data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "What is the sum of $x + y$ if the sequence $3, ~8, ~13, \\ldots, ~x, ~y, ~33$ forms an arithmetic sequence? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"split": "train",
"index": 36249,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At Jefferson High School, there are 500 students enrolled. One hundred twenty students are in the orchestra, 190 are in band, and 220 are in chorus. If only 400 students are in orchestra, band, and/or chorus, how many students are in exactly two of these groups? Let's think ste... | math | {
"style": "rule",
"ground_truth": "130"
} | {
"split": "train",
"index": 36250,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Petya bought one cake, two cupcakes and three bagels, Apya bought three cakes and a bagel, and Kolya bought six cupcakes. They all paid the same amount of money for purchases. Lena bought two cakes and two bagels. And how many cupcakes could be bought for the same amount spent ... | math | {
"style": "rule",
"ground_truth": "\\frac{13}{4}"
} | {
"split": "train",
"index": 36251,
"solution": "\nTo solve this problem, we need to determine how many cupcakes can be purchased for the same amount that Lena spent, given the prices of each pastry type.\n\nLet's denote the prices: \n- The price of one cake as \\( c \\).\n- The price of one cupcake as \\( p \\).... |
deepscaler | [
{
"role": "user",
"content": "Given the points $(2, 3)$, $(10, 9)$, and $(6, m)$, where $m$ is an integer, determine the sum of all possible values of $m$ for which the area of the triangle formed by these points is a maximum. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 36252,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Tam created the mosaic shown using a regular hexagon, squares, and equilateral triangles. If the side length of the hexagon is \\( 20 \\text{ cm} \\), what is the outside perimeter of the mosaic? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 36253,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P$ be a point on the ellipse $\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1$, $F_{1}$ and $F_{2}$ be the two foci of the ellipse, and $e$ be the eccentricity of the ellipse. Given $\\angle P F_{1} F_{2}=\\alpha$ and $\\angle P F_{2} F_{1}=\\beta$, express $\\tan \\frac{\\alp... | math | {
"style": "rule",
"ground_truth": "\\frac{1 - e}{1 + e}"
} | {
"split": "train",
"index": 36254,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate \\[\\frac{3}{\\log_8{5000^4}} + \\frac{2}{\\log_9{5000^4}},\\] giving your answer as a fraction in lowest terms. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 36255,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of $x$ such that $\\sqrt{3x + 7} = 10$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 36256,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sunshine High School is planning to order a batch of basketballs and jump ropes from an online store. After checking on Tmall, they found that each basketball is priced at $120, and each jump rope is priced at $25. There are two online stores, Store A and Store B, both offering... | math | {
"style": "rule",
"ground_truth": "5700"
} | {
"split": "train",
"index": 36257,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The midpoints of the sides of a regular octagon \\( ABCDEFGH \\) are joined to form a smaller octagon. What fraction of the area of \\( ABCDEFGH \\) is enclosed by the smaller octagon? 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": 36258,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that 2 teachers and 4 students are to be divided into 2 groups, each consisting of 1 teacher and 2 students, calculate the total number of different arrangements for the social practice activities in two different locations, A and B. Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 36259,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In quadrilateral $EFGH$, $\\angle F$ is a right angle, diagonal $\\overline{EG}$ is perpendicular to $\\overline{GH}$, $EF=20$, $FG=24$, and $GH=16$. Find the perimeter of $EFGH$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60 + 8\\sqrt{19}"
