data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "As shown in the figure below, six semicircles lie in the interior of a regular hexagon with side length 2 so that the diameters of the semicircles coincide with the sides of the hexagon. What is the area of the shaded region ---- inside the hexagon but outside all of the semicircles? Please output... | {
"ground_truth": "6\\sqrt{3} - 3\\pi",
"style": "rule"
} | {
"index": "7400",
"split": "train"
} |
math | [
{
"content": "Two identical cylindrical vessels are connected by a small tube with a valve at the bottom. Initially, the valve is closed, and water is poured into the first vessel while oil is poured into the second vessel, such that the liquid levels are equal and are $h=40$ cm. At what level will the water be... | {
"ground_truth": "32.94",
"style": "rule"
} | {
"index": "7401",
"split": "train"
} |
math | [
{
"content": "An influenza outbreak occurred in regions $A$, $B$, and $C$. The percentages of people with influenza in these three regions are $6\\%$, $5\\%$, and $4\\%$ respectively. It is assumed that the populations of these three regions are in the ratio of $3:5:2$. Now, a person is randomly selected from t... | {
"ground_truth": "\\frac{18}{51}",
"style": "rule"
} | {
"index": "7402",
"split": "train"
} |
math | [
{
"content": "For the opening home game of the baseball season, the Madd Batters minor league baseball team offered the following incentives to its fans:\n\nEvery 75th fan who entered the stadium got a coupon for a free hot dog.\n\nEvery 30th fan who entered the stadium got a coupon for a free cup of soda.\n\nE... | {
"ground_truth": "26",
"style": "rule"
} | {
"index": "7403",
"split": "train"
} |
math | [
{
"content": "Alex needs to catch a train. The train arrives randomly some time between 1:00 and 2:00, waits for 10 minutes, and then leaves. If Alex also arrives randomly between 1:00 and 2:00, what is the probability that the train will be there when Alex arrives? Please output the final answer within \\boxed... | {
"ground_truth": "\\frac{11}{72}",
"style": "rule"
} | {
"index": "7404",
"split": "train"
} |
math | [
{
"content": "What is the probability that Hannah gets fewer than 4 heads if she flips 12 coins? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{299}{4096}",
"style": "rule"
} | {
"index": "7405",
"split": "train"
} |
math | [
{
"content": "Given a sequence: $2016$, $2017$, $1$, $-2016$, $-2017$, $…$, the characteristic of this sequence is that starting from the second term, each term is equal to the sum of the term before and after it. Find the sum of the first $2017$ terms of the sequence. Please output the final answer within \\bo... | {
"ground_truth": "2016",
"style": "rule"
} | {
"index": "7406",
"split": "train"
} |
math | [
{
"content": "$\\triangle GHI$ is inscribed inside $\\triangle XYZ$ such that $G, H, I$ lie on $YZ, XZ, XY$, respectively. The circumcircles of $\\triangle GZC, \\triangle HYD, \\triangle IXF$ have centers $O_1, O_2, O_3$, respectively. Also, $XY = 26, YZ = 28, XZ = 27$, and $\\stackrel{\\frown}{YI} = \\stackre... | {
"ground_truth": "29",
"style": "rule"
} | {
"index": "7407",
"split": "train"
} |
math | [
{
"content": "In a right triangle \\(ABC\\), the legs \\(AB\\) and \\(AC\\) measure 4 and 3 respectively. Point \\(D\\) bisects the hypotenuse \\(BC\\). Find the distance between the centers of the incircles of triangles \\(ADC\\) and \\(ABD\\). Please output the final answer within \\boxed{}.",
"role": "us... | {
"ground_truth": "\\frac{5 \\sqrt{13}}{12}",
"style": "rule"
} | {
"index": "7408",
"split": "train"
} |
math | [
{
"content": "Point $C(0,p)$ lies on the $y$-axis between $Q(0,12)$ and $O(0,0)$ as shown. Determine an expression for the area of $\\triangle COB$ in terms of $p$. Your answer should be simplified as much as possible. [asy]\nsize(5cm);defaultpen(fontsize(9));\npair o = (0, 0); pair q = (0, 12); pair b = (12, 0... | {
"ground_truth": "6p",
"style": "rule"
} | {
"index": "7409",
"split": "train"
} |
math | [
{
"content": "In right triangle $MNO$, $\\tan{M}=\\frac{5}{4}$, $OM=8$, and $\\angle O = 90^\\circ$. Find $MN$. Express your answer in simplest radical form. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{41}",
"style": "rule"
} | {
"index": "7410",
"split": "train"
} |
math | [
{
