data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Find the area of the triangle with vertices $(6,5,3),$ $(3,3,1),$ and $(15,11,9).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 30649,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ $c$ be the roots of $3x^3 - 3x^2 + 11x - 8 = 0.$ Find $ab + ac + bc.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{3}"
} | {
"split": "train",
"index": 30650,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a cultural performance, there are already 10 programs arranged in the program list. Now, 3 more programs are to be added, with the requirement that the relative order of the originally scheduled 10 programs remains unchanged. How many different arrangements are there for the... | math | {
"style": "rule",
"ground_truth": "1716"
} | {
"split": "train",
"index": 30651,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence $\\{a\\_n\\}$ that satisfies: $(a\\_1=-13$, $a\\_6+a\\_8=-2)$, and $(a_{n-1}=2a\\_n-a_{n+1}\\ (n\\geqslant 2))$, find the sum of the first 13 terms of the sequence $\\left\\{\\frac{1}{a\\_n a_{n+1}}\\right\\}$. Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{13}"
} | {
"split": "train",
"index": 30652,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Draw a $2004 \\times 2004$ array of points. What is the largest integer $n$ for which it is possible to draw a convex $n$-gon whose vertices are chosen from the points in the array? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "561"
} | {
"split": "train",
"index": 30653,
"solution": "\nTo determine the largest integer \\( n \\) for which it is possible to draw a convex \\( n \\)-gon whose vertices are chosen from the points in a \\( 2004 \\times 2004 \\) array, we need to consider the properties of the convex hull and the arrangement of points.... |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all integer solutions to $|n| < |n-3| < 9$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-14"
} | {
"split": "train",
"index": 30654,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, line $l_{1}$: $kx-y+2=0$ intersects with line $l_{2}$: $x+ky-2=0$ at point $P$. When the real number $k$ varies, the maximum distance from point $P$ to the line $x-y-4=0$ is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{2}"
} | {
"split": "train",
"index": 30655,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, $\\triangle PQR$ is right-angled at $P$ and $\\angle PRQ=\\theta$. A circle with center $P$ is drawn passing through $Q$. The circle intersects $PR$ at $S$ and $QR$ at $T$. If $QT=8$ and $TR=10$, determine the value of $\\cos \\theta$. Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{7}}{3}"
} | {
"split": "train",
"index": 30656,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An isosceles right triangle is removed from each corner of a square piece of paper to form a rectangle. If $AB = 15$ units in the new configuration, what is the combined area of the four removed triangles? Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "112.5"
} | {
"split": "train",
"index": 30657,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ $c$ be complex numbers such that\n\\begin{align*}\nab + 4b &= -16, \\\\\nbc + 4c &= -16, \\\\\nca + 4a &= -16.\n\\end{align*}Enter all possible values of $abc,$ separated by commas. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 30658,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of solutions to the equation\n\\[\\tan (7 \\pi \\cos \\theta) = \\cot (7 \\pi \\sin \\theta)\\]\nwhere $\\theta \\in (0, 4 \\pi).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 30659,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that two congruent 30°-60°-90° triangles with hypotenuses of 12 are overlapped such that their hypotenuses exactly coincide, calculate the area of the overlapping region. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9 \\sqrt{3}"
} | {
"split": "train",
"index": 30660,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the number of ordered pairs of integers $(a, b)$, with $2 \\leq a, b \\leq 2021$, that satisfy the equation $$a^{\\log _{b}\\left(a^{-4}\\right)}=b^{\\log _{a}\\left(b a^{-3}\\right)}.$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "43"
} | {
"split": "train",
"index": 30661,
"solution": "Taking $\\log _{a}$ of both sides and simplifying gives $$-4 \\log _{b} a=\\left(\\log _{a} b\\right)^{2}-3 \\log _{a} b$$ Plugging in $x=\\log _{a} b$ and using $\\log _{b} a=\\frac{1}{\\log _{a} b}$ gives $$x^{3}-3 x^{2}+4=0$$ We can factor the polynomial as $(x-... |
deepscaler | [
{
"role": "user",