} | {
"split": "train",
"index": 36260,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine which integers $n > 1$ have the property that there exists an infinite sequence $a_1, a_2, a_3, \\ldots$ of nonzero integers such that the equality \\[a_k+2a_{2k}+\\ldots+na_{nk}=0\\]holds for every positive integer $k$. Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "n > 2"
} | {
"split": "train",
"index": 36261,
"solution": "Consider the problem to determine which integers \\( n > 1 \\) have the property that there exists an infinite sequence \\( a_1, a_2, a_3, \\ldots \\) of nonzero integers satisfying the equality:\n\n\\[\na_k + 2a_{2k} + \\ldots + na_{nk} = 0\n\\]\n\nfor every posit... |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all positive rational numbers that are less than 10 and that have denominator 30 when written in lowest terms. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "400"
} | {
"split": "train",
"index": 36262,
"solution": "Note that if $x$ is a solution, then $(300-x)$ is a solution. We know that $\\phi(300) = 80.$ Therefore the answer is $\\frac{80}{2} \\cdot\\frac{300}{30} = \\boxed{400}.$\n The problems on this page are copyrighted by the Mathematical Association of America's Amer... |
deepscaler | [
{
"role": "user",
"content": "Determine the difference between the sum of the first one hundred positive even integers and the sum of the first one hundred positive multiples of 3. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-5050"
} | {
"split": "train",
"index": 36263,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers from 1 to 9 are placed at the vertices of a cube such that the sum of the four numbers on each face is the same. Find the common sum. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22.5"
} | {
"split": "train",
"index": 36264,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A string of 33 pearls has its middle pearl as the largest and most valuable. The values of the remaining pearls decrease by $3000 \\mathrm{Ft}$ per pearl towards one end and by $4500 \\mathrm{Ft}$ per pearl towards the other end. How much is the middle pearl worth if the total ... | math | {
"style": "rule",
"ground_truth": "90000"
} | {
"split": "train",
"index": 36265,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the fixed point $M(1,0)$, $A$ and $B$ are two moving points on the ellipse $\\frac{x^2}{4}+y^2=1$, and $\\overrightarrow{MA} \\cdot \\overrightarrow{MB}=0$, find the minimum value of $\\overrightarrow{AM} \\cdot \\overrightarrow{AB}$. Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 36266,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\( A B C \\) with the side ratio \\( A B: A C = 5:4 \\), the angle bisector of \\( \\angle B A C \\) intersects side \\( B C \\) at point \\( L \\). Find the length of segment \\( A L \\), given that the length of the vector \\( 4 \\cdot \\overrightarrow{A B} + 5 \... | math | {
"style": "rule",
"ground_truth": "224"
} | {
"split": "train",
"index": 36267,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sir Alex plays the following game on a row of 9 cells. Initially, all cells are empty. In each move, Sir Alex is allowed to perform exactly one of the following two operations:\n[list=1]\n[*] Choose any number of the form $2^j$, where $j$ is a non-negative integer, and put it i... | math | {
"style": "rule",
"ground_truth": "2 \\sum_{i=0}^{8} \\binom{n}{i} - 1"
} | {
"split": "train",
"index": 36268,
"solution": "\nTo determine the maximum number of moves that Sir Alex could have made in this game, we need to analyze the operations and how each affects the game state.\n\nSir Alex has 9 cells initially empty. The objective is to have one cell contain the number \\(2^n\\) at ... |
deepscaler | [
{
"role": "user",
"content": "Anna walked at a constant rate. If she walked 600 metres in 4 minutes, how far did she walk in 6 minutes? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "900"
} | {
"split": "train",
"index": 36269,
"solution": "If Anna walked 600 metres in 4 minutes, then she walked $\\frac{600}{4}=150$ metres each minute. Therefore, in 6 minutes, she walked $6 \\times 150=900$ metres."