"content": "Given non-zero vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ satisfy $|\\overrightarrow{a}|=2|\\overrightarrow{b}|$, and $(\\overrightarrow{a}-\\overrightarrow{b})\\bot \\overrightarrow{b}$, then the angle between $\\overrightarrow{a}$ and $\\overrightarrow{b}$ is ______. Please output t... | {
"ground_truth": "\\frac{\\pi}{3}",
"style": "rule"
} | {
"index": "7411",
"split": "train"
} |
math | [
{
"content": "Find \\( n > m \\geq 1 \\) such that the last two digits of the decimal representation of \\( 42^{n} \\) are the same as those of \\( 42^{m} \\), and such that the sum \\( m+n \\) is minimal. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "7412",
"split": "train"
} |
math | [
{
"content": "Determine all real numbers $q$ for which the equation $x^4 -40x^2 +q = 0$ has four real solutions which form an arithmetic progression Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "144",
"style": "rule"
} | {
"index": "7413",
"split": "train"
} |
math | [
{
"content": "Fnargs are either red or blue and have 2, 3, or 4 heads. A group of six Fnargs consisting of one of each possible form (one red and one blue for each number of heads) is made to line up such that no immediate neighbors are the same color nor have the same number of heads. How many ways are there o... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "7414",
"split": "train"
} |
math | [
{
"content": "A housewife saved $2.50 in buying a dress on sale. If she spent $25 for the dress, she saved about: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9 \\%",
"style": "rule"
} | {
"index": "7415",
"split": "train"
} |
math | [
{
"content": "Candice starts driving home from work at 5:00 PM. Starting at exactly 5:01 PM, and every minute after that, Candice encounters a new speed limit sign and slows down by 1 mph. Candice's speed, in miles per hour, is always a positive integer. Candice drives for \\(2/3\\) of a mile in total. She driv... | {
"ground_truth": "5:05(PM)",
"style": "rule"
} | {
"index": "7416",
"split": "train"
} |
math | [
{
"content": "For the largest possible $n$, can you create two bi-infinite sequences $A$ and $B$ such that any segment of length $n$ from sequence $B$ is contained in $A$, where $A$ has a period of 1995, but $B$ does not have this property (it is either not periodic or has a different period)?\n\nComment: The s... | {
"ground_truth": "1995",
"style": "rule"
} | {
"index": "7417",
"split": "train"
} |
math | [
{
"content": "Point $P$ lies inside equilateral triangle $A B C$ so that $\\angle B P C=120^{\\circ}$ and $A P \\sqrt{2}=B P+C P$. $\\frac{A P}{A B}$ can be written as $\\frac{a \\sqrt{b}}{c}$, where $a, b, c$ are integers, $c$ is positive, $b$ is square-free, and $\\operatorname{gcd}(a, c)=1$. Find $100 a+10 b... | {
"ground_truth": "255",
"style": "rule"
} | {
"index": "7418",
"split": "train"
} |
math | [
{
"content": "An equilateral triangle is placed on top of a square with each side of the square equal to one side of the triangle, forming a pentagon. What percent of the area of the pentagon is the area of the equilateral triangle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{4\\sqrt{3} - 3}{13} \\times 100\\%",
"style": "rule"
} | {
"index": "7419",
"split": "train"
} |
math | [
{
"content": "What is the least positive integer with exactly $12$ positive factors? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "7420",
"split": "train"
} |
math | [
{
"content": "Let sets $X$ and $Y$ have $30$ and $25$ elements, respectively, and there are at least $10$ elements in both sets. Find the smallest possible number of elements in $X \\cup Y$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "45",
"style": "rule"
} | {
"index": "7421",
"split": "train"
} |
math | [
{
"content": "The marked price of a book was 30% less than the suggested retail price. Alice purchased the book for half the marked price at a Fiftieth Anniversary sale. What percent of the suggested retail price did Alice pay? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "35\\%",
"style": "rule"
} | {
"index": "7422",
"split": "train"
} |
math | [
{
"content": "In the new clubroom, there were only chairs and a table. Each chair had four legs, and the table had three legs. Scouts came into the clubroom. Each sat on their own chair, two chairs remained unoccupied, and the total number of legs in the room was 101.\n\nDetermine how many chairs were in the cl... | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "7423",
