data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Let $\\{a_n\\}$ be a geometric sequence with a common ratio not equal to 1, and $a_4=16$. The sum of the first $n$ terms is denoted as $S_n$, and $5S_1$, $2S_2$, $S_3$ form an arithmetic sequence. \n(1) Find the general formula for the sequence $\\{a_n\\}$. \n(2) Let $b_n = \\frac{1}{\\log_{2}a_... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "7100",
"split": "train"
} |
math | [
{
"content": "How many subsets containing three different numbers can be selected from the set \\(\\{ 89,95,99,132, 166,173 \\}\\) so that the sum of the three numbers is even? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "7101",
"split": "train"
} |
math | [
{
"content": "An aluminum part and a copper part have the same volume. The density of aluminum is $\\rho_{A} = 2700 \\, \\text{kg/m}^3$, and the density of copper is $\\rho_{M} = 8900 \\, \\text{kg/m}^3$. Find the mass of the aluminum part, given that the masses of the parts differ by $\\Delta m = 60 \\, \\text... | {
"ground_truth": "26.13",
"style": "rule"
} | {
"index": "7102",
"split": "train"
} |
math | [
{
"content": "In the diagram, semicircles are constructed on diameters $\\overline{AB}$, $\\overline{AC}$, and $\\overline{CB}$, such that they are mutually tangent. Point $C$ is at one third the distance from $A$ to $B$, so $\\overline{AC} = \\frac{1}{3} \\overline{AB}$ and $\\overline{CB} = \\frac{2}{3} \\ove... | {
"ground_truth": "1:4",
"style": "rule"
} | {
"index": "7103",
"split": "train"
} |
math | [
{
"content": "Geoff and Trevor each roll a fair eight-sided die (the sides are labeled 1 through 8). What is the probability that the product of the numbers they roll is even or a prime number? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{7}{8}",
"style": "rule"
} | {
"index": "7104",
"split": "train"
} |
math | [
{
"content": "A tourist city was surveyed, and it was found that the number of tourists per day $f(t)$ (in ten thousand people) and the time $t$ (in days) within the past month (calculated as $30$ days) approximately satisfy the function relationship $f(t)=4+ \\frac {1}{t}$. The average consumption per person $... | {
"ground_truth": "403 \\frac {1}{3}",
"style": "rule"
} | {
"index": "7105",
"split": "train"
} |
math | [
{
"content": "Square pyramid $ABCDE$ has base $ABCD$, which measures $3$ cm on a side, and altitude $AE$ perpendicular to the base, which measures $6$ cm. Point $P$ lies on $BE$, one third of the way from $B$ to $E$; point $Q$ lies on $DE$, one third of the way from $D$ to $E$; and point $R$ lies on $CE$, two t... | {
"ground_truth": "2\\sqrt{2}",
"style": "rule"
} | {
"index": "7106",
"split": "train"
} |
math | [
{
"content": "In space, there are four spheres with radii 2, 2, 3, and 3. Each sphere is externally tangent to the other three spheres. Additionally, there is a smaller sphere that is externally tangent to all four of these spheres. Find the radius of this smaller sphere. Please output the final answer within \... | {
"ground_truth": "\\frac{6}{2 - \\sqrt{26}}",
"style": "rule"
} | {
"index": "7107",
"split": "train"
} |
math | [
{
"content": "Let $a$ be the sum of the numbers: $99 \\times 0.9$ $999 \\times 0.9$ $9999 \\times 0.9$ $\\vdots$ $999\\cdots 9 \\times 0.9$ where the final number in the list is $0.9$ times a number written as a string of $101$ digits all equal to $9$ .\nFind the sum of the digits in the number $a$ ... | {
"ground_truth": "891",
"style": "rule"
} | {
"index": "7108",
"split": "train"
} |
math | [
{
"content": "Given a moving point $M$ whose distance to the point $F(0,1)$ is equal to its distance to the line $y=-1$, the trajectory of point $M$ is denoted as $C$.\n\n(1) Find the equation of the trajectory $C$.\n\n(2) Let $P$ be a point on the line $l: x-y-2=0$. Construct two tangents $PA$ and $PB$ from po... | {
"ground_truth": "\\frac{9}{2}",
"style": "rule"
} | {
"index": "7109",
"split": "train"
} |
math | [
{
"content": "Given that line $l_{1}$ passes through points $A(m,1)$ and $B(-3,4)$, and line $l_{2}$ passes through points $C(1,m)$ and $D(-1,m+1)$, for what value of $m$ do the lines $l_{1}$ and $l_{2}$\n\n$(1)$ intersect at a right angle;\n\n$(2)$ are parallel;\n\n$(3)$ the angle of inclination of $l_{1}$ is ... | {
"ground_truth": "-6",
"style": "rule"
} | {
"index": "7110",