} |
deepscaler | [
{
"role": "user",
"content": "If $r$ is positive and the line whose equation is $x + y = r$ is tangent to the circle whose equation is $x^2 + y^2 = r$, then $r$ equals Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 36270,
"solution": "1. **Identify the Geometry of the Circle and Line:**\n - The circle given by the equation $x^2 + y^2 = r$ has its center at $(0,0)$ and a radius of $\\sqrt{r}$.\n - The line given by the equation $x + y = r$ can be rewritten in standard form as $x + y - r = 0$.... |
deepscaler | [
{
"role": "user",
"content": "Darren has borrowed $100$ clams from Ethan at a $10\\%$ simple daily interest. Meanwhile, Fergie has borrowed $150$ clams from Gertie at a $5\\%$ simple daily interest. In how many days will Darren and Fergie owe the same amounts, assuming that they will not make any repayments... | math | {
"style": "rule",
"ground_truth": "20\\text{ days}"
} | {
"split": "train",
"index": 36271,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a 3 by 3 grid of unit squares, an up-right path is a path from the bottom left corner to the top right corner that travels only up and right in steps of 1 unit. For such a path $p$, let $A_{p}$ denote the number of unit squares under the path $p$. Compute the sum of $A_{p}$ ... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 36272,
"solution": "Each path consists of 3 steps up and 3 steps to the right, so there are $\\binom{6}{3}=20$ total paths. Consider the sum of the areas of the regions above all of these paths. By symmetry, this is the same as the answer to the problem. For any path, the sum of the a... |
deepscaler | [
{
"role": "user",
"content": "Find four distinct positive integers $a, b, c, d$ less than $15$ which are invertible modulo $15$. Calculate the remainder when $(abc + abd + acd + bcd)(abcd)^{-1}$ is divided by $15$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 36273,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the polar equation of circle $C$ is $\\rho=2\\cos \\theta$, and the parametric equation of line $l$ is $\\begin{cases}x= \\frac{1}{2}+ \\frac{ \\sqrt{3}}{2}t \\\\ y= \\frac{1}{2}+ \\frac{1}{2}t\\end{cases}$ (where $t$ is the parameter), and the polar coordinates of point ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 36274,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each of $2011$ boxes in a line contains two red marbles, and for $1 \\le k \\le 2011$, the box in the $k\\text{th}$ position also contains $k+1$ white marbles. Liam begins at the first box and successively draws a single marble at random from each box, in order. He stops when h... | math | {
"style": "rule",
"ground_truth": "62"
} | {
"split": "train",
"index": 36275,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the complex number $z$ such that\n\\[|z - 1| = |z + 3| = |z - i|.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1 - i"
} | {
"split": "train",
"index": 36276,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The value of \\( a \\) is chosen such that the number of roots of the first equation \\( 4^{x} - 4^{-x} = 2 \\cos a x \\) is 2007. How many roots does the second equation \\( 4^{x} + 4^{-x} = 2 \\cos a x + 4 \\) have for the same \\( a \\)? Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "4014"
} | {
"split": "train",
"index": 36277,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the mean square of \\( n \\) numbers \\( a_{1}, a_{2}, \\cdots, a_{n} \\) be defined as \\(\\left(\\frac{a_{1}^{2} + a_{2}^{2} + \\cdots + a_{n}^{2}}{n}\\right)^{\\frac{1}{2}}\\). Let \\( M \\) be the set of all values of \\( n \\) such that the mean square of the first \\(... | math | {
"style": "rule",
"ground_truth": "337"
} | {
"split": "train",
"index": 36278,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x$ be a positive integer such that $9x\\equiv 1\\pmod{25}$.\n\nWhat is the remainder when $11+x$ is divided by $25$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 36279,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the remainder when $5x^8 - 3x^7 + 4x^6 - 9x^4 + 3x^3 - 5x^2 + 8$ is divided by $3x - 6$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1020"
} | {
"split": "train",
"index": 36280,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $x^2 + y^2 = 14x + 6y + 6,$ find the largest possible value of $3x + 4y.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "73"
} | {
"split": "train",
"index": 36281,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A ball is dropped from 1000 feet high and always bounces back up half the distance it just fell. After how many bounces will the ball first reach a maximum height less than 1 foot? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 36282,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that Let \\\\(S_{n}\\\\) and \\\\(T_{n}\\\\) be the sums of the first \\\\(n\\\\) terms of the arithmetic sequences \\\\(\\{a_{n}\\}\\\\) and \\\\(\\{b_{n}\\}\\\\), respectively, and \\\\( \\frac {S_{n}}{T_{n}}= \\frac {n}{2n+1} (n∈N^{*})\\\\), determine the value of \\\\... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{23}"
} | {
"split": "train",
"index": 36283,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, each of four identical circles touch three others. The circumference of each circle is 48. Calculate the perimeter of the shaded region formed within the central area where all four circles touch. Assume the circles are arranged symmetrically like petals of a fl... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 36284,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Write $\\frac{5}{8}$ as a decimal. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.625"