"split": "train"
} |
math | [
{
"content": "Find the curve defined by the equation\n\\[\\theta = \\frac{\\pi}{3}.\\](A) Line\n(B) Circle\n(C) Parabola\n(D) Ellipse\n(E) Hyperbola\n\nEnter the letter of the correct option. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\text{(A)}",
"style": "rule"
} | {
"index": "7424",
"split": "train"
} |
math | [
{
"content": "Rationalize the denominator of $\\frac{\\sqrt{32}}{\\sqrt{16}-\\sqrt{2}}$. The answer can be written as $\\frac{A\\sqrt{B}+C}{D}$, where $A$, $B$, $C$, and $D$ are integers, $D$ is positive, and $B$ is not divisible by the square of any prime. Find the minimum possible value of $A+B+C+D$. Please ... | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "7425",
"split": "train"
} |
math | [
{
"content": "Given real numbers $x \\gt 0 \\gt y$, and $\\frac{1}{x+2}+\\frac{1}{1-y}=\\frac{1}{6}$, find the minimum value of $x-y$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "7426",
"split": "train"
} |
math | [
{
"content": "Compute the smallest positive integer $n$ such that $n + i,$ $(n + i)^2,$ and $(n + i)^3$ are the vertices of a triangle in the complex plane whose area is greater than 2015. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "7427",
"split": "train"
} |
math | [
{
"content": "Arnaldo claimed that one billion is the same as one million millions. Professor Piraldo corrected him and said, correctly, that one billion is the same as one thousand millions. What is the difference between the correct value of one billion and Arnaldo's assertion?\n(a) 1000\n(b) 999000\n(c) 1000... | {
"ground_truth": "999000000000",
"style": "rule"
} | {
"index": "7428",
"split": "train"
} |
math | [
{
"content": "Let $n$ and $k$ be integers satisfying $\\binom{2k}{2} + n = 60$ . It is known that $n$ days before Evan's 16th birthday, something happened. Compute $60-n$ .\n\n*Proposed by Evan Chen* Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "45",
"style": "rule"
} | {
"index": "7429",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, point $F$ divides side $AC$ in the ratio $1:2$. Let $E$ be the point of intersection of side $BC$ and $AG$ where $G$ is the midpoints of $BF$. The point $E$ divides side $BC$ in the ratio Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1/3",
"style": "rule"
} | {
"index": "7430",
"split": "train"
} |
math | [
{
"content": "Let $a, b, c, n$ be positive real numbers such that $\\frac{a+b}{a}=3, \\frac{b+c}{b}=4$, and $\\frac{c+a}{c}=n$. Find $n$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{7}{6}",
"style": "rule"
} | {
"index": "7431",
"split": "train"
} |
math | [
{
"content": "Your math friend Steven rolls five fair icosahedral dice (each of which is labelled $1,2, \\ldots, 20$ on its sides). He conceals the results but tells you that at least half of the rolls are 20. Assuming that Steven is truthful, what is the probability that all three remaining concealed dice show... | {
"ground_truth": "\\frac{1}{58}",
"style": "rule"
} | {
"index": "7432",
"split": "train"
} |
math | [
{
"content": "Monsieur Dupont remembered that today is their wedding anniversary and invited his wife to dine at a fine restaurant. Upon leaving the restaurant, he noticed that he had only one fifth of the money he initially took with him. He found that the centimes he had left were equal to the francs he initi... | {
"ground_truth": "7996",
"style": "rule"
} | {
"index": "7433",
"split": "train"
} |
math | [
{
"content": "Antonette gets $70\\%$ on a 10-problem test, $80\\%$ on a 20-problem test and $90\\%$ on a 30-problem test. If the three tests are combined into one 60-problem test, which percent is her overall score, rounded to the nearest percent? Please output the final answer within \\boxed{}.",
"role": "... | {
"ground_truth": "83\\%",
"style": "rule"
} | {
"index": "7434",
"split": "train"
} |
math | [
{
"content": "Flights are arranged between 13 countries. For $ k\\ge 2$ , the sequence $ A_{1} ,A_{2} ,\\ldots A_{k}$ is said to a cycle if there exist a flight from $ A_{1}$ to $ A_{2}$ , from $ A_{2}$ to $ A_{3}$ , $ \\ldots$ , from $ A_{k \\minus{} 1}$ to $ A_{k}$ , and from $ A_{k}$ to $ A_{... | {
"ground_truth": "79",
"style": "rule"
} | {
"index": "7435",