"content": "How many positive integers $n$ less than 150 have a corresponding integer $m$ not divisible by 3 such that the roots of $x^2-nx+m=0$ are consecutive positive integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 30662,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Brachycephalus frogs have three toes on each foot and two fingers on each hand. The common frog has five toes on each foot and four fingers on each hand. Some Brachycephalus and common frogs are in a bucket. Each frog has all its fingers and toes. Between them they have 122 toe... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 30663,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A number composed of ten million, three hundred thousand, and fifty is written as \\_\\_\\_\\_\\_\\_, and this number is read as \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10300050"
} | {
"split": "train",
"index": 30664,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest constant $C$ so that for all real numbers $x$, $y$, and $z$,\n\\[x^2 + y^2 + z^3 + 1 \\ge C(x + y + z).\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 30665,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Polina custom makes jewelry for a jewelry store. Each piece of jewelry consists of a chain, a stone, and a pendant. The chains can be silver, gold, or iron. Polina has stones - cubic zirconia, emerald, quartz - and pendants in the shape of a star, sun, and moon. Polina is happy... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 30666,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the four roots of the equation $(x^{2}-mx+27)(x^{2}-nx+27)=0$ form a geometric sequence with 1 as the first term, find $|m-n|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 30667,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many distinct four-digit numbers are divisible by 5 and have 45 as their last two digits? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 30668,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nA circle touches the extensions of two sides \\(AB\\) and \\(AD\\) of square \\(ABCD\\), and the point of tangency cuts off a segment of length \\(2 + \\sqrt{5 - \\sqrt{5}}\\) cm from vertex \\(A\\). From point \\(C\\), two tangents are drawn to this circle. Find the side len... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{\\sqrt{5} - 1} \\cdot \\sqrt[4]{125}}{5}"
} | {
"split": "train",
"index": 30669,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two real numbers $1<p<q$ so that $\\frac{1}{p} + \\frac{1}{q} = 1$ and $pq = \\frac{9}{2}$, what is $q$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "q = 3"
} | {
"split": "train",
"index": 30670,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are $27$ unit cubes. We are marking one point on each of the two opposing faces, two points on each of the other two opposing faces, and three points on each of the remaining two opposing faces of each cube. We are constructing a $3\\times 3 \\times 3$ cube with these... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 30671,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We need to arrange the performance order for 4 singing programs and 2 skit programs. The requirement is that there must be exactly 3 singing programs between the 2 skit programs. The number of possible arrangements is \\_\\_\\_\\_\\_\\_ . (Answer with a number) Let's think step... | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 30672,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A six-digit number is formed by the digits 1, 2, 3, 4, with two pairs of repeating digits, where one pair of repeating digits is not adjacent, and the other pair is adjacent. Calculate the number of such six-digit numbers. Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "432"
} | {
"split": "train",
"index": 30673,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Harry Potter can do any of the three tricks arbitrary number of times: $i)$ switch $1$ plum and $1$ pear with $2$ apples $ii)$ switch $1$ pear and $1$ apple with $3$ plums $iii)$ switch $1$ apple and $1$ plum with $4$ pears \nIn the beginning, Harry had ... | math | {
"style": "rule",
"ground_truth": "2025"
} | {
"split": "train",
"index": 30674,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two spheres with radii $36$ and one sphere with radius $13$ are each externally tangent to the other two spheres and to two different planes $\\mathcal{P}$ and $\\mathcal{Q}$. The intersection of planes $\\mathcal{P}$ and $\\mathcal{Q}$ is the line $\\ell$. The distance from li... | math | {
"style": "rule",
"ground_truth": "335"
} | {
"split": "train",
"index": 30675,