} | {
"split": "train",
"index": 36285,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "One hundred bricks, each measuring $3''\\times12''\\times20''$, are to be stacked to form a tower 100 bricks tall. Each brick can contribute $3''$, $12''$, or $20''$ to the total height of the tower. However, each orientation must be used at least once. How many different tower... | math | {
"style": "rule",
"ground_truth": "187"
} | {
"split": "train",
"index": 36286,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle with a radius of 3 is inscribed in a right trapezoid, where the shorter base is 4. Find the length of the longer base of the trapezoid. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 36287,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 3 boys and 3 girls, making a total of 6 students standing in a row.\n\n(1) If the three girls must stand together, find the total number of different arrangements.\n(2) If boy A cannot stand at either end, and among the 3 girls, exactly two girls stand together, find ... | math | {
"style": "rule",
"ground_truth": "288"
} | {
"split": "train",
"index": 36288,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a parlor game, the magician asks one of the participants to think of a three digit number $(abc)$ where $a$, $b$, and $c$ represent digits in base $10$ in the order indicated. The magician then asks this person to form the numbers $(acb)$, $(bca)$, $(bac)$, $(cab)$, and $(cb... | math | {
"style": "rule",
"ground_truth": "358"
} | {
"split": "train",
"index": 36289,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the area of the smallest square that can completely enclose a circle of radius 5 units? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 36290,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Acme Corporation has released a new version of its vowel soup where each vowel (A, E, I, O, U) appears six times, and additionally, each bowl contains one wildcard character that can represent any vowel. How many six-letter \"words\" can be formed from a bowl of this new Acme E... | math | {
"style": "rule",
"ground_truth": "46656"
} | {
"split": "train",
"index": 36291,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A list of seven positive integers has a median of 5 and a mean of 15. What is the maximum possible value of the list's largest element? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "87"
} | {
"split": "train",
"index": 36292,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point $Q$ is randomly placed in the interior of the right triangle $XYZ$ with $XY = 10$ units and $XZ = 6$ units. What is the probability that the area of triangle $QYZ$ is less than one-third of the area of triangle $XYZ$? Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 36293,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\sin{\\alpha} = -\\frac{3}{5}$, $\\sin{\\beta} = \\frac{12}{13}$, and $\\alpha \\in (\\pi, \\frac{3\\pi}{2})$, $\\beta \\in (\\frac{\\pi}{2}, \\pi)$, find the values of $\\sin({\\alpha - \\beta})$, $\\cos{2\\alpha}$, and $\\tan{\\frac{\\beta}{2}}$. Let's think step... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 36294,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Integers less than $4010$ but greater than $3000$ have the property that their units digit is the sum of the other digits and also the full number is divisible by 3. How many such integers exist? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 36295,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are a batch of wooden strips with lengths of \\(1, 2, 3, 4, 5, 6, 7, 8, 9, 10,\\) and 11 centimeters, with an adequate quantity of each length. If you select 3 strips appropriately to form a triangle with the requirement that the base is 11 centimeters long, how many diff... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 36296,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A palindrome is a number that reads the same forward and backward. How many three-digit perfect squares are palindromes? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 36297,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point $A(-2,-4)$ outside the parabola $y^{2}=2px (p > 0)$ is connected to a line $l$: $\\begin{cases} x=-2+ \\frac{\\sqrt{2}}{2}t \\\\ y=-4+ \\frac{\\sqrt{2}}{2}t \\end{cases} (t$ is a parameter, $t \\in \\mathbb{R})$ intersecting the parabola at points $M_{1}$ and $M_{2}$. T... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{10}"
} | {
"split": "train",
"index": 36298,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A motorcycle travels due west at $\\frac{5}{8}$ mile per minute on a long straight road. At the same time, a circular storm, whose radius is $60$ miles, moves southwest at $\\frac{1}{2}$ mile per minute. At time $t=0$, the center of the storm is $100$ miles due north of the mot... | math | {
"style": "rule",
"ground_truth": "160"
} | {
"split": "train",
"index": 36299,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The digits 2, 4, 6, and 8 are each used once to create two 2-digit numbers. Find the smallest possible difference between the two 2-digit numbers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 36300,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many real numbers $a$ does the quadratic equation $x^2 + ax + 12a = 0$ have only integer roots for $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 36301,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sum of the binomial coefficients of all terms in the expansion of ${(a{x}^{2}+\\frac{1}{x})}^{n}$ is $128$, and the sum of all coefficients is $-1$.<br/>$(1)$ Find the values of $n$ and $a$;<br/>$(2)$ Find the constant term in the expansion of $(2x-\\frac{1}{{x}^... | math | {