"split": "train"
} |
math | [
{
"content": "If $f(x)$ is a real valued function of the real variable $x$, and $f(x)$ is not identically zero, and for all $a$ and $b$, $f(a+b)+f(a-b)=2f(a)+2f(b)$, then for all $x$ and $y$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$f(-x)=f(x)$",
"style": "rule"
} | {
"index": "7436",
"split": "train"
} |
math | [
{
"content": "If $7^{4x}=343$, what is the value of $7^{4x-3}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "7437",
"split": "train"
} |
math | [
{
"content": "The slope angle of the tangent line to the curve $y=\\frac{1}{2}x^2+2$ at the point $\\left(-1, \\frac{5}{2}\\right)$ is what? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3\\pi}{4}",
"style": "rule"
} | {
"index": "7438",
"split": "train"
} |
math | [
{
"content": "Find a positive integer that is divisible by 20 and whose cube root is a number between 8.2 and 8.3. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "560",
"style": "rule"
} | {
"index": "7439",
"split": "train"
} |
math | [
{
"content": "Let\n\\[ f(x) = \\left\\{ \n\\begin{array}{cl} \n2ax + 4, & \\text{if } x > 2, \\\\\nx^2-2 & \\text{if } -2 \\le x \\le 2, \\\\\n3x - c, & \\text{if } x < -2. \n\\end{array} \n\\right. \\]\nFind $a+c$ if the piecewise function is continuous. Please output the final answer within \\boxed{}.",
"... | {
"ground_truth": "-\\frac{17}{2}",
"style": "rule"
} | {
"index": "7440",
"split": "train"
} |
math | [
{
"content": "Karl was attempting to calculate economic figures. He found the following equation to be true:\\[fp-w=10000\\]If $f=5$ and $w=5+125i$, what is $p$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2001+25i",
"style": "rule"
} | {
"index": "7441",
"split": "train"
} |
math | [
{
"content": "Let \\(a\\) and \\(b\\) be positive integers such that \\(90 < a + b < 99\\) and \\(0.9 < \\frac{a}{b} < 0.91\\). Find \\(ab\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2346",
"style": "rule"
} | {
"index": "7442",
"split": "train"
} |
math | [
{
"content": "1. Let $[x]$ denote the greatest integer less than or equal to the real number $x$. Given a sequence of positive integers $\\{a_{n}\\}$ such that $a_{1} = a$, and for any positive integer $n$, the sequence satisfies the recursion\n$$\na_{n+1} = a_{n} + 2 \\left[\\sqrt{a_{n}}\\right].\n$$\n(1) If $... | {
"ground_truth": "82",
"style": "rule"
} | {
"index": "7443",
"split": "train"
} |
math | [
{
"content": "What is the greatest common factor of 90, 135, and 225? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "45",
"style": "rule"
} | {
"index": "7444",
"split": "train"
} |
math | [
{
"content": "Simplify first, then evaluate: $(1-\\frac{a}{a+1})\\div \\frac{{a}^{2}-1}{{a}^{2}+2a+1}$, where $a=\\sqrt{2}+1$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "7445",
"split": "train"
} |
math | [
{
"content": "Given vectors $\\overrightarrow {OA} = (1, -2)$, $\\overrightarrow {OB} = (4, -1)$, $\\overrightarrow {OC} = (m, m+1)$.\n(1) If $\\overrightarrow {AB} \\parallel \\overrightarrow {OC}$, find the value of the real number $m$;\n(2) If $\\triangle ABC$ is a right-angled triangle, find the value of th... | {
"ground_truth": "\\frac{5}{2}",
"style": "rule"
} | {
"index": "7446",
"split": "train"
} |
math | [
{
"content": "Knowing that segment \\( CD \\) has a length of 6 and its midpoint is \\( M \\), two triangles \\( \\triangle ACD \\) and \\( \\triangle BCD \\) are constructed on the same side with \\( CD \\) as a common side and both having a perimeter of 16, such that \\( \\angle AMB = 90^{\\circ} \\). Find th... | {
"ground_truth": "400/41",
"style": "rule"
} | {
"index": "7447",
"split": "train"
} |
math | [
{
"content": "If $\\cos 60^{\\circ} = \\cos 45^{\\circ} \\cos \\theta$ with $0^{\\circ} \\leq \\theta \\leq 90^{\\circ}$, what is the value of $\\theta$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "45^{\\circ}",
"style": "rule"
} | {
"index": "7448",
"split": "train"
} |
math | [
{
"content": "An archipelago consists of $N \\geqslant 7$ islands. Any two islands are connected by at most one bridge. It is known that no more than 5 bridges lead from each island, and among any 7 islands, there are always at least two connected by a bridge. What is the maximum possible value of $N$? Please o... | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "7449",
"split": "train"
