data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "For real numbers $t,$ the point of intersection of the lines $x + 2y = 7t + 3$ and $x - y = 2t - 2$ is plotted. All the plotted points lie on a line. Find the slope of this line. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{11}",
"style": "rule"
} | {
"index": "40000",
"split": "train"
} |
math | [
{
"content": "If $\\tan{\\alpha}$ and $\\tan{\\beta}$ are the roots of $x^2 - px + q = 0$, and $\\cot{\\alpha}$ and $\\cot{\\beta}$ are the roots of $x^2 - rx + s = 0$, then $rs$ is necessarily Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{p}{q^2}",
"style": "rule"
} | {
"index": "40001",
"split": "train"
} |
math | [
{
"content": "Given that $a$ and $b$ are constants, and $a \\neq 0$, $f(x) = ax^2 + bx$, $f(2) = 0$.\n\n$(1)$ If the equation $f(x) - x = 0$ has a unique real root, find the expression for the function $f(x)$;\n\n$(2)$ When $a = 1$, find the maximum and minimum values of the function $f(x)$ in the interval $[-1... | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "40002",
"split": "train"
} |
math | [
{
"content": "Given $|x+2|+|1-x|=9-|y-5|-|1+y|$, find the maximum and minimum values of $x+y$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "40003",
"split": "train"
} |
math | [
{
"content": "A plane $P$ slices through a cube of volume 1 with a cross-section in the shape of a regular hexagon. This cube also has an inscribed sphere, whose intersection with $P$ is a circle. What is the area of the region inside the regular hexagon but outside the circle? Please output the final answer wi... | {
"ground_truth": "(3 \\sqrt{3}-\\pi) / 4",
"style": "rule"
} | {
"index": "40004",
"split": "train"
} |
math | [
{
"content": "If $r$ and $s$ are the roots of $x^2-px+q=0$, then $r^2+s^2$ equals: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "p^2-2q",
"style": "rule"
} | {
"index": "40005",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively, and $B$ is an acute angle. If $\\frac{\\sin A}{\\sin B} = \\frac{5c}{2b}$, $\\sin B = \\frac{\\sqrt{7}}{4}$, and $S_{\\triangle ABC} = \\frac{5\\sqrt{7}}{4}$, find the value of $b$. Please output the fi... | {
"ground_truth": "\\sqrt{14}",
"style": "rule"
} | {
"index": "40006",
"split": "train"
} |
math | [
{
"content": "When drawing a histogram of the lifespans of 1000 people, if the class interval is uniformly 20, and the height of the vertical axis for the age range 60 to 80 years is 0.03, calculate the number of people aged 60 to 80. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "600",
"style": "rule"
} | {
"index": "40007",
"split": "train"
} |
math | [
{
"content": "Given that the parametric equations of the line $l$ are $\\left\\{{\\begin{array}{l}{x=1+\\frac{1}{2}t}\\\\{y=\\sqrt{3}+\\frac{{\\sqrt{3}}}{2}t}\\end{array}}\\right.$ (where $t$ is a parameter), establish a polar coordinate system with the origin as the pole and the non-negative x-axis as the pola... | {
"ground_truth": "\\frac{3\\sqrt{3} - 1}{2}",
"style": "rule"
} | {
"index": "40008",
"split": "train"
} |
math | [
{
"content": "The terms of an arithmetic sequence add to $715$. The first term of the sequence is increased by $1$, the second term is increased by $3$, the third term is increased by $5$, and in general, the $k$th term is increased by the $k$th odd positive integer. The terms of the new sequence add to $836$. ... | {
"ground_truth": "195",
"style": "rule"
} | {
"index": "40009",
"split": "train"
} |
math | [
{
"content": "Find the smallest natural number such that when it is multiplied by 9, the resulting number consists of the same digits in a different order. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1089",
"style": "rule"
} | {
"index": "40010",
"split": "train"
} |
math | [
{
"content": "What is the remainder when $1 + 2 + 3 + 4 + \\dots + 9 + 10$ is divided by 8? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "40011",
"split": "train"
} |
math | [
{
"content": "Let $a$ and $b$ be real numbers. One of the roots of $x^3 + ax + b = 0$ is $1 + i \\sqrt{3}.$ Find $a + b.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "40012",
"split": "train"
} |
math | [
{