"style": "rule",
"ground_truth": "448"
} | {
"split": "train",
"index": 36302,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Joe will randomly select two letters from the word CAMP, four letters from the word HERBS, and three letters from the word GLOW. What is the probability that he will have all of the letters from the word PROBLEM? Express your answer as a common fraction. Let's think step by s... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{30}"
} | {
"split": "train",
"index": 36303,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For some real numbers $a$ and $b$, the equation $9x^3 + 5ax^2 + 4bx + a = 0$ has three distinct positive roots. If the sum of the base-2 logarithms of the roots is 4, what is the value of $a$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-144"
} | {
"split": "train",
"index": 36304,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given any two positive integers, a certain operation (denoted by the operator $\\oplus$) is defined as follows: when $m$ and $n$ are both positive even numbers or both positive odd numbers, $m \\oplus n = m + n$; when one of $m$ and $n$ is a positive even number and the other i... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 36305,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute: $\\displaystyle \\frac{66,\\!666^4}{22,\\!222^4}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "81"
} | {
"split": "train",
"index": 36306,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a \\lt 0$, the graph of the function $f\\left(x\\right)=a^{2}\\sin 2x+\\left(a-2\\right)\\cos 2x$ is symmetric with respect to the line $x=-\\frac{π}{8}$. Find the maximum value of $f\\left(x\\right)$. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{2}"
} | {
"split": "train",
"index": 36307,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five dice with faces numbered 1 through 6 are stacked in a similar manner to the original problem. Ten of the thirty faces are visible, leaving twenty faces hidden. The visible numbers are 1, 2, 2, 3, 3, 3, 4, 4, 5, and 6. What is the total number of dots NOT visible in this vi... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 36308,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the side \\( BC \\) of triangle \\( ABC \\), points \\( A_1 \\) and \\( A_2 \\) are marked such that \\( BA_1 = 6 \\), \\( A_1A_2 = 8 \\), and \\( CA_2 = 4 \\). On the side \\( AC \\), points \\( B_1 \\) and \\( B_2 \\) are marked such that \\( AB_1 = 9 \\) and \\( CB_2 = 6 ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 36309,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the product of the prime numbers less than 20? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9699690"
} | {
"split": "train",
"index": 36310,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define $a * b$ as $2a - b^2$. If $a * 5 = 9$, what is the value of $a$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 36311,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A child spends their time drawing pictures of Native Americans (referred to as \"Indians\") and Eskimos. Each drawing depicts either a Native American with a teepee or an Eskimo with an igloo. However, the child sometimes makes mistakes and draws a Native American with an igloo... | math | {
"style": "rule",
"ground_truth": "7/8"
} | {
"split": "train",
"index": 36312,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with left and right foci $F\\_1$ and $F\\_2$, respectively. Point $A(4,2\\sqrt{2})$ lies on the ellipse, and $AF\\_2$ is perpendicular to the $x$-axis.\n\n1. Find the equation of the ellipse.\n2. A lin... | math | {
"style": "rule",
"ground_truth": "8\\sqrt{2}"
} | {
"split": "train",
"index": 36313,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The image shows a 3x3 grid where each cell contains one of the following characters: 华, 罗, 庚, 杯, 数, 学, 精, 英, and 赛. Each character represents a different number from 1 to 9, and these numbers satisfy the following conditions:\n1. The sum of the four numbers in each \"田\" (four ... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 36314,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A prime number $ q $ is called***'Kowai'***number if $ q = p^2 + 10$ where $q$ , $p$ , $p^2-2$ , $p^2-8$ , $p^3+6$ are prime numbers. WE know that, at least one ***'Kowai'*** number can be found. Find the summation of all ***'Kowai'*** numbers.\n Let's think step by ... | math | {
"style": "rule",
"ground_truth": "59"
} | {
"split": "train",
"index": 36315,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, $AB$ is a line segment. What is the value of $x$?\n\n[asy]\ndraw((0,0)--(10,0),black+linewidth(1));\ndraw((4,0)--(4,8),black+linewidth(1));\ndraw((4,0)--(3.5,0)--(3.5,0.5)--(4,0.5)--cycle,black+linewidth(1));\ndraw((4,0)--(9,7),black+linewidth(1));\nlabel(\"$A$... | math | {
"style": "rule",
"ground_truth": "38"
} | {
"split": "train",
"index": 36316,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the sum of the series $(3+13+23+33+43)+(11+21+31+41+51)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "270"
} | {
"split": "train",
"index": 36317,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many perfect squares divide $2^{3} \\cdot 3^{5} \\cdot 5^{7} \\cdot 7^{9}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 36318,
"solution": "The number of such perfect squares is $2 \\cdot 3 \\cdot 4 \\cdot 5$, since the exponent of each prime can be any nonnegative even number less than the given exponent."
