the day... | {
"ground_truth": "2/3",
"style": "rule"
} | {
"index": "34053",
"split": "train"
} |
math | [
{
"content": "For a nonnegative integer $n$, let $r_7(3n)$ represent the remainder when $3n$ is divided by $7$. Determine the $22^{\\text{nd}}$ entry in an ordered list of all nonnegative integers $n$ that satisfy $$r_7(3n)\\le 4~.$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "29",
"style": "rule"
} | {
"index": "34054",
"split": "train"
} |
math | [
{
"content": "There is a list of seven numbers. The average of the first four numbers is $5$, and the average of the last four numbers is $8$. If the average of all seven numbers is $6\\frac{4}{7}$, then the number common to both sets of four numbers is Please output the final answer within \\boxed{}.",
"ro... | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "34055",
"split": "train"
} |
math | [
{
"content": "Given the set $A=\\{2,3,4,8,9,16\\}$, if $a\\in A$ and $b\\in A$, the probability that the event \"$\\log_{a}b$ is not an integer but $\\frac{b}{a}$ is an integer\" occurs is $\\_\\_\\_\\_\\_\\_$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{18}",
"style": "rule"
} | {
"index": "34056",
"split": "train"
} |
math | [
{
"content": "The amount $2.5$ is split into two nonnegative real numbers uniformly at random, for instance, into $2.143$ and $.357$, or into $\\sqrt{3}$ and $2.5-\\sqrt{3}$. Then each number is rounded to its nearest integer, for instance, $2$ and $0$ in the first case above, $2$ and $1$ in the second. What is... | {
"ground_truth": "\\frac{3}{5}",
"style": "rule"
} | {
"index": "34057",
"split": "train"
} |
math | [
{
"content": "On the sides \\(A B, B C, C D\\) and \\(A D\\) of the convex quadrilateral \\(A B C D\\) are points \\(M, N, K\\) and \\(L\\) respectively, such that \\(A M: M B = 3: 2\\), \\(C N: N B = 2: 3\\), \\(C K = K D\\) and \\(A L: L D = 1: 2\\). Find the ratio of the area of the hexagon \\(M B N K D L\\)... | {
"ground_truth": "4/5",
"style": "rule"
} | {
"index": "34058",
"split": "train"
} |
math | [
{
"content": "A square sheet of paper $ABCD$ is folded straight in such a way that point $B$ hits to the midpoint of side $CD$ . In what ratio does the fold line divide side $BC$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5/3",
"style": "rule"
} | {
"index": "34059",
"split": "train"
} |
math | [
{
"content": "Two equal circles in the same plane cannot have the following number of common tangents. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "34060",
"split": "train"
} |
math | [
{
"content": "Let $\\triangle ABC$ have sides $a$, $b$, $c$ opposite angles $A$, $B$, $C$ respectively, given that $a^{2}+2b^{2}=c^{2}$, then $\\dfrac {\\tan C}{\\tan A}=$ ______ ; the maximum value of $\\tan B$ is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac { \\sqrt {3}}{3}",
"style": "rule"
} | {
"index": "34061",
"split": "train"
} |
math | [
{
"content": "In a trapezoid $ABCD$ with bases $\\overline{AB} \\parallel \\overline{CD}$ and $\\overline{BC} \\perp \\overline{CD}$, suppose that $CD = 10$, $\\tan C = 2$, and $\\tan D = 1$. Calculate the length of $AB$ and determine the area of the trapezoid. Please output the final answer within \\boxed{}.",... | {
"ground_truth": "300",
"style": "rule"
} | {
"index": "34062",
"split": "train"
} |
math | [
{
"content": "Alice drew a regular $2021$-gon in the plane. Bob then labeled each vertex of the $2021$-gon with a real number, in such a way that the labels of consecutive vertices differ by at most $1$. Then, for every pair of non-consecutive vertices whose labels differ by at most $1$, Alice drew a diagonal c... | {
"ground_truth": "2018",
"style": "rule"
} | {
"index": "34063",
"split": "train"
} |
math | [
{