"content": "Triangle $ABC$ has vertices at $A(5,8)$, $B(3,-2)$, and $C(6,1)$. The point $D$ with coordinates $(m,n)$ is chosen inside the triangle so that the three small triangles $ABD$, $ACD$ and $BCD$ all have equal areas. What is the value of $10m + n$? Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "49",
"style": "rule"
} | {
"index": "40013",
"split": "train"
} |
math | [
{
"content": "$A$ and $B$ are $46$ kilometers apart. Person A rides a bicycle from point $A$ to point $B$ at a speed of $15$ kilometers per hour. One hour later, person B rides a motorcycle along the same route from point $A$ to point $B$ at a speed of $40$ kilometers per hour. \n$(1)$ After how many hours can... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "40014",
"split": "train"
} |
math | [
{
"content": "Right $\\triangle PQR$ has sides $PQ = 5$, $QR = 12$, and $PR = 13$. Rectangle $ABCD$ is inscribed in $\\triangle PQR$ such that $A$ and $B$ are on $\\overline{PR}$, $D$ on $\\overline{PQ}$, and $C$ on $\\overline{QR}$. If the height of the rectangle (parallel to side $\\overline{PQ}$) is half its... | {
"ground_truth": "7.5",
"style": "rule"
} | {
"index": "40015",
"split": "train"
} |
math | [
{
"content": "A Yule log is shaped like a right cylinder with height $10$ and diameter $5$ . Freya cuts it parallel to its bases into $9$ right cylindrical slices. After Freya cut it, the combined surface area of the slices of the Yule log increased by $a\\pi$ . Compute $a$ . Please output the final answ... | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "40016",
"split": "train"
} |
math | [
{
"content": "Ms. Garcia weighed the packages in three different pairings and obtained weights of 162, 164, and 168 pounds. Find the total weight of all four packages. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "247",
"style": "rule"
} | {
"index": "40017",
"split": "train"
} |
math | [
{
"content": "Two points are taken on the lateral edge of a pyramid, dividing the edge into three equal parts. Planes parallel to the base are drawn through these points. Find the volume of the section of the pyramid between these planes, given that the volume of the entire pyramid is 1. Please output the final... | {
"ground_truth": "\\frac{7}{27}",
"style": "rule"
} | {
"index": "40018",
"split": "train"
} |
math | [
{
"content": "How many positive integers less than 100, which are also not divisible by 3, have an even number of positive divisors? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "59",
"style": "rule"
} | {
"index": "40019",
"split": "train"
} |
math | [
{
"content": "In triangle $\\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $b\\tan A = (2c-b)\\tan B$. \n$(1)$ Find angle $A$; \n$(2)$ If $\\overrightarrow{m}=(0,-1)$ and $\\overrightarrow{n}=(\\cos B, 2\\cos^2\\frac{C}{2})$, find the minimum value of $|\\o... | {
"ground_truth": "\\frac{\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "40020",
"split": "train"
} |
math | [
{
"content": "Given the hyperbola $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$) with its right focus at point $F$, and a point $P$ on the left branch of the hyperbola. Also given is that $PF$ is tangent to the circle $x^2 + y^2 = a^2$ at point $M$, where $M$ is precisely the midpoint of the line ... | {
"ground_truth": "\\sqrt{5}",
"style": "rule"
} | {
"index": "40021",
"split": "train"
} |
math | [
{
"content": "There are 5 balls of the same shape and size in a bag, including 3 red balls and 2 yellow balls. Now, balls are randomly drawn from the bag one at a time until two different colors of balls are drawn. Let the random variable $\\xi$ be the number of balls drawn at this time. Find $E(\\xi)=$____. Pl... | {
"ground_truth": "\\frac{5}{2}",
"style": "rule"
} | {
"index": "40022",
"split": "train"
} |
math | [
{
"content": "Let $n$ equal the number of sides in a regular polygon. For $3\\leq n < 10$, how many values of $n$ result in a regular polygon where the common degree measure of the interior angles is not an integer? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "40023",
"split": "train"
} |
math | [
{