} |
math | [
{
"content": "Find $y$ if $\\dfrac{y^2 - 9y + 8}{y-1} + \\dfrac{3y^2 +16y-12 }{3y -2} = -3$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{1}{2}",
"style": "rule"
} | {
"index": "7450",
"split": "train"
} |
math | [
{
"content": "Simplify $\\sqrt{28x} \\cdot \\sqrt{15x} \\cdot \\sqrt{21x}$. Express your answer in simplest radical form in terms of $x$.\n\nNote: When entering a square root with more than one character, you must use parentheses or brackets. For example, you should enter $\\sqrt{14}$ as \"sqrt(14)\" or \"sqrt... | {
"ground_truth": "42x\\sqrt{5x}",
"style": "rule"
} | {
"index": "7451",
"split": "train"
} |
math | [
{
"content": "For any sequence of real numbers $A=\\{a_1, a_2, a_3, \\ldots\\}$, define $\\triangle A$ as the sequence $\\{a_2 - a_1, a_3 - a_2, a_4 - a_3, \\ldots\\}$, where the $n$-th term is $a_{n+1} - a_n$. Assume that all terms of the sequence $\\triangle (\\triangle A)$ are $1$ and $a_{18} = a_{2017} = 0$... | {
"ground_truth": "1000",
"style": "rule"
} | {
"index": "7452",
"split": "train"
} |
math | [
{
"content": "Students are in classroom with $n$ rows. In each row there are $m$ tables. It's given that $m,n \\geq 3$ . At each table there is exactly one student. We call neighbours of the student students sitting one place right, left to him, in front of him and behind him. Each student shook hands with... | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "7453",
"split": "train"
} |
math | [
{
"content": "Let \\( a \\) be a real number, and let \\( m(a) \\) denote the minimum value of the function \\( f(x) = x^2 - ax + \\frac{a}{2} \\) for \\( 0 \\leq x \\leq 1 \\). Find the maximum value of \\( m(a) \\) as \\( a \\) varies. Please output the final answer within \\boxed{}.",
"role": "user"
}
... | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "7454",
"split": "train"
} |
math | [
{
"content": "Max and Minnie each add up sets of three-digit positive integers. Each of them adds three different three-digit integers whose nine digits are all different. Max creates the largest possible sum. Minnie creates the smallest possible sum. What is the difference between Max's sum and Minnie's sum? P... | {
"ground_truth": "1845",
"style": "rule"
} | {
"index": "7455",
"split": "train"
} |
math | [
{
"content": "The surface of a 3x3x3 Rubik's cube consists of 54 cells. What is the maximum number of cells that can be marked such that no two marked cells share a common vertex? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "7456",
"split": "train"
} |
math | [
{
"content": "In Zuminglish-Advanced, all words still consist only of the letters $M, O,$ and $P$; however, there is a new rule that any occurrence of $M$ must be immediately followed by $P$ before any $O$ can occur again. Also, between any two $O's$, there must appear at least two consonants. Determine the num... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "7457",
"split": "train"
} |
math | [
{
"content": "For certain real values of $a, b, c,$ and $d_{},$ the equation $x^4+ax^3+bx^2+cx+d=0$ has four non-real roots. The product of two of these roots is $13+i$ and the sum of the other two roots is $3+4i,$ where $i^2 = -1.$ Find $b.$ Please output the final answer within \\boxed{}.",
"role": "user"... | {
"ground_truth": "51",
"style": "rule"
} | {
"index": "7458",
"split": "train"
} |
math | [
{
"content": "In a circle, parallel chords of lengths 2, 3, and 4 determine central angles of $\\alpha$, $\\beta$, and $\\alpha + \\beta$ radians, respectively, where $\\alpha + \\beta < \\pi$. If $\\cos \\alpha$, which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum o... | {
"ground_truth": "49",
"style": "rule"
} | {
"index": "7459",
"split": "train"
} |
math | [
{
"content": "Let $ABC$ be a triangle with $BC = a$ , $CA = b$ , and $AB = c$ . The $A$ -excircle is tangent to $\\overline{BC}$ at $A_1$ ; points $B_1$ and $C_1$ are similarly defined.\n\nDetermine the number of ways to select positive integers $a$ , $b$ , $c$ such that\n\n\n- the numbers $-a... | {
"ground_truth": "125000",
"style": "rule"
} | {
"index": "7460",
"split": "train"
} |
math | [
{
"content": "The probability that students A and B stand together among three students A, B, and C lined up in a row is what? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "7461",
"split": "train"
} |