} |
deepscaler | [
{
"role": "user",
"content": "The angle between the slant height of a cone and the base plane is $30^{\\circ}$. The lateral surface area of the cone is $3 \\pi \\sqrt{3}$ square units. Determine the volume of a regular hexagonal pyramid inscribed in the cone. Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "\\frac{27 \\sqrt{2}}{8}"
} | {
"split": "train",
"index": 36319,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number of games won by six basketball teams are displayed in the graph, but the names of the teams are missing. The following clues provide information about the teams:\n\n1. The Hawks won more games than the Falcons.\n2. The Warriors won more games than the Knights, but fe... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 36320,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the Cesaro sum of the 50-term sequence \\((b_1,\\dots,b_{50})\\) is 500, what is the Cesaro sum of the 51-term sequence \\((2,b_1,\\dots,b_{50})\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "492"
} | {
"split": "train",
"index": 36321,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are exactly $77,000$ ordered quadruplets $(a, b, c, d)$ such that $\\gcd(a, b, c, d) = 77$ and $\\operatorname{lcm}(a, b, c, d) = n$. What is the smallest possible value for $n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27,720"
} | {
"split": "train",
"index": 36322,
"solution": "1. **Define normalized variables**: Let $A = \\frac{a}{77}$, $B = \\frac{b}{77}$, $C = \\frac{c}{77}$, and $D = \\frac{d}{77}$. This transformation ensures that $\\gcd(A, B, C, D) = 1$. The problem then translates to finding the least common multiple of $A, B, C, D... |
deepscaler | [
{
"role": "user",
"content": "Given a real number $a$ satisfying ${a}^{\\frac{1}{2}}\\leqslant 3$ and $\\log _{a}3\\leqslant \\frac{1}{2}$. \n$(1)$ Find the range of real number $a$; \n$(2)$ If $a \\gt 1$, $f\\left(x\\right)=mx^{a}+\\ln \\left(1+x\\right)^{a}-a\\ln \\left(1-x\\right)-2\\left(m\\in R\\righ... | math | {
"style": "rule",
"ground_truth": "-13"
} | {
"split": "train",
"index": 36323,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Using the letters $A$, $M$, $O$, $S$, and $U$, we can form five-letter \"words\". If these \"words\" are arranged in alphabetical order, then the \"word\" $USAMO$ occupies position Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "115"
} | {
"split": "train",
"index": 36324,
"solution": "To find the position of the word \"USAMO\" in the alphabetical order of all possible permutations of the letters $A$, $M$, $O$, $S$, and $U$, we can proceed as follows:\n\n1. **Count the permutations starting with each letter before 'U':**\n - The letters before ... |
deepscaler | [
{
"role": "user",
"content": "For any real number \\( x \\), let \\([x]\\) denote the greatest integer less than or equal to \\( x \\). When \\( 0 \\leqslant x \\leqslant 100 \\), how many different integers are in the range of the function \\( f(x) = [2x] + [3x] + [4x] + [5x] \\)? Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "101"
} | {
"split": "train",
"index": 36325,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Orvin went to the store with just enough money to buy $30$ balloons. When he arrived, he discovered that the store had a special sale on balloons: buy $1$ balloon at the regular price and get a second at $\\frac{1}{3}$ off the regular price. What is the greatest number of ballo... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 36326,
"solution": "1. **Assume the cost of each balloon**: Let's assume each balloon costs $3$ dollars. This assumption simplifies calculations and does not affect the generality of the problem since we are interested in the number of balloons, not the actual cost.\n\n2. **Calculate ... |
deepscaler | [
{
"role": "user",
"content": "Given in a cube ABCD-A1B1C1D1 with edge length 1, P is a moving point inside the cube (including the surface), \nif $x + y + z = s$, and $0 \\leq x \\leq y \\leq z \\leq 1$, then the volume of the geometric body formed by all possible positions of point P is $\\_\\_\\_\\_\\_\\... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"split": "train",