"content": "In triangle $\\triangle ABC$, $AC=2$, $D$ is the midpoint of $AB$, $CD=\\frac{1}{2}BC=\\sqrt{7}$, $P$ is a point on $CD$, and $\\overrightarrow{AP}=m\\overrightarrow{AC}+\\frac{1}{3}\\overrightarrow{AB}$. Find $|\\overrightarrow{AP}|$. Please output the final answer within \\boxed{}.",
"role":... | {
"ground_truth": "\\frac{2\\sqrt{13}}{3}",
"style": "rule"
} | {
"index": "40024",
"split": "train"
} |
math | [
{
"content": "In a triangle, two angles measure 45 degrees and 60 degrees. The side opposite the 45-degree angle measures 8 units. Calculate the sum of the lengths of the other two sides. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "19.3",
"style": "rule"
} | {
"index": "40025",
"split": "train"
} |
math | [
{
"content": "Vanya had a certain amount of cookies; he ate some, and then Tanya came to visit him and they divided the remaining cookies equally. It turned out that Vanya ate five times more cookies than Tanya did. What fraction of all the cookies did Vanya eat by the time Tanya arrived? Please output the fina... | {
"ground_truth": "\\frac{5}{7}",
"style": "rule"
} | {
"index": "40026",
"split": "train"
} |
math | [
{
"content": "Someone wrote down two numbers $5^{2020}$ and $2^{2020}$ consecutively. How many digits will the resulting number contain? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2021",
"style": "rule"
} | {
"index": "40027",
"split": "train"
} |
math | [
{
"content": "What is $2\\frac{1}{4}$ divided by $\\frac{3}{5}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3 \\frac{3}{4}",
"style": "rule"
} | {
"index": "40028",
"split": "train"
} |
math | [
{
"content": "The sequence $\\{a\\_n\\}$ satisfies $(a_{n+1}-1)(1-a_{n})=a_{n}$, $a_{8}=2$, then $S_{2017}=$ _____ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac {2017}{2}",
"style": "rule"
} | {
"index": "40029",
"split": "train"
} |
math | [
{
"content": "Let $x$ be Walter's age in 1996. If Walter was one-third as old as his grandmother in 1996, then his grandmother's age in 1996 is $3x$. The sum of their birth years is $3864$, and the sum of their ages in 1996 plus $x$ and $3x$ is $1996+1997$. Express Walter's age at the end of 2001 in terms of $x... | {
"ground_truth": "37",
"style": "rule"
} | {
"index": "40030",
"split": "train"
} |
math | [
{
"content": "In a certain group, the probability that each member uses mobile payment is $p$, and the payment methods of each member are independent of each other. Let $X$ be the number of members in the group of $10$ who use mobile payment, $D\\left(X\\right)=2.4$, $P\\left(X=4\\right) \\lt P\\left(X=6\\right... | {
"ground_truth": "0.6",
"style": "rule"
} | {
"index": "40031",
"split": "train"
} |
math | [
{
"content": "Call a positive integer monotonous if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, $3$, $23578$, and $987620$ are monotonous, but $88$, $7434$, and $23557$ are not. How many monotonous positi... | {
"ground_truth": "1524",
"style": "rule"
} | {
"index": "40032",
"split": "train"
} |
math | [
{
"content": "In a triangle, two of the side lengths are 7 and 8, and the angle between them is $120^\\circ.$ Find the length of the third side. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "40033",
"split": "train"
} |
math | [
{
"content": "Add together the numbers $7.56$ and $4.29$. Write the result as a decimal. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11.85",
"style": "rule"
} | {
"index": "40034",
"split": "train"
} |
math | [
{
"content": "Given $O$ as the circumcenter of $\\triangle ABC$ and $D$ as the midpoint of $BC$. If $\\overrightarrow{AO} \\cdot \\overrightarrow{AD}=4$ and $BC=2 \\sqrt{6}$, then find the length of $AD$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": "40035",
"split": "train"
} |
math | [
{
"content": "Xiao Li was doing a subtraction problem and mistook the tens digit 7 for a 9 and the ones digit 3 for an 8, resulting in a difference of 76. The correct difference is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "51",
"style": "rule"
} | {
"index": "40036",
"split": "train"
} |
math | [
{