math | [
{
"content": "If $3^{2x}+9=10\\left(3^{x}\\right)$, then the value of $(x^2+1)$ is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1 or 5",
"style": "rule"
} | {
"index": "7462",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively, and $\\cos A= \\frac{ \\sqrt{6}}{3}$.\n(1) Find $\\tan 2A$;\n(2) If $\\cos B= \\frac{ 2\\sqrt{2}}{3}, c=2\\sqrt{2}$, find the area of $\\triangle ABC$. Please output the final answer within \\b... | {
"ground_truth": "\\frac{2\\sqrt{2}}{3}",
"style": "rule"
} | {
"index": "7463",
"split": "train"
} |
math | [
{
"content": "Two dice are differently designed. The first die has faces numbered $1$, $1$, $2$, $2$, $3$, and $5$. The second die has faces numbered $2$, $3$, $4$, $5$, $6$, and $7$. What is the probability that the sum of the numbers on the top faces of the two dice is $3$, $7$, or $8$?\nA) $\\frac{11}{36}$\n... | {
"ground_truth": "\\frac{13}{36}",
"style": "rule"
} | {
"index": "7464",
"split": "train"
} |
math | [
{
"content": "For a positive integer $M$, if there exist integers $a$, $b$, $c$ and $d$ so that:\n\\[ M \\leq a < b \\leq c < d \\leq M+49, \\qquad ad=bc \\]\nthen we call $M$ a GOOD number, if not then $M$ is BAD. Please find the greatest GOOD number and the smallest BAD number. Please output the final answer ... | {
"ground_truth": "576",
"style": "rule"
} | {
"index": "7465",
"split": "train"
} |
math | [
{
"content": "Three squares, with side-lengths 2, are placed together edge-to-edge to make an L-shape. The L-shape is placed inside a rectangle so that all five vertices of the L-shape lie on the rectangle, one of them at the midpoint of an edge, as shown.\nWhat is the area of the rectangle?\n\nA 16\nB 18\nC 20... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "7466",
"split": "train"
} |
math | [
{
"content": "Given a complex number $z$ satisfying the equation $|z-1|=|z+2i|$ (where $i$ is the imaginary unit), find the minimum value of $|z-1-i|$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{9\\sqrt{5}}{10}",
"style": "rule"
} | {
"index": "7467",
"split": "train"
} |
math | [
{
"content": "Given $ \\tan \\left( \\alpha + \\frac{\\pi}{4} \\right) = -\\frac{1}{2} $ and $\\frac{\\pi}{2} < \\alpha < \\pi$, calculate the value of $ \\frac{\\sin 2\\alpha - 2\\cos^2 \\alpha}{\\sin \\left(\\alpha - \\frac{\\pi}{4} \\right)} $. Please output the final answer within \\boxed{}.",
"role": "... | {
"ground_truth": "-\\frac{2 \\sqrt{5}}{5}",
"style": "rule"
} | {
"index": "7468",
"split": "train"
} |
math | [
{
"content": "If the polynomial $x^{2}+x^{10}=a_{0}+a_{1}(x+1)+\\cdots+a_{9}(x+1)^{9}+a_{10}(x+1)^{10}$, find the value of $a_{9}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-10",
"style": "rule"
} | {
"index": "7469",
"split": "train"
} |
math | [
{
"content": "On a chessboard, $n$ white rooks and $n$ black rooks are arranged such that rooks of different colors do not attack each other. Find the maximum possible value of $n$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "7470",
"split": "train"
} |
math | [
{
"content": "Compute $$100^{2}+99^{2}-98^{2}-97^{2}+96^{2}+95^{2}-94^{2}-93^{2}+\\ldots+4^{2}+3^{2}-2^{2}-1^{2}$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10100",
"style": "rule"
} | {
"index": "7471",
"split": "train"
} |
math | [
{
"content": "A school organizes a social practice activity during the summer vacation and needs to allocate 8 grade 10 students to company A and company B evenly. Among these students, two students with excellent English scores cannot be allocated to the same company, and neither can the three students with st... | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "7472",
"split": "train"
} |
math | [
{
"content": "Let $\\alpha$ and $\\beta$ be real numbers. Find the minimum value of\n\\[(3 \\cos \\alpha + 4 \\sin \\beta - 10)^2 + (3 \\sin \\alpha + 4 \\cos \\beta - 12)^2.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(\\sqrt{244} - 7)^2",
"style": "rule"
} | {
"index": "7473",
"split": "train"
} |
math | [
{
"content": "Neznaika is drawing closed paths inside a $5 \\times 8$ rectangle, traveling along the diagonals of $1 \\times 2$ rectangles. In the illustration, an example of a path passing through 12 such diagonals is shown. Help Neznaika draw the longest possible path. Please output the final answer within \\... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "7474",