"content": "Player A and player B are two basketball players shooting from the same position independently, with shooting accuracies of $\\dfrac{1}{2}$ and $p$ respectively, and the probability of player B missing both shots is $\\dfrac{1}{16}$.\n- (I) Calculate the probability that player A hits at least one... | {
"ground_truth": "\\dfrac{3}{8}",
"style": "rule"
} | {
"index": "34064",
"split": "train"
} |
math | [
{
"content": "The eccentricity of the ellipse $\\frac {x^{2}}{9}+ \\frac {y^{2}}{4+k}=1$ is $\\frac {4}{5}$. Find the value of $k$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "34065",
"split": "train"
} |
math | [
{
"content": "The probability of getting rain on any given day in August in Beach Town is \\(\\frac{1}{5}\\). What is the probability that it rains on at most 3 days in the first week of August? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.813",
"style": "rule"
} | {
"index": "34066",
"split": "train"
} |
math | [
{
"content": "Trapezoid $ABCD$ has $\\overline{AB}\\parallel\\overline{CD}, BC=CD=43$, and $\\overline{AD}\\perp\\overline{BD}$. Let $O$ be the intersection of the diagonals $\\overline{AC}$ and $\\overline{BD}$, and let $P$ be the midpoint of $\\overline{BD}$. Given that $OP=11$, the length of $AD$ can be writ... | {
"ground_truth": "194",
"style": "rule"
} | {
"index": "34067",
"split": "train"
} |
math | [
{
"content": "Find the product of all constants \\(t\\) such that the quadratic \\(x^2 + tx + 12\\) can be factored in the form \\((x+a)(x+b)\\), where \\(a\\) and \\(b\\) are integers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "530816",
"style": "rule"
} | {
"index": "34068",
"split": "train"
} |
math | [
{
"content": "The Cookie Monster encounters a cookie whose boundary is the equation $x^2+y^2 + 21 = 4x + 18 y$ and is very confused. He wants to know if this cookie is a lunch-sized cookie or a snack-sized cookie. What is the radius of this cookie? Please output the final answer within \\boxed{}.",
"role": ... | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "34069",
"split": "train"
} |
math | [
{
"content": "What is the smallest positive perfect square that is divisible by both 2 and 3? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "34070",
"split": "train"
} |
math | [
{
"content": "Two positive integers differ by 6 and their product is 135. What is the larger integer? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "34071",
"split": "train"
} |
math | [
{
"content": "The desired number is greater than 400 and less than 500. Find it if the sum of its digits is 9 and it is equal to 47/36 of the number obtained by reversing its digits. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "423",
"style": "rule"
} | {
"index": "34072",
"split": "train"
} |
math | [
{
"content": "Let $n$ be the integer such that $0 \\le n < 31$ and $3n \\equiv 1 \\pmod{31}$. What is $\\left(2^n\\right)^3 - 2 \\pmod{31}$?\n\nExpress your answer as an integer from $0$ to $30$, inclusive. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "34073",
"split": "train"
} |
math | [
{
"content": "For a real number $y$, find the maximum value of\n\\[\n\\frac{y^6}{y^{12} + 3y^9 - 9y^6 + 27y^3 + 81}.\n\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{27}",
"style": "rule"
} | {
"index": "34074",
"split": "train"
} |
math | [
{
"content": "Given $$a_{n}= \\frac {n(n+1)}{2}$$, remove all the numbers in the sequence $\\{a_n\\}$ that can be divided by 2, and arrange the remaining numbers in ascending order to form the sequence $\\{b_n\\}$. Find the value of $b_{21}$. Please output the final answer within \\boxed{}.",
"role": "user"... | {
"ground_truth": "861",
"style": "rule"
} | {
"index": "34075",
"split": "train"
} |
math | [
{
"content": "The increasing sequence \\( T = 2, 3, 5, 6, 7, 8, 10, 11, \\ldots \\) consists of all positive integers which are not perfect squares. What is the 2012th term of \\( T \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2057",