"content": "Given a circle and $2006$ points lying on this circle. Albatross colors these $2006$ points in $17$ colors. After that, Frankinfueter joins some of the points by chords such that the endpoints of each chord have the same color and two different chords have no common points (not even a common... | {
"ground_truth": "117",
"style": "rule"
} | {
"index": "40037",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $a^{2}+b^{2}+4 \\sqrt {2}=c^{2}$ and $ab=4$, find the minimum value of $\\frac {\\sin C}{\\tan ^{2}A\\cdot \\sin 2B}$. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "\\frac {3 \\sqrt {2}}{2}+2",
"style": "rule"
} | {
"index": "40038",
"split": "train"
} |
math | [
{
"content": "Suppose that $\\frac{2}{3}$ of $10$ bananas are worth as much as $8$ oranges. How many oranges are worth as much as $\\frac{1}{2}$ of $5$ bananas? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "40039",
"split": "train"
} |
math | [
{
"content": "Given that the sequence $\\{a\\_n\\}$ is an arithmetic progression, and the sequence $\\{b\\_n\\}$ satisfies $b\\_n=a\\_n a_{n+1} \\cdot a_{n+2} (n \\in \\mathbb{N}^*)$, let $S\\_n$ be the sum of the first $n$ terms of $\\{b\\_n\\}$. If $a_{12}=\\frac{3}{8} a_{5} > 0$, find the value of $n$ when $... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "40040",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x) = \\sin(2x + \\frac{\\pi}{3}) - \\sqrt{3}\\sin(2x - \\frac{\\pi}{6})$,\n\n(1) Find the smallest positive period of the function $f(x)$ and its intervals of monotonic increase;\n\n(2) When $x \\in \\left[-\\frac{\\pi}{6}, \\frac{\\pi}{3}\\right]$, find the maximum and minim... | {
"ground_truth": "-\\sqrt{3}",
"style": "rule"
} | {
"index": "40041",
"split": "train"
} |
math | [
{
"content": "Simplify $(-3-2i)-(1+4i)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-4-6i",
"style": "rule"
} | {
"index": "40042",
"split": "train"
} |
math | [
{
"content": "A certain township responded to the call of \"green waters and green mountains are mountains of gold and silver\" and tailored the town to be a \"ecological fruit characteristic town\" according to local conditions. After investigation, it was found that the yield $W$ (unit: kg) of a certain fruit... | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "40043",
"split": "train"
} |
math | [
{
"content": "The quadratic $x^2+1300x+1300$ can be written in the form $(x+b)^2+c$, where $b$ and $c$ are constants. What is $\\frac{c}{b}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-648",
"style": "rule"
} | {
"index": "40044",
"split": "train"
} |
math | [
{
"content": "If $a-1=b+2=c-3=d+4$, which of the four quantities $a,b,c,d$ is the largest? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$c$",
"style": "rule"
} | {
"index": "40045",
"split": "train"
} |
math | [
{
"content": "The sum of the distances from one vertex of a square with sides of length $2$ to the midpoints of each of the sides of the square is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$2+2\\sqrt{5}$",
"style": "rule"
} | {
"index": "40046",
"split": "train"
} |
math | [
{
"content": "Given $\\cos\\left(\\alpha- \\frac{\\beta}{2}\\right) = -\\frac{1}{9}$ and $\\sin\\left(\\frac{\\alpha}{2} - \\beta\\right) = \\frac{2}{3}$, with $\\frac{\\pi}{2} < \\alpha < \\pi$ and $0 < \\beta < \\frac{\\pi}{2}$, find $\\cos(\\alpha + \\beta)$. Please output the final answer within \\boxed{}."... | {
"ground_truth": "-\\frac{239}{729}",
"style": "rule"
} | {
"index": "40047",
"split": "train"
} |
math | [
{
"content": "Find the product of all constants $t$ such that the quadratic $x^2 + tx - 10$ 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": "729",
"style": "rule"
} | {
"index": "40048",
"split": "train"
} |
math | [
{
"content": "A cylindrical glass vessel is heated to a temperature of $t = 1200^{\\circ}$C, causing part of the air to be expelled. After covering the vessel with a well-sealing glass plate, it is left to cool to room temperature at $t^{\\prime}=16^{\\circ}$C. The remaining air inside becomes less dense, and t... | {
"ground_truth": "19.6\\%",
"style": "rule"
} | {
"index": "40049",
"split": "train"
} |
math | [
{