"split": "train"
} |
math | [
{
"content": "What is the sum of all the prime numbers between 1 and 10? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "7111",
"split": "train"
} |
math | [
{
"content": "If a person A is taller or heavier than another person B, then we note that A is *not worse than* B. In 100 persons, if someone is *not worse than* other 99 people, we call him *excellent boy*. What's the maximum value of the number of *excellent boys*? Please output the final answer within \\boxe... | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "7112",
"split": "train"
} |
math | [
{
"content": "A jar has $10$ red candies and $10$ blue candies. Terry picks two candies at random, then Mary picks two of the remaining candies at random. Given that the probability that they get the same color combination, irrespective of order, is $m/n,$ where $m$ and $n$ are relatively prime positive integer... | {
"ground_truth": "441",
"style": "rule"
} | {
"index": "7113",
"split": "train"
} |
math | [
{
"content": "In the circle with center $O$, the measure of $\\angle RIP$ is $36^\\circ$ and $OR=10$ cm. Find the number of centimeters in the length of arc $RP$. Express your answer in terms of $\\pi$. [asy]\ndraw((1,0)..(0,1)..(-1,0)..(0,-1)..cycle);\ndraw((0,0)--(4/5,3/5)--(-1,0)--(4/5,-3/5)--cycle);\nlabel(... | {
"ground_truth": "4\\pi",
"style": "rule"
} | {
"index": "7114",
"split": "train"
} |
math | [
{
"content": "Given the hyperbola $\\frac {x^{2}}{a^{2}}- \\frac {y^{2}}{b^{2}}=1$ ($a>0$, $b>0$), a line passing through the origin with an inclination angle of $\\frac {\\pi}{3}$ intersects the left and right branches of the hyperbola at points P and Q respectively. A circle with the segment PQ as its diamete... | {
"ground_truth": "\\sqrt{(\\sqrt{3}+1)^2}",
"style": "rule"
} | {
"index": "7115",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $M$ is the midpoint of $BC$ with $BM = 2$. $AM = c - b$. Find the maximum area of $\\triangle ABC$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{3}",
"style": "rule"
} | {
"index": "7116",
"split": "train"
} |
math | [
{
"content": "Here is a fairly simple puzzle: EH is four times greater than OY. AY is four times greater than OH. Find the sum of all four. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "150",
"style": "rule"
} | {
"index": "7117",
"split": "train"
} |
math | [
{
"content": "Let \\( A = (-4, 0) \\), \\( B = (-1, 2) \\), \\( C = (1, 2) \\), and \\( D = (4, 0) \\). Suppose that point \\( P \\) satisfies \n\\[ PA + PD = 10 \\quad \\text{and} \\quad PB + PC = 10. \\]\nFind the \\( y \\)-coordinate of \\( P \\), when simplified, which can be expressed in the form \\( \\fra... | {
"ground_truth": "35",
"style": "rule"
} | {
"index": "7118",
"split": "train"
} |
math | [
{
"content": "Compute the exact value of the expression $\\left|3 - |9 - \\pi^2| \\right|$. Write your answer using only integers, $\\pi$, and necessary mathematical operations, without any absolute value signs. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12 - \\pi^2",
"style": "rule"
} | {
"index": "7119",
"split": "train"
} |
math | [
{
"content": "Rectangle \\(PQRS\\) is divided into 60 identical squares, as shown. The length of the diagonal of each of these squares is 2. The length of \\(QS\\) is closest to Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "7120",
"split": "train"
} |
math | [
{
"content": "Points $A$ , $B$ , and $O$ lie in the plane such that $\\measuredangle AOB = 120^\\circ$ . Circle $\\omega_0$ with radius $6$ is constructed tangent to both $\\overrightarrow{OA}$ and $\\overrightarrow{OB}$ . For all $i \\ge 1$ , circle $\\omega_i$ with radius $r_i$ is constructed... | {
"ground_truth": "233",
"style": "rule"
} | {
"index": "7121",
"split": "train"
} |
math | [
{
"content": "Let $D$ be the circle with equation $x^2 - 8x + y^2 + 10y = -14$. A translation of the coordinate system moves the center of $D$ to $(1, -2)$. Determine $a+b+r$ after the translation, where $(a,b)$ is the center and $r$ is the radius of the translated circle. Please output the final answer within ... | {
"ground_truth": "-1 + \\sqrt{27}",
"style": "rule"
} | {
"index": "7122",
"split": "train"
} |
math | [
{