"split": "train"
} |
math | [
{
"content": "Let n rational numbers $x_{1}$, $x_{2}$, $\\ldots$, $x_{n}$ satisfy $|x_{i}| \\lt 1$ for $i=1,2,\\ldots,n$, and $|x_{1}|+|x_{2}|+\\ldots+|x_{n}|=19+|x_{1}+x_{2}+\\ldots+x_{n}|$. Calculate the minimum value of $n$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "7475",
"split": "train"
} |
math | [
{
"content": "Given a point $Q$ on a rectangular piece of paper $DEF$, where $D, E, F$ are folded onto $Q$. Let $Q$ be a fold point of $\\triangle DEF$ if the creases, which number three unless $Q$ is one of the vertices, do not intersect within the triangle. Suppose $DE=24, DF=48,$ and $\\angle E=90^\\circ$. D... | {
"ground_truth": "147",
"style": "rule"
} | {
"index": "7476",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=x+ \\frac{1}{x}$, and $g(x)=f^{2}(x)-af(x)+2a$ has four distinct zeros $x\\_1$, $x\\_2$, $x\\_3$, $x\\_4$, determine the value of $[2-f(x\\_1)]\\cdot[2-f(x\\_2)]\\cdot[2-f(x\\_3)]\\cdot[2-f(x\\_4)]$. Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "7477",
"split": "train"
} |
math | [
{
"content": "Let \\( A \\) be the set of any 20 points on the circumference of a circle. Joining any two points in \\( A \\) produces one chord of this circle. Suppose every three such chords are not concurrent. Find the number of regions within the circle which are divided by all these chords. Please output t... | {
"ground_truth": "5036",
"style": "rule"
} | {
"index": "7478",
"split": "train"
} |
math | [
{
"content": "On a table, there are 210 different cards each with a number from the sequence \\(2, 4, 6, \\ldots, 418, 420\\) (each card has exactly one number, and each number appears exactly once). In how many ways can 2 cards be chosen so that the sum of the numbers on the chosen cards is divisible by \\(7\\... | {
"ground_truth": "3135",
"style": "rule"
} | {
"index": "7479",
"split": "train"
} |
math | [
{
"content": "Two identical rectangular crates are packed with cylindrical pipes, using different methods. Each pipe has diameter 10 cm. A side view of the first four rows of each of the two different methods of packing is shown below.\n\n[asy]\ndraw(circle((1,1),1),black+linewidth(1));\ndraw(circle((3,1),1),b... | {
"ground_truth": "190-100\\sqrt{3}",
"style": "rule"
} | {
"index": "7480",
"split": "train"
} |
math | [
{
"content": "If $8 \\cdot 2^x = 5^{y + 8}$, then when $y = -8$, $x =$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "7481",
"split": "train"
} |
math | [
{
"content": "Malcolm can run a race at a speed of 6 minutes per mile, while Joshua runs at 8 minutes per mile. In a 10-mile race, how many minutes after Malcolm crosses the finish line will Joshua cross the finish line if they start the race together? Please output the final answer within \\boxed{}.",
"rol... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "7482",
"split": "train"
} |
math | [
{
"content": "In Dr. Strange's laboratory, there are some bacteria. Each day, 11 bacteria are eliminated, and each night, 5 bacteria are added. If there are 50 bacteria on the morning of the first day, on which day will all the bacteria be eliminated? Please output the final answer within \\boxed{}.",
"role... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "7483",
"split": "train"
} |
math | [
{
"content": "The polygon enclosed by the solid lines in the figure consists of 4 congruent squares joined edge-to-edge. One more congruent square is attached to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a cube with one face missing? Please out... | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "7484",
"split": "train"
} |
math | [
{
"content": "Let $m>1$ be an integer. Find the smallest positive integer $n$, such that for any integers $a_1,a_2,\\ldots ,a_n; b_1,b_2,\\ldots ,b_n$ there exists integers $x_1,x_2,\\ldots ,x_n$ satisfying the following two conditions: \n\ni) There exists $i\\in \\{1,2,\\ldots ,n\\}$ such that $x_i$ and $m$ ar... | {
"ground_truth": "2\\omega(m) + 1",
"style": "rule"
} | {
"index": "7485",
"split": "train"
} |
math | [
{
"content": "Evaluate the following expression:\n\n$$\\frac { \\sqrt {3}tan12 ° -3}{sin12 ° (4cos ^{2}12 ° -2)}$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-4 \\sqrt {3}",
"style": "rule"
} | {
"index": "7486",
"split": "train"
} |
math | [