"content": "Jennifer is hiking in the mountains. She walks northward for 3 miles, then turns 45 degrees eastward and walks 5 miles. How far is she from her starting point? Express your answer in simplest radical form. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{34 + 15\\sqrt{2}}",
"style": "rule"
} | {
"index": "40050",
"split": "train"
} |
math | [
{
"content": "On the base $AC$ of an isosceles triangle $ABC (AB=BC)$, a point $M$ is marked. It is known that $AM=7$, $MB=3$, and $\\angle BMC=60^{\\circ}$. Find the length of segment $AC$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "40051",
"split": "train"
} |
math | [
{
"content": "Determine the value of the following product with a short calculation:\n\n$$\n\\frac{6 \\cdot 27^{12}+2 \\cdot 81^{9}}{8000000^{2}} \\cdot \\frac{80 \\cdot 32^{3} \\cdot 125^{4}}{9^{19}-729^{6}}\n$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "40052",
"split": "train"
} |
math | [
{
"content": "Mr. Zhou places all the integers from $1$ to $225$ into a $15$ by $15$ grid. He places $1$ in the middle square (eighth row and eighth column) and places other numbers one by one clockwise, as shown in part in the diagram below. What is the sum of the greatest number and the least number that appe... | {
"ground_truth": "367",
"style": "rule"
} | {
"index": "40053",
"split": "train"
} |
math | [
{
"content": "Find the smallest positive integer $n$ such that, if there are initially $2n$ townspeople and 1 goon, then the probability the townspeople win is greater than $50\\%$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "40054",
"split": "train"
} |
math | [
{
"content": "In triangle $\\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and it is given that $(2a-c)\\cos B = b\\cos C$. \n\n(1) Find the size of angle $B$; \n(2) If the area of $\\triangle ABC$ is $T$ and $b=\\sqrt{3}$, find the value of $a+c$; Please outpu... | {
"ground_truth": "\\sqrt{ \\frac{30}{2} }",
"style": "rule"
} | {
"index": "40055",
"split": "train"
} |
math | [
{
"content": "For how many integers $n$ between $1$ and $50$, inclusive, is $\\frac{(n^2-1)!}{(n!)^n}$ an integer? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "34",
"style": "rule"
} | {
"index": "40056",
"split": "train"
} |
math | [
{
"content": "The members of a club are arranged in a rectangular formation. When they are arranged in 10 rows, there are 4 positions unoccupied in the formation. When they are arranged in 11 rows, there are 5 positions unoccupied. How many members are in the club if the membership is between 150 and 300? Pleas... | {
"ground_truth": "226",
"style": "rule"
} | {
"index": "40057",
"split": "train"
} |
math | [
{
"content": "$Q$ is the point of intersection of the diagonals of one face of a cube whose edges have length 2 units. Calculate the length of $QR$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{6}",
"style": "rule"
} | {
"index": "40058",
"split": "train"
} |
math | [
{
"content": "Simplify first, then evaluate: $(1-\\frac{2}{x+1})÷\\frac{x^2-1}{2x+2}$, where $x=\\pi ^{0}+1$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "40059",
"split": "train"
} |
math | [
{
"content": "Base prime representation of a natural number is defined using the exponents of its prime factorization as follows. Each place in a base prime represents a prime number, and it is occupied by the corresponding exponent of that prime, starting on the right side with the smallest prime number and pr... | {
"ground_truth": "220",
"style": "rule"
} | {
"index": "40060",
"split": "train"
} |
math | [
{
"content": "Given that the area of the parallelogram $ABCD$ is $240$, $E$ and $H$ are the midpoints of sides $AD$ and $AB$ respectively. Point $G$ is on side $BC$ such that $BG = 2GC$, and point $F$ is on side $CD$ such that $DF = 3FC$. Point $K$ is on side $AC$ such that the area of $\\triangle EKF$ is 33. F... | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "40061",
"split": "train"
} |
math | [
{
"content": "How many four-digit positive integers are multiples of 7? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1286",
"style": "rule"
} | {
"index": "40062",
"split": "train"
} |
math | [
{