"content": "A car license plate contains three letters and three digits, for example, A123BE. The allowed letters are А, В, Е, К, М, Н, О, Р, С, Т, У, Х (a total of 12 letters) and all digits except the combination 000. Kira considers a license plate lucky if the second letter is a vowel, the second digit is ... | {
"ground_truth": "359999",
"style": "rule"
} | {
"index": "7123",
"split": "train"
} |
math | [
{
"content": "A shop advertises everything is \"half price in today's sale.\" In addition, a coupon gives a 20% discount on sale prices. Using the coupon, the price today represents what percentage off the original price? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "7124",
"split": "train"
} |
math | [
{
"content": "If \\(a\\) copies of a right-angled isosceles triangle with hypotenuse \\(\\sqrt{2} \\, \\mathrm{cm}\\) can be assembled to form a trapezium with perimeter equal to \\(b \\, \\mathrm{cm}\\), find the least possible value of \\(b\\). (Give the answer in surd form.) Please output the final answer wi... | {
"ground_truth": "4 + 2\\sqrt{2}",
"style": "rule"
} | {
"index": "7125",
"split": "train"
} |
math | [
{
"content": "Nine balls, numbered $1, 2, \\cdots, 9$, are placed in a bag. These balls are identical except for their numbers. Person A draws a ball from the bag, which has the number $a$. After placing it back, person B draws another ball from the bag, which has the number $b$. The probability that the inequa... | {
"ground_truth": "$\\frac{61}{81}$",
"style": "rule"
} | {
"index": "7126",
"split": "train"
} |
math | [
{
"content": "A regular tetrahedron has two spheres, one inscribed within it and the other circumscribed around it. Between each face of the tetrahedron and the circumscribed sphere, there are four smaller spheres centered on each face. Given a point \\( P \\) within the circumscribed sphere, the probability th... | {
"ground_truth": "0.2",
"style": "rule"
} | {
"index": "7127",
"split": "train"
} |
math | [
{
"content": "Simplify: $\\frac{{x}^{2}-4}{{x}^{2}-4x+4}+\\frac{x}{{x}^{2}-x}\\div \\frac{x-2}{x-1}$, then choose a value of $x$ from the integers in the range $-1\\leqslant x\\leqslant 3$ to substitute and evaluate. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{2}{3}",
"style": "rule"
} | {
"index": "7128",
"split": "train"
} |
math | [
{
"content": "The expression $\\cos 2x + \\cos 6x + \\cos 10x + \\cos 14x$ can be written in the equivalent form\n\\[a \\cos bx \\cos cx \\cos dx\\] for some positive integers $a$, $b$, $c$, and $d$. Find $a+b+c+d$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "7129",
"split": "train"
} |
math | [
{
"content": "A rectangular table of size \\( x \\) cm by 80 cm is covered with identical sheets of paper of size 5 cm by 8 cm. The first sheet is placed in the bottom-left corner, and each subsequent sheet is placed one centimeter higher and one centimeter to the right of the previous one. The last sheet is pl... | {
"ground_truth": "77",
"style": "rule"
} | {
"index": "7130",
"split": "train"
} |
math | [
{
"content": "A point \\((x, y)\\) is a distance of 14 units from the \\(x\\)-axis. It is a distance of 8 units from the point \\((1, 8)\\). Given that \\(x > 1\\), what is the distance \\(n\\) from this point to the origin? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "7131",
"split": "train"
} |
math | [
{
"content": "Find the number of integers $ c$ such that $ \\minus{}2007 \\leq c \\leq 2007$ and there exists an integer $ x$ such that $ x^2 \\plus{} c$ is a multiple of $ 2^{2007}$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "670",
"style": "rule"
} | {
"index": "7132",
"split": "train"
} |
math | [
{
"content": "By what common fraction does $0.\\overline{06}$ exceed $0.06$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{1650}",
"style": "rule"
} | {
"index": "7133",
"split": "train"
} |
math | [
{
"content": "Points $A_{1}$ and $C_{1}$ are located on the sides $BC$ and $AB$ of triangle $ABC$. Segments $AA_{1}$ and $CC_{1}$ intersect at point $M$.\n\nIn what ratio does line $BM$ divide side $AC$, if $AC_{1}: C_{1}B = 2: 3$ and $BA_{1}: A_{1}C = 1: 2$? Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "1:3",
"style": "rule"
} | {
"index": "7134",
"split": "train"
} |
math | [
{
"content": "Given a rhombus \\(ABCD\\) with diagonals equal to 3 cm and 4 cm. From the vertex of the obtuse angle \\(B\\), draw the altitudes \\(BE\\) and \\(BF\\). Calculate the area of the quadrilateral \\(BFDE\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4.32",