"solution": "The isosceles triangle of centers $O_1 O_2 O$ ($O$ is the center of sphere of radii $13$) has sides $O_1 O = O_2 O = 36 + 13 = 49,$ and $O_1 O_2 = 36 + 36 = 72.$\n Let $N$ be the midpoint $O_1 O_2$.\n The isosceles triangle of points of tangency $T_1 T_2 T$ has sid... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, point $M$ lies inside the triangle such that $\\angle MBA = 30^\\circ$ and $\\angle MAB = 10^\\circ$. Given that $\\angle ACB = 80^\\circ$ and $AC = BC$, find $\\angle AMC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 30676,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Grunters play the Screamers 6 times. The Grunters have a 60% chance of winning any given game. If a game goes to overtime, the probability of the Grunters winning changes to 50%. There is a 10% chance that any game will go into overtime. What is the probability that the Gru... | math | {
"style": "rule",
"ground_truth": "\\frac{823543}{10000000}"
} | {
"split": "train",
"index": 30677,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "License plates from different states follow different alpha-numeric formats, which dictate which characters of a plate must be letters and which must be numbers. Florida has license plates with an alpha-numeric format like the one pictured. North Dakota, on the other hand, has ... | math | {
"style": "rule",
"ground_truth": "28121600"
} | {
"split": "train",
"index": 30678,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let us call a ticket with a number from 000000 to 999999 excellent if the difference between some two neighboring digits of its number is 5. Find the number of excellent tickets. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "409510"
} | {
"split": "train",
"index": 30679,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that Jo and Blair take turns counting from 1, with Jo adding 2 to the last number said and Blair subtracting 1 from the last number said, determine the 53rd number said. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "79"
} | {
"split": "train",
"index": 30680,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define $H_n = 1+\\frac{1}{2}+\\cdots+\\frac{1}{n}$ . Let the sum of all $H_n$ that are terminating in base 10 be $S$ . If $S = m/n$ where m and n are relatively prime positive integers, find $100m+n$ .\n\n*Proposed by Lewis Chen* Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "9920"
} | {
"split": "train",
"index": 30681,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be a set of $6$ integers taken from $\\{1,2,\\dots,12\\}$ with the property that if $a$ and $b$ are elements of $S$ with $a<b$, then $b$ is not a multiple of $a$. What is the least possible value of an element in $S$? Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 30682,
"solution": "To solve this problem, we need to ensure that no element in the set $S$ is a multiple of any other element in $S$. We start by partitioning the set $\\{1, 2, \\dots, 12\\}$ into subsets where each element in a subset is a multiple of the smallest element in that su... |
deepscaler | [
{
"role": "user",
"content": "Simplify: $i^{0}+i^{1}+\\cdots+i^{2009}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1+i"
} | {
"split": "train",
"index": 30683,
"solution": "By the geometric series formula, the sum is equal to $\\frac{i^{2010}-1}{i-1}=\\frac{-2}{i-1}=1+i$."
} |
deepscaler | [
{
"role": "user",
"content": "There are two types of camels: dromedary camels with one hump on their back and Bactrian camels with two humps. Dromedary camels are taller, with longer limbs, and can walk and run in the desert; Bactrian camels have shorter and thicker limbs, suitable for walking in deserts an... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 30684,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "One of the roots of\n\\[ax^3 + 3x^2 + bx - 65 = 0,\\]is $-2 - 3i,$ where $a$ and $b$ are real numbers. Find the real root of this cubic polynomial. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"split": "train",
"index": 30685,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If a point $(-4,a)$ lies on the terminal side of an angle of $600^{\\circ}$, determine the value of $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-4 \\sqrt{3}"
} | {
"split": "train",
"index": 30686,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function \\(f(x) = 5x^2 - 15x - 2\\) has a minimum value when x is negative. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-13.25"
} | {
"split": "train",