"index": 36327,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the maximum value that can be taken by the sum\n\n$$\n\\left|x_{1}-1\\right|+\\left|x_{2}-2\\right|+\\ldots+\\left|x_{63}-63\\right|\n$$\n\nif $x_{1}, x_{2}, \\ldots, x_{63}$ are some permutation of the numbers $1, 2, 3, \\ldots, 63$? Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "1984"
} | {
"split": "train",
"index": 36328,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a five-digit integer in the form $AB,BCA$, where $A$, $B$, and $C$ are distinct digits. What is the largest possible value of $AB,BCA$ that is divisible by 7? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "98,879"
} | {
"split": "train",
"index": 36329,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $x$ is a multiple of $3456$, what is the greatest common divisor of $f(x)=(5x+3)(11x+2)(14x+7)(3x+8)$ and $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 36330,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the point of intersection of the asymptotes of the graph of\n\\[y = \\frac{x^2 - 4x + 3}{x^2 - 4x + 4}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(2,1)"
} | {
"split": "train",
"index": 36331,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The circumcenter of a regular tetrahedron \\( ABCD \\) is \\( O \\). If \\( E \\) is the midpoint of \\( BC \\), what is the measure of the dihedral angle between \\( A-BO-E \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}\\pi"
} | {
"split": "train",
"index": 36332,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x_0$ be a zero of the function $f(x)=\\sin \\pi x$, and it satisfies $|x_0|+f(x_0+ \\frac{1}{2}) < 33$. Calculate the number of such zeros. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "65"
} | {
"split": "train",
"index": 36333,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)={x^3}+\\frac{{{{2023}^x}-1}}{{{{2023}^x}+1}}+5$, if real numbers $a$ and $b$ satisfy $f(2a^{2})+f(b^{2}-2)=10$, then the maximum value of $a\\sqrt{1+{b^2}}$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{2}}{4}"
} | {
"split": "train",
"index": 36334,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a kindergarten, there are two small Christmas trees and five children. The teachers want to divide the children into two circles around each of the Christmas trees, with at least one child in each circle. The teachers can distinguish between the children but not between the ... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 36335,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A teacher must divide 221 apples evenly among 403 students. What is the minimal number of pieces into which she must cut the apples? (A whole uncut apple counts as one piece.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "611"
} | {
"split": "train",
"index": 36336,
"solution": "Consider a bipartite graph, with 221 vertices representing the apples and 403 vertices representing the students; each student is connected to each apple that she gets a piece of. The number of pieces then equals the number of edges in the graph. Each student gets ... |
deepscaler | [
{
"role": "user",
"content": "Let $f$ be a polynomial with integer coefficients such that the greatest common divisor of all its coefficients is 1. For any $n \\in \\mathbb{N}, f(n)$ is a multiple of 85. Find the smallest possible degree of $f$. Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 36337,
"solution": "Notice that, if $p$ is a prime and $g$ is a polynomial with integer coefficients such that $g(n) \\equiv 0(\\bmod p)$ for some $n$, then $g(n+m p)$ is divisible by $p$ as well for any integer multiple $m p$ of $p$. Therefore, it suffices to find the smallest possib... |
deepscaler | [
{
"role": "user",
"content": "A bus ticket costs 1 yuan each. Xiaoming and 6 other children are lining up to buy tickets. Each of the 6 children has only 1 yuan, while Xiaoming has a 5-yuan note. The seller has no change. In how many ways can they line up so that the seller can give Xiaoming change when he ... | math | {
"style": "rule",
"ground_truth": "10800"
} | {
"split": "train",
"index": 36338,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the line $y=ax$ intersects the circle $C:x^2+y^2-2ax-2y+2=0$ at points $A$ and $B$, and $\\Delta ABC$ is an equilateral triangle, then the area of circle $C$ is __________. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6\\pi"
} | {
"split": "train",