"style": "rule"
} | {
"index": "7135",
"split": "train"
} |
math | [
{
"content": "A five-digit integer will be chosen at random from all possible positive five-digit integers. What is the probability that the number's units digit will be less than 5? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "7136",
"split": "train"
} |
math | [
{
"content": "Evaluate the sum\n$$\n\\cos \\left(\\frac{2\\pi}{18}\\right) + \\cos \\left(\\frac{4\\pi}{18}\\right) + \\cdots + \\cos \\left(\\frac{34\\pi}{18}\\right).\n$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "7137",
"split": "train"
} |
math | [
{
"content": "Find all \\( t \\) such that \\( x-t \\) is a factor of \\( 10x^2 + 21x - 10 \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{5}{2}",
"style": "rule"
} | {
"index": "7138",
"split": "train"
} |
math | [
{
"content": "Given two linear functions $f(x)$ and $g(x)$ such that the graphs of $y=f(x)$ and $y=g(x)$ are parallel lines, not parallel to the coordinate axes. Find the minimum value of the function $(g(x))^{2}+f(x)$, if the minimum value of the function $(f(x))^{2}+g(x)$ is -6. Please output the final answer... | {
"ground_truth": "\\frac{11}{2}",
"style": "rule"
} | {
"index": "7139",
"split": "train"
} |
math | [
{
"content": "Find the number of positive integers $n$ that satisfy\n\\[(n - 2)(n - 4)(n - 6) \\dotsm (n - 98) < 0.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "47",
"style": "rule"
} | {
"index": "7140",
"split": "train"
} |
math | [
{
"content": "There is an unlimited supply of congruent equilateral triangles made of colored paper. Each triangle is a solid color with the same color on both sides of the paper. A large equilateral triangle is constructed from four of these paper triangles as shown. Two large triangles are considered distin... | {
"ground_truth": "336",
"style": "rule"
} | {
"index": "7141",
"split": "train"
} |
math | [
{
"content": "Find all real numbers \\(k\\) such that the inequality\n\\[ \na^{3} + b^{3} + c^{3} + d^{3} + 1 \\geq k(a + b + c + d)\n\\]\nholds for any \\(a, b, c, d \\in [-1, +\\infty)\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{4}",
"style": "rule"
} | {
"index": "7142",
"split": "train"
} |
math | [
{
"content": "Let \\(\\Gamma_{1}\\) and \\(\\Gamma_{2}\\) be two circles externally tangent to each other at \\(N\\) that are both internally tangent to \\(\\Gamma\\) at points \\(U\\) and \\(V\\), respectively. A common external tangent of \\(\\Gamma_{1}\\) and \\(\\Gamma_{2}\\) is tangent to \\(\\Gamma_{1}\\)... | {
"ground_truth": "\\frac{\\left(Rr_{1}+Rr_{2}-2r_{1}r_{2}\\right)2\\sqrt{r_{1}r_{2}}}{\\left|r_{1}-r_{2}\\right|\\sqrt{\\left(R-r_{1}\\right)\\left(R-r_{2}\\right)}}",
"style": "rule"
} | {
"index": "7143",
"split": "train"
} |
math | [
{
"content": "Given a four-digit number $\\overline{A B C D}$ that satisfies the following properties: $\\overline{A B}$, $\\overline{B C}$, and $\\overline{C D}$ are all perfect squares (a perfect square is a number that can be expressed as the square of an integer, such as $4 = 2^2$ and $81 = 9^2$). What is t... | {
"ground_truth": "13462",
"style": "rule"
} | {
"index": "7144",
"split": "train"
} |
math | [
{
"content": "At a large gathering hosted by Benjamin Franklin, each man shakes hands with every other attendee except their own spouse and except Benjamin Franklin, who is also present. No handshakes take place between women, and Franklin only shakes hands with the women. If 15 married couples attended the gat... | {
"ground_truth": "225",
"style": "rule"
} | {
"index": "7145",
"split": "train"
} |
math | [
{
"content": "If one of the 13 provinces or territories is chosen at random, what is the probability that it joined Canadian Confederation between 1890 and 1969? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$\\frac{4}{13}$",
"style": "rule"
} | {
"index": "7146",
"split": "train"
} |
math | [
{
"content": "Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 6.$ Find the minimum value of\n\\[\\frac{x^2 + y^2}{x + y} + \\frac{x^2 + z^2}{x + z} + \\frac{y^2 + z^2}{y + z}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "7147",
"split": "train"
} |
math | [
{