"index": 30687,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When the length of a rectangle is increased by $20\\%$ and the width increased by $10\\%$, by what percent is the area increased? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32 \\%"
} | {
"split": "train",
"index": 30688,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the operation $x*y$ be defined as $x*y = (x+1)(y+1)$. The operation $x^{*2}$ is defined as $x^{*2} = x*x$. Calculate the value of the polynomial $3*(x^{*2}) - 2*x + 1$ when $x=2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 30689,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x - y = 6$ and $x + y = 12$, what is the value of $y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 30690,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\sin \\alpha$ is a root of the equation $5x^{2}-7x-6=0$, find:\n$(1)$ The value of $\\frac {\\cos (2\\pi-\\alpha)\\cos (\\pi+\\alpha)\\tan ^{2}(2\\pi-\\alpha)}{\\cos ( \\frac {\\pi}{2}+\\alpha)\\sin (2\\pi-\\alpha)\\cot ^{2}(\\pi-\\alpha)}$.\n$(2)$ In $\\triangle A... | math | {
"style": "rule",
"ground_truth": "-2- \\sqrt {3}"
} | {
"split": "train",
"index": 30691,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the coefficient of the $x^{3}$ term in the expansion of $(x^{2}-x+1)^{10}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-210"
} | {
"split": "train",
"index": 30692,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A man, standing on a lawn, is wearing a circular sombrero of radius 3 feet. The hat blocks the sunlight, causing the grass directly under it to die instantly. If the man walks in a circle of radius 5 feet, what area of dead grass will result? Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "60\\pi"
} | {
"split": "train",
"index": 30693,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the least common multiple of 6, 8, and 10? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 30694,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sphere is inscribed in a right circular cylinder. The height of the cylinder is 12 inches, and the diameter of its base is 10 inches. Find the volume of the inscribed sphere. Express your answer in terms of $\\pi$. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "\\frac{500}{3} \\pi"
} | {
"split": "train",
"index": 30695,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Twelve chess players played a round-robin tournament. Each player then wrote 12 lists. In the first list, only the player himself was included, and in the $(k+1)$-th list, the players included those who were in the $k$-th list as well as those whom they defeated. It turned out ... | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 30696,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the plane containing $\\triangle PAD$ is perpendicular to the plane containing rectangle $ABCD$, and $PA = PD = AB = 2$, with $\\angle APD = 60^\\circ$. If points $P, A, B, C, D$ all lie on the same sphere, find the surface area of this sphere. Let's think step by st... | math | {
"style": "rule",
"ground_truth": "\\frac{28}{3}\\pi"
} | {
"split": "train",
"index": 30697,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers $a_1,$ $a_2,$ $a_3,$ $b_1,$ $b_2,$ $b_3,$ $c_1,$ $c_2,$ $c_3$ are equal to the numbers $1,$ $2,$ $3,$ $\\dots,$ $9$ in some order. Find the smallest possible value of\n\\[a_1 a_2 a_3 + b_1 b_2 b_3 + c_1 c_2 c_3.\\] Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "214"
} | {
"split": "train",
"index": 30698,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify the product \\[\\frac{9}{3}\\cdot\\frac{15}{9}\\cdot\\frac{21}{15} \\dotsm \\frac{3n+6}{3n} \\dotsm \\frac{3003}{2997}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1001"
} | {
"split": "train",
"index": 30699,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A water tower in the shape of a cylinder has radius 10 m and height 30 m. A spiral staircase, with constant slope, circles once around the outside of the water tower. A vertical ladder of height 5 m then extends to the top of the tower. What is the total distance along the stai... | math | {
"style": "rule",
"ground_truth": "72.6 \\mathrm{~m}"
} | {
"split": "train",
"index": 30700,
"solution": "To calculate the total distance, we add the length of the vertical ladder (5 m) to the length of the spiral staircase. The spiral staircase wraps once around the tower. Since the tower has radius 10 m, then its circumference is $2 \\times \\pi \\times 10=20 \\pi \\... |
deepscaler | [
{
"role": "user",
"content": "Given a constant function on the interval $(0,1)$, $f(x)$, which satisfies: when $x \\notin \\mathbf{Q}$, $f(x)=0$; and when $x=\\frac{p}{q}$ (with $p, q$ being integers, $(p, q)=1, 0<p<q$), $f(x)=\\frac{p+1}{q}$. Determine the maximum value of $f(x)$ on the interval $\\left(\\... | math | {