"style": "rule"
} | {
"index": "34076",
"split": "train"
} |
math | [
{
"content": "A list of $2018$ positive integers has a unique mode, which occurs exactly $10$ times. What is the least number of distinct values that can occur in the list? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "225",
"style": "rule"
} | {
"index": "34077",
"split": "train"
} |
math | [
{
"content": "Given two circles $x^{2}+y^{2}=4$ and $x^{2}+y^{2}-2y-6=0$, find the length of their common chord. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{3}",
"style": "rule"
} | {
"index": "34078",
"split": "train"
} |
math | [
{
"content": "In triangle $XYZ$, $E$ lies on $\\overline{YZ}$ and $G$ lies on $\\overline{XY}$. Let $\\overline{XE}$ and $\\overline{YG}$ intersect at $Q.$\n\nIf $XQ:QE = 5:2$ and $GQ:QY = 3:4$, find $\\frac{XG}{GY}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{4}{3}",
"style": "rule"
} | {
"index": "34079",
"split": "train"
} |
math | [
{
"content": "A game show offers a contestant three prizes A, B and C, each of which is worth a whole number of dollars from $$ 1$ to $$ 9999$ inclusive. The contestant wins the prizes by correctly guessing the price of each prize in the order A, B, C. As a hint, the digits of the three prices are given. On a p... | {
"ground_truth": "420",
"style": "rule"
} | {
"index": "34080",
"split": "train"
} |
math | [
{
"content": "C is the complex numbers. \\( f : \\mathbb{C} \\to \\mathbb{R} \\) is defined by \\( f(z) = |z^3 - z + 2| \\). What is the maximum value of \\( f \\) on the unit circle \\( |z| = 1 \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{13}",
"style": "rule"
} | {
"index": "34081",
"split": "train"
} |
math | [
{
"content": "Convert -630° to radians. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{7\\pi}{2}",
"style": "rule"
} | {
"index": "34082",
"split": "train"
} |
math | [
{
"content": "In an $11 \\times 11$ table, integers from 0 to 10 are placed (naturally, numbers can repeat, and not necessarily all listed numbers occur). It is known that in every $3 \\times 2$ or $2 \\times 3$ rectangle, the sum of the numbers is 10. Find the smallest possible value of the sum of the numbers ... | {
"ground_truth": "200",
"style": "rule"
} | {
"index": "34083",
"split": "train"
} |
math | [
{
"content": "Find $n$ such that $2^6 \\cdot 3^3 \\cdot n = 10!$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2100",
"style": "rule"
} | {
"index": "34084",
"split": "train"
} |
math | [
{
"content": "Given the geometric sequence $\\{a_{n}\\}$, $a_{2}$ and $a_{18}$ are the two roots of the equation $x^{2}+15x+16=0$, find the value of $a_{3}a_{10}a_{17}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-64",
"style": "rule"
} | {
"index": "34085",
"split": "train"
} |
math | [
{
"content": "If five people are selected at random from a group of ten men and five women, what is the probability that at least one woman is selected? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{917}{1001}",
"style": "rule"
} | {
"index": "34086",
"split": "train"
} |
math | [
{
"content": "A *palindromic table* is a $3 \\times 3$ array of letters such that the words in each row and column read the same forwards and backwards. An example of such a table is shown below. \n\\[ \\begin{array}[h]{ccc}\n O & M & O \n N & M & N \n O & M & O\n\\end{array} \\]\nHow many palindromic tabl... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "34087",
"split": "train"
} |
math | [
{
"content": "In triangle \\(ABC\\), \\(AC = 18 \\, \\text{cm}\\) and \\(BC = 21 \\, \\text{cm}\\). Point \\(K\\) is the midpoint of side \\(BC\\), and point \\(M\\) is the midpoint of side \\(AB\\). Point \\(N\\) lies on side \\(AC\\) such that \\(AN = 6 \\, \\text{cm}\\). Additionally, \\(MN = KN\\). Find the... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "34088",
"split": "train"
} |