"index": 36339,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two arithmetic sequences $\\{a_n\\}$ and $\\{b_n\\}$, whose sums of the first $n$ terms are $A_n$ and $B_n$ respectively, and $\\frac {A_{n}}{B_{n}} = \\frac {7n+1}{4n+27}$, determine $\\frac {a_{6}}{b_{6}}$. Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "\\frac{78}{71}"
} | {
"split": "train",
"index": 36340,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $E$ be a three-dimensional ellipsoid. For a plane $p$, let $E(p)$ be the projection of $E$ onto the plane $p$. The minimum and maximum areas of $E(p)$ are $9 \\pi$ and $25 \\pi$, and there exists a $p$ where $E(p)$ is a circle of area $16 \\pi$. If $V$ is the volume of $E$,... | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 36341,
"solution": "Let the three radii of $E$ be $a<b<c$. We know that $ab=9$ and $bc=25$. Consider the plane $p$ where projection $E(p)$ has area $9 \\pi$. Fixing $p$, rotate $E$ on the axis passing through the radius with length $b$ until $E(p)$ has area $25 \\pi$. The projection o... |
deepscaler | [
{
"role": "user",
"content": "The sequences of positive integers $1,a_2, a_3,...$ and $1,b_2, b_3,...$ are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let $c_n=a_n+b_n$. There is an integer $k$ such that $c_{k-1}=100$ and $c_{k+1}=1000$. Find $c_k$. Let's think step... | math | {
"style": "rule",
"ground_truth": "262"
} | {
"split": "train",
"index": 36342,
"solution": "Since all the terms of the sequences are integers, and 100 isn't very big, we should just try out the possibilities for $b_2$. When we get to $b_2=9$ and $a_2=91$, we have $a_4=271$ and $b_4=729$, which works, therefore, the answer is $b_3+a_3=81+181=\\boxed{262}$.... |
deepscaler | [
{
"role": "user",
"content": "An error of $.05''$ is made in the measurement of a line $25''$ long, and an error of $.3''$ is made in the measurement of a line $150''$ long. In comparison with the relative error of the first measurement, find the relative error of the second measurement. Let's think step by... | math | {
"style": "rule",
"ground_truth": "0.2\\%"
} | {
"split": "train",
"index": 36343,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Quadrilateral $ABCD$ is a trapezoid, $AD = 15$, $AB = 50$, $BC = 20$, and the altitude is $12$. What is the area of the trapezoid? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "750"
} | {
"split": "train",
"index": 36344,
"solution": "1. **Identify the Components of the Trapezoid**:\n Given that $ABCD$ is a trapezoid with $AB$ and $CD$ as the parallel sides, and the altitude (height) from $AB$ to $CD$ is $12$. The lengths of the sides are $AD = 15$, $AB = 50$, $BC = 20$.\n\n2. **Draw Altitudes... |
deepscaler | [
{
"role": "user",
"content": "In square $PQRS$ with side length 2, each of $P, Q, R$, and $S$ is the centre of a circle with radius 1. What is the area of the shaded region? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4-\\pi"
} | {
"split": "train",
"index": 36345,
"solution": "The area of the shaded region is equal to the area of square $PQRS$ minus the combined areas of the four unshaded regions inside the square. Since square $PQRS$ has side length 2, its area is $2^{2}=4$. Since $PQRS$ is a square, then the angle at each of $P, Q, R$,... |
deepscaler | [
{
"role": "user",
"content": "Positive integers $a$ and $b$ are each less than $6$. What is the smallest possible value for $2 \\cdot a - a \\cdot b$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-15"
} | {
"split": "train",
"index": 36346,
"solution": "1. **Expression Simplification**:\n Start by simplifying the given expression:\n \\[\n 2a - ab = a(2 - b)\n \\]\n This shows that the expression depends on the values of $a$ and $b$ through the product $a(2 - b)$.\n\n2. **Determine the Range of Values**:\... |
deepscaler | [
{
"role": "user",
"content": "The integers \\( n \\) and \\( m \\) satisfy the inequalities \\( 3n - m < 5 \\), \\( n + m > 26 \\), and \\( 3m - 2n < 46 \\). What can \\( 2n + m \\) be equal to? Indicate all possible values. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 36347,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute\n\\[\\cos^2 0^\\circ + \\cos^2 1^\\circ + \\cos^2 2^\\circ + \\dots + \\cos^2 90^\\circ.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{91}{2}"
} | {
"split": "train",
"index": 36348,
"solution": ""
} |
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