"style": "rule",
"ground_truth": "\\frac{16}{17}"
} | {
"split": "train",
"index": 30701,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Circle $I$ passes through the center of, and is tangent to, circle $II$. The area of circle $I$ is $4$ square inches. \nThen the area of circle $II$, in square inches, is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 30702,
"solution": "1. **Understanding the Relationship Between the Circles**:\n Given that Circle $I$ is tangent to Circle $II$ and passes through the center of Circle $II$, we can deduce that the diameter of Circle $I$ is equal to the radius of Circle $II$. This is because the poi... |
deepscaler | [
{
"role": "user",
"content": "Brad has a cylindrical barrel with radius 10 inches and height 15 inches. He wants to fill it from a well, using a bucket in the shape of a hemisphere with a radius of 10 inches. How many trips must Brad make to the well in order to fill the barrel? Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 30703,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $\\left|{-4+\\frac{7}{6}i}\\right|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{25}{6}"
} | {
"split": "train",
"index": 30704,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If \\(\\lceil{\\sqrt{x}}\\rceil=20\\), how many possible integer values of \\(x\\) are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "39"
} | {
"split": "train",
"index": 30705,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $(x^2+1)(2x+1)^9 = a_0 + a_1(x+2) + a_2(x+2)^2 + \\ldots + a_{11}(x+2)^{11}$, find the value of $a_0 + a_1 + a_2 + \\ldots + a_{11}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 30706,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many integer values of $n$ between 1 and 990 inclusive does the decimal representation of $\\frac{n}{1000}$ terminate? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "990"
} | {
"split": "train",
"index": 30707,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that circle $C$ passes through points $P(0,-4)$, $Q(2,0)$, and $R(3,-1)$. \n$(1)$ Find the equation of circle $C$. \n$(2)$ If the line $l: mx+y-1=0$ intersects circle $C$ at points $A$ and $B$, and $|AB|=4$, find the value of $m$. Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"split": "train",
"index": 30708,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The year 2009 has a unique property: by rearranging the digits of the number 2009, it is impossible to form a smaller four-digit number (numbers do not start with zero). In which future year will this property first repeat again? Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "2022"
} | {
"split": "train",
"index": 30709,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be constants. Suppose that the equation \\[\\frac{(x+a)(x+b)(x+12)}{(x+3)^2} = 0\\]has exactly $3$ distinct roots, while the equation \\[\\frac{(x+2a)(x+3)(x+6)}{(x+b)(x+12)} = 0\\]has exactly $1$ distinct root. Compute $100a + b.$ Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "156"
} | {
"split": "train",
"index": 30710,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a right-angled triangle, one of whose acute angles is $\\alpha$. Find the ratio of the radii of the circumscribed and inscribed circles and determine for which value of $\\alpha$ this ratio will be the smallest. Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2} + 1"
} | {
"split": "train",
"index": 30711,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\det \\mathbf{A} = 5,$ then find $\\det (\\mathbf{A^3}).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "125"
} | {
"split": "train",
"index": 30712,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A particle is placed on the parabola $y = x^2- x -6$ at a point $P$ whose $y$-coordinate is $6$. It is allowed to roll along the parabola until it reaches the nearest point $Q$ whose $y$-coordinate is $-6$. The horizontal distance traveled by the particle (the numerical value o... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 30713,
"solution": "To solve this problem, we need to find the $x$-coordinates of points $P$ and $Q$ on the parabola $y = x^2 - x - 6$ where $y = 6$ and $y = -6$, respectively, and then calculate the horizontal distance between these points.\n\n1. **Finding $x$-coordinates for $P$ whe... |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the values of $x$ which satisfy $x^2 +1992x = 1993$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1992"
} | {
"split": "train",