"content": "In a country with 15 cities, some of which are connected by airlines belonging to three different companies, it is known that even if any one of the airlines ceases operations, it will still be possible to travel between any two cities (possibly with transfers) using the remaining two companies' f... | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "40063",
"split": "train"
} |
math | [
{
"content": "Lily and Sarah are playing a game. They each choose a real number at random between -1 and 1. They then add the squares of their numbers together. If the result is greater than or equal to 1, Lily wins, and if the result is less than 1, Sarah wins. What is the probability that Sarah wins? Please o... | {
"ground_truth": "\\frac{\\pi}{4}",
"style": "rule"
} | {
"index": "40064",
"split": "train"
} |
math | [
{
"content": "The triangular plot of ACD lies between Aspen Road, Brown Road and a railroad. Main Street runs east and west, and the railroad runs north and south. The numbers in the diagram indicate distances in miles. The width of the railroad track can be ignored. How many square miles are in the plot of ... | {
"ground_truth": "4.5",
"style": "rule"
} | {
"index": "40065",
"split": "train"
} |
math | [
{
"content": "Let $S$ be the increasing sequence of positive integers whose binary representation has exactly $8$ ones. Let $N$ be the 1000th number in $S$. Find the remainder when $N$ is divided by $1000$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "40066",
"split": "train"
} |
math | [
{
"content": "The Elvish language consists of 4 words: \"elara\", \"quen\", \"silva\", and \"nore\". In a sentence, \"elara\" cannot come directly before \"quen\", and \"silva\" cannot come directly before \"nore\"; all other word combinations are grammatically correct (including sentences with repeated words).... | {
"ground_truth": "48",
"style": "rule"
} | {
"index": "40067",
"split": "train"
} |
math | [
{
"content": "A store has three types of boxes containing marbles in large, medium, and small sizes, respectively holding 13, 11, and 7 marbles. If someone wants to buy 20 marbles, it can be done without opening the boxes (1 large box plus 1 small box). However, if someone wants to buy 23 marbles, a box must be... | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "40068",
"split": "train"
} |
math | [
{
"content": "A hectoliter barrel is full of wine; 1 liter of liquid flows out in 24 hours, which is replaced by the inflow of 1 liter of water. After how much time will the contents of the barrel be half wine and half water? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "69",
"style": "rule"
} | {
"index": "40069",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=x^{3}+ax^{2}+bx+a^{2}$ has an extremum of $10$ at $x=1$, find the value of $f(2)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "40070",
"split": "train"
} |
math | [
{
"content": "Let $\\{b_k\\}$ be a sequence of integers such that $b_1=2$ and $b_{m+n}=b_m+b_n+mn^2,$ for all positive integers $m$ and $n.$ Find $b_{12}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "98",
"style": "rule"
} | {
"index": "40071",
"split": "train"
} |
math | [
{
"content": "A fair die (a cube toy with 1, 2, 3, 4, 5, 6 dots on each face) is rolled twice. Let 'm' be the number of dots that appear on the first roll, and 'n' be the number of dots that appear on the second roll. The probability that m = kn (k ∈ N*) is _____. Please output the final answer within \\boxed{}... | {
"ground_truth": "\\frac{7}{18}",
"style": "rule"
} | {
"index": "40072",
"split": "train"
} |
math | [
{
"content": "If the real numbers x and y satisfy \\((x-3)^{2}+4(y-1)^{2}=4\\), find the maximum and minimum values of \\(\\frac{x+y-3}{x-y+1}\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "40073",
"split": "train"
} |
math | [
{
"content": "A novel is to be recorded onto compact discs and takes 528 minutes to read aloud. Each disc can now hold up to 45 minutes of reading. Assuming that the smallest possible number of discs is used and that each disc contains the same length of reading, calculate the number of minutes of reading that ... | {
"ground_truth": "44",
"style": "rule"
} | {
"index": "40074",
"split": "train"
} |
math | [
{