"content": "A circle with a radius of 6 cm is tangent to three sides of a rectangle. The area of the circle is half the area of the rectangle. What is the length of the longer side of the rectangle, in centimeters? Express your answer in terms of $\\pi$. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "6\\pi",
"style": "rule"
} | {
"index": "7148",
"split": "train"
} |
math | [
{
"content": "The regular octagon $ABCDEFGH$ has its center at $J$. Each of the vertices and the center are to be associated with one of the digits $1$ through $9$, with each digit used once, in such a way that the sums of the numbers on the lines $AJE$, $BJF$, $CJG$, and $DJH$ are equal. In how many ways can t... | {
"ground_truth": "1152",
"style": "rule"
} | {
"index": "7149",
"split": "train"
} |
math | [
{
"content": "A certain product in a shopping mall sells an average of 30 items per day, with a profit of 50 yuan per item. In order to reduce inventory quickly, the mall decides to take appropriate price reduction measures. After investigation, it was found that for each item, for every 1 yuan reduction in pri... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "7150",
"split": "train"
} |
math | [
{
"content": "Given numbers $5, 6, 7, 8, 9, 10, 11, 12, 13$ are written in a $3\\times3$ array, with the condition that two consecutive numbers must share an edge. If the sum of the numbers in the four corners is $32$, calculate the number in the center of the array. Please output the final answer within \\boxe... | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "7151",
"split": "train"
} |
math | [
{
"content": "Find the sum $$\\frac{2^{1}}{4^{1}-1}+\\frac{2^{2}}{4^{2}-1}+\\frac{2^{4}}{4^{4}-1}+\\frac{2^{8}}{4^{8}-1}+\\cdots$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "7152",
"split": "train"
} |
math | [
{
"content": "Out of a batch of 5000 electric lamps, a self-random non-repeating sample of 300 was selected. The average burning time of the lamps in the sample was found to be 1450 hours, and the variance was 40,000. What is the probability that the average burning time of the lamps in the entire batch falls w... | {
"ground_truth": "0.99964",
"style": "rule"
} | {
"index": "7153",
"split": "train"
} |
math | [
{
"content": "Determine the sum of all real numbers $x$ satisfying\n\\[(x^2-4x+2)^{x^2-5x+2} = 1.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "7154",
"split": "train"
} |
math | [
{
"content": "In square $ABCE$, $AF=2FE$ and $CD=2DE$. What is the ratio of the area of $\\triangle BFD$ to the area of square $ABCE$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{18}",
"style": "rule"
} | {
"index": "7155",
"split": "train"
} |
math | [
{
"content": "A lock code is made up of four digits that satisfy the following rules: - At least one digit is a 4, but neither the second digit nor the fourth digit is a 4. - Exactly one digit is a 2, but the first digit is not 2. - Exactly one digit is a 7. - The code includes a 1, or the code includes a 6, or... | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "7156",
"split": "train"
} |
math | [
{
"content": "What is the units digit of $13^{2003}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "7157",
"split": "train"
} |
math | [
{
"content": "When simplified, $(x^{-1}+y^{-1})^{-1}$ is equal to: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{xy}{x+y}",
"style": "rule"
} | {
"index": "7158",
"split": "train"
} |
math | [
{
"content": "Before Ashley started a three-hour drive, her car's odometer reading was 29792, a palindrome. (A palindrome is a number that reads the same way from left to right as it does from right to left). At her destination, the odometer reading was another palindrome. If Ashley never exceeded the speed lim... | {
"ground_truth": "70 \\frac{1}{3}",
"style": "rule"
} | {
"index": "7159",
"split": "train"
} |
math | [
{
"content": "If $2^n$ divides $5^{256} - 1$ , what is the largest possible value of $n$ ? $ \n\\textbf{a)}\\ 8\n\\qquad\\textbf{b)}\\ 10\n\\qquad\\textbf{c)}\\ 11\n\\qquad\\textbf{d)}\\ 12\n\\qquad\\textbf{e)}\\ \\text{None of above}\n$ Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "7160",
"split": "train"
} |
math | [
{
"content": "In triangle \\(ABC\\), the sides \\(a\\), \\(b\\), and \\(c\\) are opposite to angles \\(A\\), \\(B\\), and \\(C\\) respectively, with \\(a-2b=0\\).\n1. If \\(B= \\dfrac{\\pi}{6}\\), find \\(C\\).\n2. If \\(C= \\dfrac{2}{3}\\pi\\) and \\(c=14\\), find the area of \\(\\triangle ABC\\). Please outpu... | {