"index": 30714,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nGiven positive real numbers \\( a, b, c, d \\) that satisfy the equalities\n\n\\[ a^{2}+d^{2}-ad = b^{2}+c^{2}+bc \\quad \\text{and} \\quad a^{2}+b^{2} = c^{2}+d^{2}, \\]\n\nfind all possible values of the expression \\( \\frac{ab+cd}{ad+bc} \\). Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 30715,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a_0 = 3,$ $b_0 = 4,$ and\n\\[a_{n + 1} = \\frac{a_n^2}{b_n} \\quad \\text{and} \\quad b_{n + 1} = \\frac{b_n^2}{a_n}\\] for all $n \\ge 0.$ Calculate $b_7 = \\frac{4^p}{3^q}$ for some integers $p$ and $q.$ Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "(1094,1093)"
} | {
"split": "train",
"index": 30716,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\\frac{\\cos B}{b} + \\frac{\\cos C}{c} = \\frac{\\sin A}{\\sqrt{3} \\sin C}$.\n\n(1) Find the value of $b$.\n(2) If $\\cos B + \\sqrt{3} \\sin B = 2$, find the maximum ar... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{3}}{4}"
} | {
"split": "train",
"index": 30717,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all integers $m$ and $n$ such that the fifth power of $m$ minus the fifth power of $n$ is equal to $16mn$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(m, n) = (0, 0) \\text{ and } (m, n) = (-2, 2)"
} | {
"split": "train",
"index": 30718,
"solution": "\nWe are tasked with finding all integer pairs \\((m, n)\\) such that:\n\n\\[\nm^5 - n^5 = 16mn\n\\]\n\n**Step 1: Algebraic Manipulation**\n\nWe begin by rewriting the given equation as:\n\n\\[\nm^5 - n^5 - 16mn = 0\n\\]\n\n**Step 2: Factorization**\n\nUsing the id... |
deepscaler | [
{
"role": "user",
"content": "If the odd function \\( y=f(x) \\) defined on \\( \\mathbf{R} \\) is symmetrical about the line \\( x=1 \\), and when \\( 0 < x \\leqslant 1 \\), \\( f(x)=\\log_{3}x \\), find the sum of all real roots of the equation \\( f(x)=-\\frac{1}{3}+f(0) \\) in the interval \\( (0,10) \... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 30719,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P(x)$ be a polynomial such that\n\\[P(x) = P(0) + P(1) x + P(2) x^2\\]and $P(-1) = 1.$ Find $P(x).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "x^2 - x - 1"
} | {
"split": "train",
"index": 30720,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $c_{1}$: $\\frac{x^{2}}{8} + \\frac{y^{2}}{4} = 1$ with left and right focal points $F_{1}$ and $F_{2}$, a line $l_{1}$ is drawn through point $F_{1}$ perpendicular to the x-axis. A line $l_{2}$ intersects $l_{1}$ perpendicularly at point $P$. The perpendicula... | math | {
"style": "rule",
"ground_truth": "\\frac{64}{9}"
} | {
"split": "train",
"index": 30721,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Based on the data provided, what were the average daily high and low temperatures in Addington from September 15th, 2008 to September 21st, 2008, inclusive? The high temperatures in degrees Fahrenheit for these days are 49, 62, 58, 57, 46, 55, 60 and the corresponding low tempe... | math | {
"style": "rule",
"ground_truth": "42.6"
} | {
"split": "train",
"index": 30722,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $y$ if $\\dfrac{y^2 - 9y + 8}{y-1} + \\dfrac{3y^2 +16y-12 }{3y -2} = -3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"split": "train",
"index": 30723,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Container A holds 4 red balls and 6 green balls; container B holds 8 red balls and 6 green balls; container C holds 8 red balls and 6 green balls; container D holds 3 red balls and 7 green balls. A container is selected at random and then a ball is randomly selected from that c... | math | {
"style": "rule",
"ground_truth": "\\frac{13}{28}"
} | {
"split": "train",
"index": 30724,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many ways can $47$ be written as the sum of two primes? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 30725,
"solution": "To solve the problem of expressing $47$ as the sum of two primes, we need to consider the properties of prime numbers and the parity (odd or even nature) of the numbers involved.\n\n1. **Understanding the Parity of Primes**: \n - All prime numbers greater than $2... |
deepscaler | [
{
"role": "user",
"content": "A regular octagon is inscribed in a circle and another regular octagon is circumscribed about the same circle. What is the ratio of the area of the larger octagon to the area of the smaller octagon? Express your answer as a common fraction. Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "4 - 2\\sqrt{2}"
} | {
"split": "train",