math | [
{
"content": "The sides of a triangle are 5, 6, and 7. Find the area of the orthogonal projection of the triangle onto a plane that forms an angle equal to the smallest angle of the triangle with the plane of the triangle. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{30 \\sqrt{6}}{7}",
"style": "rule"
} | {
"index": "34089",
"split": "train"
} |
math | [
{
"content": "Let $\\theta=\\frac{2\\pi}{2015}$ , and suppose the product \\[\\prod_{k=0}^{1439}\\left(\\cos(2^k\\theta)-\\frac{1}{2}\\right)\\] can be expressed in the form $\\frac{b}{2^a}$ , where $a$ is a non-negative integer and $b$ is an odd integer (not necessarily positive). Find $a+b$ .\n\n*2017 ... | {
"ground_truth": "1441",
"style": "rule"
} | {
"index": "34090",
"split": "train"
} |
math | [
{
"content": "BoatWorks built 3 canoes in January of this year and then each subsequent calendar month they built twice the number of canoes they had built the previous month. How many total canoes were built by BoatWorks by the end of March of this year? Please output the final answer within \\boxed{}.",
"... | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "34091",
"split": "train"
} |
math | [
{
"content": "Find the probability that a randomly selected 8-digit number composed of 0s and 1s has the sum of the digits in even positions equal to the sum of the digits in odd positions. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "35/128",
"style": "rule"
} | {
"index": "34092",
"split": "train"
} |
math | [
{
"content": "Gilda has a bag of marbles. She gives $20\\%$ of them to her friend Pedro. Then Gilda gives $10\\%$ of what is left to another friend, Ebony. Finally, Gilda gives $25\\%$ of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for he... | {
"ground_truth": "54",
"style": "rule"
} | {
"index": "34093",
"split": "train"
} |
math | [
{
"content": "Find the greatest common divisor of $8!$ and $(6!)^2.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5760",
"style": "rule"
} | {
"index": "34094",
"split": "train"
} |
math | [
{
"content": "If \\( 6x + t = 4x - 9 \\), what is the value of \\( x + 4 \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-4",
"style": "rule"
} | {
"index": "34095",
"split": "train"
} |
math | [
{
"content": "Ten children were given 100 pieces of macaroni each on their plates. Some children didn't want to eat and started playing. With one move, one child transfers one piece of macaroni from their plate to each of the other children's plates. What is the minimum number of moves needed such that all the ... | {
"ground_truth": "45",
"style": "rule"
} | {
"index": "34096",
"split": "train"
} |
math | [
{
"content": "Given a square region with a side length of 1 meter, and a total of 5120 beans within the square with 4009 beans within the inscribed circle, determine the approximate value of pi rounded to three decimal places. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3.13",
"style": "rule"
} | {
"index": "34097",
"split": "train"
} |
math | [
{
"content": "Amy works for 36 hours per week for 10 weeks during the summer, making $\\$3000$. If she works for 30 weeks during the school year at the same rate of pay and needs to make another $\\$3000$, how many hours per week must she work? Please output the final answer within \\boxed{}.",
"role": "use... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "34098",
"split": "train"
} |
math | [
{
"content": "Right $ \\triangle ABC$ has $ AB \\equal{} 3$ , $ BC \\equal{} 4$ , and $ AC \\equal{} 5$ . Square $ XYZW$ is inscribed in $ \\triangle ABC$ with $ X$ and $ Y$ on $ \\overline{AC}$ , $ W$ on $ \\overline{AB}$ , and $ Z$ on $ \\overline{BC}$ . What is the side length of the squa... | {
"ground_truth": "\\frac {60}{37}",
"style": "rule"
} | {
"index": "34099",
"split": "train"
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.