"content": "Let $c$ be a real number randomly selected from the interval $[-20,20]$. Then, $p$ and $q$ are two relatively prime positive integers such that $\\frac{p}{q}$ is the probability that the equation $x^4 + 36c^2 = (9c^2 - 15c)x^2$ has at least two distinct real solutions. Find the value of $p + q$. P... | {
"ground_truth": "29",
"style": "rule"
} | {
"index": "40075",
"split": "train"
} |
math | [
{
"content": "Suppose that $a$ is inversely proportional to $b$. Let $a_1,a_2$ be two nonzero values of $a$ such that $\\frac{a_1}{a_2}=\\frac{2}{3}$. Let the corresponding $b$ values be $b_1,b_2$. If $b_1,b_2$ are nonzero, find the value of $\\frac{b_1}{b_2}$. Please output the final answer within \\boxed{}... | {
"ground_truth": "\\frac{3}{2}",
"style": "rule"
} | {
"index": "40076",
"split": "train"
} |
math | [
{
"content": "Subtract $555.55$ from $888.88.$ Express the result as a decimal to the nearest hundredth. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "333.33",
"style": "rule"
} | {
"index": "40077",
"split": "train"
} |
math | [
{
"content": "A $3 \\times 3$ square is partitioned into $9$ unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated $90^{\\circ}$ clockwise about its center, and every white square in a position formerl... | {
"ground_truth": "\\frac{49}{512}",
"style": "rule"
} | {
"index": "40078",
"split": "train"
} |
math | [
{
"content": "At the New-Vasyuki currency exchange, 11 tugriks are traded for 14 dinars, 22 rupees for 21 dinars, 10 rupees for 3 thalers, and 5 crowns for 2 thalers. How many tugriks can be exchanged for 13 crowns? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "40079",
"split": "train"
} |
math | [
{
"content": "Alfred and Bonnie play a game in which they take turns tossing a fair coin. The winner of a game is the first person to obtain a head. Alfred and Bonnie play this game several times with the stipulation that the loser of a game goes first in the next game. Suppose that Alfred goes first in the fir... | {
"ground_truth": "93",
"style": "rule"
} | {
"index": "40080",
"split": "train"
} |
math | [
{
"content": "Let $a_n = \\frac{10^n-1}{9}$. Define $d_n$ to be the greatest common divisor of $a_n$ and $a_{n+1}$. What is the maximum possible value that $d_n$ can take on? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "40081",
"split": "train"
} |
math | [
{
"content": "What is the ones digit of $1^{2009} + 2^{2009} + 3^{2009} + \\cdots + 2009^{2009}?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "40082",
"split": "train"
} |
math | [
{
"content": "There are 55 points marked on a plane: the vertices of a regular 54-gon and its center. Petya wants to color a set of three marked points in red so that the colored points form the vertices of a regular triangle. In how many ways can Petya do this? Please output the final answer within \\boxed{}."... | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "40083",
"split": "train"
} |
math | [
{
"content": "What is the minimum (identical) number of pencils that need to be placed in each of the 6 boxes so that in any 4 boxes there are pencils of any of the 26 pre-specified colors (assuming there are enough pencils available)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "40084",
"split": "train"
} |
math | [
{
"content": "A positive integer $n$ not exceeding $100$ is chosen such that if $n\\le 60$, then the probability of choosing $n$ is $q$, and if $n > 60$, then the probability of choosing $n$ is $2q$. Find the probability that a perfect square is chosen.\n- **A)** $\\frac{1}{35}$\n- **B)** $\\frac{2}{35}$\n- **C... | {
"ground_truth": "\\frac{3}{35}",
"style": "rule"
} | {
"index": "40085",
"split": "train"
} |
math | [
{
"content": "The number of roots satisfying the equation $\\sqrt{5 - x} = x\\sqrt{5 - x}$ is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "40086",
"split": "train"
} |
math | [
{
"content": "You have a whole cake in your pantry. On your first trip to the pantry, you eat one-third of the cake. On each successive trip, you eat one-third of the remaining cake. After four trips to the pantry, what fractional part of the cake have you eaten? Please output the final answer within \\boxed{}.... | {