"index": 30726,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x) = 2^x + 3^x$ . For how many integers $1 \\leq n \\leq 2020$ is $f(n)$ relatively prime to all of $f(0), f(1), \\dots, f(n-1)$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 30727,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$A$ and $B$ together can do a job in $2$ days; $B$ and $C$ can do it in four days; and $A$ and $C$ in $2\\frac{2}{5}$ days.\nThe number of days required for A to do the job alone is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 30728,
"solution": "1. **Define the rates of work**: Let $r_A$, $r_B$, and $r_C$ be the rates at which $A$, $B$, and $C$ can complete the job per day, respectively. The units are $\\frac{\\text{job}}{\\text{day}}$.\n\n2. **Set up the equations based on the given information**:\n - $... |
deepscaler | [
{
"role": "user",
"content": "Let $W,X,Y$ and $Z$ be four different digits selected from the set\n$\\{ 1,2,3,4,5,6,7,8,9\\}.$\nIf the sum $\\dfrac{W}{X} + \\dfrac{Y}{Z}$ is to be as small as possible, then $\\dfrac{W}{X} + \\dfrac{Y}{Z}$ must equal Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "\\frac{25}{72}"
} | {
"split": "train",
"index": 30729,
"solution": "To minimize the sum $\\frac{W}{X} + \\frac{Y}{Z}$, where $W, X, Y, Z$ are distinct digits from the set $\\{1, 2, 3, 4, 5, 6, 7, 8, 9\\}$, we need to choose $W$ and $Y$ to be as small as possible and $X$ and $Z$ to be as large as possible. This is because the smalle... |
deepscaler | [
{
"role": "user",
"content": "If $\\det \\mathbf{B} = -3,$ then find $\\det (\\mathbf{B}^5).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-243"
} | {
"split": "train",
"index": 30730,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Arrange the 7 numbers $39, 41, 44, 45, 47, 52, 55$ in a sequence such that the sum of any three consecutive numbers is a multiple of 3. What is the maximum value of the fourth number in all such arrangements? Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "47"
} | {
"split": "train",
"index": 30731,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the greatest common divisor of $8!$ and $(6!)^2.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2880"
} | {
"split": "train",
"index": 30732,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The solid shown has a square base of side length $s$. The upper edge is parallel to the base and has length $2s$. All other edges have length $s$. Given that $s=6\\sqrt{2}$, what is the volume of the solid?\n[asy] size(180); import three; pathpen = black+linewidth(0.65); pointp... | math | {
"style": "rule",
"ground_truth": "288"
} | {
"split": "train",
"index": 30733,
"solution": "Draw an altitude from a vertex of the square base to the top edge. By using $30,60, 90$ triangle ratios, we obtain that the altitude has a length of $3 \\sqrt{6}$, and that little portion that hangs out has a length of $3\\sqrt2$. This is a triangular pyramid with ... |
deepscaler | [
{
"role": "user",
"content": "Let $\\alpha$ and $\\beta$ be conjugate complex numbers such that $\\frac{\\alpha}{\\beta^3}$ is a real number and $|\\alpha - \\beta| = 6$. Find $|\\alpha|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{2}"
} | {
"split": "train",
"index": 30734,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the rectangular coordinate system $xOy$, with $O$ as the pole and the positive semi-axis of $x$ as the polar axis, the polar coordinate system is established. The polar coordinate equation of the curve $C$ is $\\rho=2\\sin\\theta+2a\\cos\\theta$ ($a>0$); the parameter equati... | math | {
"style": "rule",
"ground_truth": "a=2"
} | {
"split": "train",
"index": 30735,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point $O$ is the center of the regular octagon $ABCDEFGH$, and $X$ is the midpoint of the side $\\overline{AB}.$ What fraction of the area of the octagon is shaded? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{16}"
} | {
"split": "train",
"index": 30736,
"solution": "\n#### Step 1: Understanding the Geometry\nThe regular octagon $ABCDEFGH$ is symmetric, and each side is equal. The center $O$ of the octagon is equidistant from all vertices. Since $X$ is the midpoint of $\\overline{AB}$, the line segment $\\overline{OX}$ bisects ... |
deepscaler | [
{
"role": "user",
"content": "A polyhedron has faces that all either triangles or squares. No two square faces share an edge, and no two triangular faces share an edge. What is the ratio of the number of triangular faces to the number of square faces? Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "4:3"
} | {
"split": "train",