"ground_truth": "14 \\sqrt{3}",
"style": "rule"
} | {
"index": "7161",
"split": "train"
} |
math | [
{
"content": "In the diagram, the area of square \\( QRST \\) is 36. Also, the length of \\( PQ \\) is one-half of the length of \\( QR \\). What is the perimeter of rectangle \\( PRSU \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "7162",
"split": "train"
} |
math | [
{
"content": "Given that \\( P \\) is the vertex of a right circular cone with an isosceles right triangle as its axial cross-section, \\( PA \\) is a generatrix of the cone, and \\( B \\) is a point on the base of the cone. Find the maximum value of \\(\\frac{PA + AB}{PB}\\). Please output the final answer wit... | {
"ground_truth": "\\sqrt{4 + 2\\sqrt{2}}",
"style": "rule"
} | {
"index": "7163",
"split": "train"
} |
math | [
{
"content": "If $x^2+x+4 = y - 4$ and $x = -7$, what is the value of $y$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "7164",
"split": "train"
} |
math | [
{
"content": "Sherry and Val are playing a game. Sherry has a deck containing 2011 red cards and 2012 black cards, shuffled randomly. Sherry flips these cards over one at a time, and before she flips each card over, Val guesses whether it is red or black. If Val guesses correctly, she wins 1 dollar; otherwise, ... | {
"ground_truth": "\\frac{1}{4023}",
"style": "rule"
} | {
"index": "7165",
"split": "train"
} |
math | [
{
"content": "What is the perimeter of the figure shown? [asy]\n\ndraw((0,0)--(0,2)--(1,2+sqrt(3))--(2,2)--(4,2)--(4,0)--(3,-sqrt(3))--(2,0)--cycle);\n\ndraw((-.2,1)--(.2,1));\n\ndraw((3.8,1)--(4.2,1));\n\ndraw((1,-.2)--(1,.2));\n\ndraw((3,1.8)--(3,2.2));\n\ndraw((0.15,0)--(0.15,0.15)--(0,0.15));\n\ndraw((3.85,... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "7166",
"split": "train"
} |
math | [
{
"content": "Find the maximal $x$ such that the expression $4^{27} + 4^{1000} + 4^x$ is the exact square. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1972",
"style": "rule"
} | {
"index": "7167",
"split": "train"
} |
math | [
{
"content": "The Lakers are playing the Celtics in the NBA finals. To win the NBA finals, a team must win 4 games before the other team does. If the Celtics win each game with probability $\\dfrac{2}{3}$ and there are no ties, what is the probability that the Lakers will win the NBA finals but that the contest... | {
"ground_truth": "\\frac{160}{2187}",
"style": "rule"
} | {
"index": "7168",
"split": "train"
} |
math | [
{
"content": "Simplify\n\\[\\frac{1}{\\log_{15} 2 + 1} + \\frac{1}{\\log_{10} 3 + 1} + \\frac{1}{\\log_6 5 + 1}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "7169",
"split": "train"
} |
math | [
{
"content": "Find all real values of $x$ that satisfy $\\frac{1}{x+1} + \\frac{3}{x+7} \\ge \\frac23.$ (Give your answer in interval notation.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(-7, -4] \\cup (-1, 2]",
"style": "rule"
} | {
"index": "7170",
"split": "train"
} |
math | [
{
"content": "What is the sum of the tens digit and the units digit in the decimal representation of $9^{2004}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "7171",
"split": "train"
} |
math | [
{
"content": "Continue the problem statement \"Petya and Kolya went to the forest to pick mushrooms. In the first hour, Petya collected twice as many mushrooms as Kolya...\" so that its solution is determined by the expression $30: 3+4$. Please output the final answer within \\boxed{}.",
"role": "user"
}
... | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "7172",
"split": "train"
} |
math | [
{
"content": "Given that the lateral surface of a cone is the semicircle with a radius of $2\\sqrt{3}$, find the radius of the base of the cone. If the vertex of the cone and the circumference of its base lie on the surface of a sphere $O$, determine the volume of the sphere. Please output the final answer with... | {
"ground_truth": "\\frac{32\\pi}{3}",
"style": "rule"
} | {
"index": "7173",
"split": "train"
} |
math | [
{
"content": "Find the minimum value of $| \\sin x + \\cos x + \\tan x + \\cot x + \\sec x + \\csc x |$ for real numbers $x$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{2} - 1",
"style": "rule"
} | {
"index": "7174",
"split": "train"
} |
math | [
{