"ground_truth": "\\frac{40}{81}",
"style": "rule"
} | {
"index": "40087",
"split": "train"
} |
math | [
{
"content": "Given that $W,X,Y$ and $Z$ are four different digits selected from the set ${1,2,3,4,5,6,7,8,9}$, find the value of $\\frac{W}{X} + \\frac{Y}{Z}$ when it is as small as possible. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{25}{72}",
"style": "rule"
} | {
"index": "40088",
"split": "train"
} |
math | [
{
"content": "What is the least possible positive four-digit palindrome that is divisible by 4? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1881",
"style": "rule"
} | {
"index": "40089",
"split": "train"
} |
math | [
{
"content": "Determine the largest positive integer \\( n \\) with \\( n < 500 \\) for which \\( 6048 \\cdot 28^n \\) is a perfect cube (that is, it is equal to \\( m^3 \\) for some positive integer \\( m \\) ). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "497",
"style": "rule"
} | {
"index": "40090",
"split": "train"
} |
math | [
{
"content": "A line \\( y = -\\frac{2}{3}x + 6 \\) crosses the \\( x \\)-axis at \\( P \\) and the \\( y \\)-axis at \\( Q \\). Point \\( T(r,s) \\) is on the line segment \\( PQ \\). If the area of \\( \\triangle POQ \\) is four times the area of \\( \\triangle TOP \\), what is the value of \\( r+s \\)? Pleas... | {
"ground_truth": "8.25",
"style": "rule"
} | {
"index": "40091",
"split": "train"
} |
math | [
{
"content": "Given that $\\log_{10}\\sin x + \\log_{10}\\cos x= -2$ and that $\\log_{10}(\\sin x+\\cos x)=\\frac{1}{2}(\\log_{10}m-2)$, find $m$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "102",
"style": "rule"
} | {
"index": "40092",
"split": "train"
} |
math | [
{
"content": "There are 20 people in my club. 8 of them are left-handed. 15 of them like jazz music. 2 of them are right-handed and dislike jazz music. How many club members are left-handed and like jazz music? Assume people are either left-handed or right-handed, but not both. Please output the final answe... | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "40093",
"split": "train"
} |
math | [
{
"content": "The base of a pyramid is a square. The height of the pyramid intersects the diagonal of the base. Find the maximum volume of such a pyramid if the perimeter of the diagonal cross-section that contains the height of the pyramid is 5. Please output the final answer within \\boxed{}.",
"role": "u... | {
"ground_truth": "\\frac{\\sqrt{5}}{3}",
"style": "rule"
} | {
"index": "40094",
"split": "train"
} |
math | [
{
"content": "Let $A$ and $B$ be real numbers such that $\\frac{A}{x-5}+B(x+1)=\\frac{-3x^2+12x+22}{x-5}$. What is $A+B$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "40095",
"split": "train"
} |
math | [
{
"content": "Let \\( x \\) be a real number such that \\( x + \\frac{1}{x} = 5 \\). Define \\( S_m = x^m + \\frac{1}{x^m} \\). Determine the value of \\( S_6 \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12098",
"style": "rule"
} | {
"index": "40096",
"split": "train"
} |
math | [
{
"content": "The water tank in the diagram is in the shape of an inverted right circular cone. The radius of its base is 20 feet, and its height is 100 feet. The water in the tank fills 50% of the tank's total capacity. Find the height of the water in the tank, which can be expressed in the form \\( a\\sqrt[3]... | {
"ground_truth": "52",
"style": "rule"
} | {
"index": "40097",
"split": "train"
} |
math | [
{
"content": "A sequence $(a_n)$ is defined as follows: $a_1 = 1,$ $a_2 = \\frac{1}{2},$ and\n\\[a_n = \\frac{1 - a_{n - 1}}{2a_{n - 2}}\\]for all $n \\ge 3.$ Find $a_{120}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{20}{41}",
"style": "rule"
} | {
"index": "40098",
"split": "train"
} |
math | [
{
"content": "For $a$ a positive real number, let $x_{1}, x_{2}, x_{3}$ be the roots of the equation $x^{3}-a x^{2}+a x-a=0$. Determine the smallest possible value of $x_{1}^{3}+x_{2}^{3}+x_{3}^{3}-3 x_{1} x_{2} x_{3}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-4",
"style": "rule"
} | {
"index": "40099",
"split": "train"
} |
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