"index": 30737,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f:X\\rightarrow X$, where $X=\\{1,2,\\ldots ,100\\}$, be a function satisfying:\n1) $f(x)\\neq x$ for all $x=1,2,\\ldots,100$;\n2) for any subset $A$ of $X$ such that $|A|=40$, we have $A\\cap f(A)\\neq\\emptyset$.\nFind the minimum $k$ such that for any such function $f$,... | math | {
"style": "rule",
"ground_truth": "69"
} | {
"split": "train",
"index": 30738,
"solution": "\nLet \\( f: X \\rightarrow X \\), where \\( X = \\{1, 2, \\ldots, 100\\} \\), be a function satisfying:\n1. \\( f(x) \\neq x \\) for all \\( x = 1, 2, \\ldots, 100 \\);\n2. For any subset \\( A \\) of \\( X \\) such that \\( |A| = 40 \\), we have \\( A \\cap f(A) ... |
deepscaler | [
{
"role": "user",
"content": "On the sides of triangle \\(ABC\\), points were marked: 10 on side \\(AB\\), 11 on side \\(BC\\), and 12 on side \\(AC\\). None of the vertices of the triangle were marked. How many triangles with vertices at the marked points exist? Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "4951"
} | {
"split": "train",
"index": 30739,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Grisha wrote 100 numbers on the board. Then he increased each number by 1 and noticed that the product of all 100 numbers did not change. He increased each number by 1 again, and again the product of all the numbers did not change, and so on. Grisha repeated this procedure $k$ ... | math | {
"style": "rule",
"ground_truth": "99"
} | {
"split": "train",
"index": 30740,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ that satisfy $|\\overrightarrow{a}| = \\sqrt{3}$, $|\\overrightarrow{b}| = 2$, and $|\\overrightarrow{a} + 2\\overrightarrow{b}| = \\sqrt{7}$.\n(I) Find $\\overrightarrow{a} \\cdot \\overrightarrow{b}$;\n(II) If vect... | math | {
"style": "rule",
"ground_truth": "\\frac{20}{9}"
} | {
"split": "train",
"index": 30741,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Chloe chooses a real number uniformly at random from the interval $[0, 3000]$. Independently, Max chooses a real number uniformly at random from the interval $[0, 4500]$. Determine the probability that Max's number is at least twice as large as Chloe's number. Let's think step ... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{8}"
} | {
"split": "train",
"index": 30742,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The line $y=2b$ intersects the left and right branches of the hyperbola $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1 \\ (a > 0, b > 0)$ at points $B$ and $C$ respectively, with $A$ being the right vertex and $O$ the origin. If $\\angle AOC = \\angle BOC$, then calculate the eccentr... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{19}}{2}"
} | {
"split": "train",
"index": 30743,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $ISI$ club each member is on two committees and any two committees have exactly one member in common . There are 5 committees . How many members does $ISI$ club have???? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 30744,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Circular arcs of radius 3 inches form a continuous pattern as shown. What is the area, in square inches, of the shaded region in a 2-foot length of this pattern? Each arc completes half of a circle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18\\pi"
} | {
"split": "train",
"index": 30745,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the area of the gray region, in square units, if the radius of the larger circle is four times the radius of the smaller circle and the diameter of the smaller circle is 2 units? Express your answer in terms of $\\pi$. [asy]\nsize(150);\npair A, B;\nA=(0,0);\nB=(-4,1);\... | math | {
"style": "rule",
"ground_truth": "15\\pi"
} | {
"split": "train",
"index": 30746,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At a meeting of cactus enthusiasts, 80 cactophiles presented their collections, each consisting of cacti of different species. It turned out that no single species of cactus is found in all collections simultaneously, but any 15 people have cacti of the same species. What is th... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 30747,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Through vertex $A$ of parallelogram $ABCD$, a line is drawn that intersects diagonal $BD$, side $CD$, and line $BC$ at points $E$, $F$, and $G$, respectively. Find the ratio $BE:ED$ if $FG:FE=4$. Round your answer to the nearest hundredth if needed. Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "2.24"
} | {
"split": "train",
"index": 30748,
"solution": ""
} |
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