"content": "Paint is to be mixed so that the ratio of red paint to white paint is 3 to 2. If Ben wants to make 30 cans of the mixture and all cans hold the same volume of paint, how many cans of red paint will he need? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "7175",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=ax^{3}+bx+1$, and $f(-2)=3$, find $f(2)=$ _____ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "7176",
"split": "train"
} |
math | [
{
"content": "What is the smallest positive integer that has eight positive odd integer divisors and sixteen positive even integer divisors? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "420",
"style": "rule"
} | {
"index": "7177",
"split": "train"
} |
math | [
{
"content": "Given a frustum $ABCD-A_{1}B_{1}C_{1}D_{1}$ with a rectangular lower base, where $AB=2A_{1}B_{1}$, the height is $3$, and the volume of the frustum is $63$, find the minimum value of the perimeter of the upper base $A_{1}B_{1}C_{1}D_{1}$. Please output the final answer within \\boxed{}.",
"rol... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "7178",
"split": "train"
} |
math | [
{
"content": "Triangle $ABC$ is right angled at $A$ . The circle with center $A$ and radius $AB$ cuts $BC$ and $AC$ internally at $D$ and $E$ respectively. If $BD = 20$ and $DC = 16$ , determine $AC^2$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "936",
"style": "rule"
} | {
"index": "7179",
"split": "train"
} |
math | [
{
"content": "How many integers $-13 \\leq n \\leq 13$ satisfy $(n-3)(n+5)(n+7) < 0$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "7180",
"split": "train"
} |
math | [
{
"content": "In the triangle shown, for $\\angle A$ to be the largest angle of the triangle, it must be that $m<x<n$. What is the least possible value of $n-m$, expressed as a common fraction? [asy]\ndraw((0,0)--(1,0)--(.4,.5)--cycle);\nlabel(\"$A$\",(.4,.5),N); label(\"$B$\",(1,0),SE); label(\"$C$\",(0,0),SW)... | {
"ground_truth": "\\frac{17}{6}",
"style": "rule"
} | {
"index": "7181",
"split": "train"
} |
math | [
{
"content": "Calculate the following expressions:\n1. $(-51) + (-37)$;\n2. $(+2) + (-11)$;\n3. $(-12) + (+12)$;\n4. $8 - 14$;\n5. $15 - (-8)$;\n6. $(-3.4) + 4.3$;\n7. $|-2\\frac{1}{4}| + (-\\frac{1}{2})$;\n8. $-4 \\times 1\\frac{1}{2}$;\n9. $-3 \\times (-6)$. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "7182",
"split": "train"
} |
math | [
{
"content": "Let $P$ be the number to partition 2013 into an ordered tuple of prime numbers? What is $\\log _{2}(P)$? If your answer is $A$ and the correct answer is $C$, then your score on this problem will be $\\left\\lfloor\\frac{125}{2}\\left(\\min \\left(\\frac{C}{A}, \\frac{A}{C}\\right)-\\frac{3}{5}\\ri... | {
"ground_truth": "614.519...",
"style": "rule"
} | {
"index": "7183",
"split": "train"
} |
math | [
{
"content": "Five test scores have a mean (average score) of $90$, a median (middle score) of $91$ and a mode (most frequent score) of $94$. The sum of the two lowest test scores is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "171",
"style": "rule"
} | {
"index": "7184",
"split": "train"
} |
math | [
{
"content": "Three lines are drawn parallel to each of the three sides of $\\triangle ABC$ so that the three lines intersect in the interior of $ABC$ . The resulting three smaller triangles have areas $1$ , $4$ , and $9$ . Find the area of $\\triangle ABC$ .\n\n[asy]\ndefaultpen(linewidth(0.7)); size(12... | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "7185",
"split": "train"
} |
math | [
{
"content": "Four princesses thought of two-digit numbers, and Ivan thought of a four-digit number. After they wrote their numbers in a row in some order, the result was 132040530321. Find Ivan's number. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5303",
"style": "rule"
} | {
"index": "7186",
"split": "train"
} |
math | [
{
"content": "A person named Jia and their four colleagues each own a car with license plates ending in 9, 0, 2, 1, and 5, respectively. To comply with the local traffic restriction rules from the 5th to the 9th day of a certain month (allowing cars with odd-ending numbers on odd days and even-ending numbers on... | {
"ground_truth": "80",
"style": "rule"
} | {
"index": "7187",
"split": "train"
} |
math | [
{