{
"content": "Find the smallest, positive five-digit multiple of $18$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10008",
"style": "rule"
} | {
"index": "7487",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate plane $(xOy)$, the curve $y=x^{2}-6x+1$ intersects the coordinate axes at points that lie on circle $C$.\n(1) Find the equation of circle $C$;\n(2) Given point $A(3,0)$, and point $B$ is a moving point on circle $C$, find the maximum value of $\\overrightarrow{OA} \\cdo... | {
"ground_truth": "\\frac{36\\sqrt{37}}{37}",
"style": "rule"
} | {
"index": "7488",
"split": "train"
} |
math | [
{
"content": "Suppose that point $D$ lies on side $B C$ of triangle $A B C$ such that $A D$ bisects $\\angle B A C$, and let $\\ell$ denote the line through $A$ perpendicular to $A D$. If the distances from $B$ and $C$ to $\\ell$ are 5 and 6 , respectively, compute $A D$. Please output the final answer within \... | {
"ground_truth": "\\frac{60}{11}",
"style": "rule"
} | {
"index": "7489",
"split": "train"
} |
math | [
{
"content": "Three balls marked $1,2$ and $3$ are placed in an urn. One ball is drawn, its number is recorded, and then the ball is returned to the urn. This process is repeated and then repeated once more, and each ball is equally likely to be drawn on each occasion. If the sum of the numbers recorded is $6$,... | {
"ground_truth": "\\frac{1}{7}",
"style": "rule"
} | {
"index": "7490",
"split": "train"
} |
math | [
{
"content": "How many integers are solutions to the equation $$(x-2)^{(25-x^2)}=1?$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "7491",
"split": "train"
} |
math | [
{
"content": "Vinny wrote down all the single-digit base-$b$ numbers and added them in base $b$, getting $34_b$.\n\nWhat is $b$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "7492",
"split": "train"
} |
math | [
{
"content": "Let \\(p\\), \\(q\\), \\(r\\), and \\(s\\) be the roots of the polynomial \\[x^4 + 10x^3 + 20x^2 + 15x + 6 = 0.\\] Find the value of \\[\\frac{1}{pq} + \\frac{1}{pr} + \\frac{1}{ps} + \\frac{1}{qr} + \\frac{1}{qs} + \\frac{1}{rs}.\\] Please output the final answer within \\boxed{}.",
"role": "... | {
"ground_truth": "\\frac{10}{3}",
"style": "rule"
} | {
"index": "7493",
"split": "train"
} |
math | [
{
"content": "Distribute 5 students into 3 groups: Group A, Group B, and Group C, with the condition that Group A must have at least two people, and Groups B and C must have at least one person each. How many different distribution schemes exist? Please output the final answer within \\boxed{}.",
"role": "u... | {
"ground_truth": "80",
"style": "rule"
} | {
"index": "7494",
"split": "train"
} |
math | [
{
"content": "You are standing at a pole and a snail is moving directly away from the pole at $1 \\mathrm{~cm} / \\mathrm{s}$. When the snail is 1 meter away, you start 'Round 1'. In Round $n(n \\geq 1)$, you move directly toward the snail at $n+1 \\mathrm{~cm} / \\mathrm{s}$. When you reach the snail, you imme... | {
"ground_truth": "5050",
"style": "rule"
} | {
"index": "7495",
"split": "train"
} |
math | [
{
"content": "Find the 6-digit number beginning and ending in the digit 2 that is the product of three consecutive even integers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "287232",
"style": "rule"
} | {
"index": "7496",
"split": "train"
} |
math | [
{
"content": "A circle $\\Gamma$ with center $O$ has radius 1. Consider pairs $(A, B)$ of points so that $A$ is inside the circle and $B$ is on its boundary. The circumcircle $\\Omega$ of $O A B$ intersects $\\Gamma$ again at $C \\neq B$, and line $A C$ intersects $\\Gamma$ again at $X \\neq C$. The pair $(A, B... | {
"ground_truth": "\\frac{3 \\pi}{4}",
"style": "rule"
} | {
"index": "7497",
"split": "train"
} |
math | [
{
"content": "Find the remainder when $3^{3^{3^3}}$ is divided by 1000. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "387",
"style": "rule"
} | {
"index": "7498",
"split": "train"
} |
math | [
{
"content": "A line $l$ passing through the focus of the parabola $y=4x^2$ intersects the parabola at points $A(x_1, y_1)$ and $B(x_2, y_2)$. If $y_1+y_2=2$, then the length of segment $AB$ equals \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{17}{8}",
"style": "rule"
} | {
"index": "7499",
"split": "train"
} |
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