"content": "A secret facility is in the shape of a rectangle measuring $200 \\times 300$ meters. There is a guard at each of the four corners outside the facility. An intruder approached the perimeter of the secret facility from the outside, and all the guards ran towards the intruder by the shortest paths al... | {
"ground_truth": "150",
"style": "rule"
} | {
"index": "7188",
"split": "train"
} |
math | [
{
"content": "In the number $2016 * * * * 02 *$, you need to replace each of the 5 asterisks with any of the digits $0, 2, 4, 6, 7, 8$ (digits may repeat) so that the resulting 11-digit number is divisible by 6. How many ways can this be done? Please output the final answer within \\boxed{}.",
"role": "user... | {
"ground_truth": "2160",
"style": "rule"
} | {
"index": "7189",
"split": "train"
} |
math | [
{
"content": "In Perfectville, the streets are all $20$ feet wide and the blocks they enclose are all squares of side length $400$ feet, as shown. Sarah runs around the block on the $400$-foot side of the street, while Sam runs on the opposite side of the street. How many more feet than Sarah does Sam run for... | {
"ground_truth": "160",
"style": "rule"
} | {
"index": "7190",
"split": "train"
} |
math | [
{
"content": "Alice and Bob take turns removing balls from a bag containing 10 black balls and 10 white balls, with Alice going first. Alice always removes a black ball if there is one, while Bob removes one of the remaining balls uniformly at random. Once all balls have been removed, the expected number of bla... | {
"ground_truth": "4519",
"style": "rule"
} | {
"index": "7191",
"split": "train"
} |
math | [
{
"content": "The digits 1, 3, 4, and 5 are arranged randomly to form a four-digit number. What is the probability that the number is odd? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{4}",
"style": "rule"
} | {
"index": "7192",
"split": "train"
} |
math | [
{
"content": "A function, $f$, has $f(2)=5$ and $f(3)=7$. In addition, $f$ has the property that $f(m)+f(n)=f(mn)$ for all positive integers $m$ and $n$. What is the value of $f(12)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "7193",
"split": "train"
} |
math | [
{
"content": "Bill can buy mags, migs, and mogs for $\\$3$, $\\$4$, and $\\$8$ each, respectively. What is the largest number of mogs he can purchase if he must buy at least one of each item and will spend exactly $\\$100$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "7194",
"split": "train"
} |
math | [
{
"content": "A company purchases 400 tons of a certain type of goods annually. Each purchase is of $x$ tons, and the freight cost is 40,000 yuan per shipment. The annual total storage cost is 4$x$ million yuan. To minimize the sum of the annual freight cost and the total storage cost, find the value of $x$. Pl... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "7195",
"split": "train"
} |
math | [
{
"content": "For the quadrilateral shown, how many different whole numbers could be the length of the diagonal represented by the dashed line?\n\n[asy]\ndraw((0,0)--(5,5)--(12,1)--(7,-8)--cycle,linewidth(0.7));\ndraw((0,0)--(12,1),dashed);\nlabel(\"8\",(2.5,2.5),NW);\nlabel(\"10\",(8.5,3),NE);\nlabel(\"16\",(9... | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "7196",
"split": "train"
} |
math | [
{
"content": "If the ratio of the legs of a right triangle is $3: 4$, then the ratio of the corresponding segments of the hypotenuse made by a perpendicular upon it from the vertex is:\nA) $\\frac{16}{9}$\nB) $\\frac{9}{16}$\nC) $\\frac{3}{4}$\nD) $\\frac{4}{3}$ Please output the final answer within \\boxed{}."... | {
"ground_truth": "\\frac{16}{9}",
"style": "rule"
} | {
"index": "7197",
"split": "train"
} |
math | [
{
"content": "Find the maximum value of the expression \\( (\\sin x + \\sin 2y + \\sin 3z)(\\cos x + \\cos 2y + \\cos 3z) \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4.5",
"style": "rule"
} | {
"index": "7198",
"split": "train"
} |
math | [
{
"content": "Joe has exactly enough paint to paint the surface (excluding the bases) of a cylinder with radius 3 and height 4. It turns out this is also exactly enough paint to paint the entire surface of a cube. The volume of this cube is \\( \\frac{48}{\\sqrt{K}} \\). What is \\( K \\)? Please output the fin... | {
"ground_truth": "\\frac{36}{\\pi^3}",
"style": "rule"
} | {
"index": "7199",
"split": "train"
} |
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