data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Determine all real numbers $ a$ such that the inequality $ |x^2 + 2ax + 3a|\\le2$ has exactly one solution in $ x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1, 2"
} | {
"split": "train",
"index": 39849,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When skipping rope, the midpoint of the rope can be considered to move along the same circle. If Xiaoguang takes 0.5 seconds to complete a \"single skip\" and 0.6 seconds to complete a \"double skip,\" what is the ratio of the speed of the midpoint of the rope during a \"single... | math | {
"style": "rule",
"ground_truth": "3/5"
} | {
"split": "train",
"index": 39850,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the minimum value of the expression:\n\\[(15 - x)(8 - x)(15 + x)(8 + x).\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-6480.25"
} | {
"split": "train",
"index": 39851,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $PQR$, $QR = 24$. An incircle of the triangle trisects the median $PS$ from $P$ to side $QR$. Given that the area of the triangle is $k \\sqrt{p}$, where $k$ and $p$ are integers and $p$ is square-free, find $k+p$. Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "106"
} | {
"split": "train",
"index": 39852,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $M = 36 \\cdot 36 \\cdot 77 \\cdot 330$. Find the ratio of the sum of the odd divisors of $M$ to the sum of the even divisors of $M$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1 : 62"
} | {
"split": "train",
"index": 39853,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Square $XYZW$ has area $144$. Point $P$ lies on side $\\overline{XW}$, such that $XP = 2WP$. Points $Q$ and $R$ are the midpoints of $\\overline{ZP}$ and $\\overline{YP}$, respectively. Quadrilateral $XQRW$ has an area of $20$. Calculate the area of triangle $RWP$. Let's think ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 39854,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The operation $\\&$ is defined as $a \\& b = (a+b)(a-b)$. What is the value of $6 \\& 3$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 39855,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integers $n$ satisfy \\[(n + 8)(n - 3)(n-12)<0\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 39856,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The hyperbola $C:\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$ $(a > 0,b > 0)$ has an asymptote perpendicular to the line $x+2y+1=0$. Let $F_1$ and $F_2$ be the foci of $C$, and let $A$ be a point on the hyperbola. If $|F_1A|=2|F_2A|$, then $\\cos \\angle AF_2F_1=$ __________. Let's t... | math | {
"style": "rule",
"ground_truth": "\\dfrac{\\sqrt{5}}{5}"
} | {
"split": "train",
"index": 39857,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The polynomial $P$ is a quadratic with integer coefficients. For every positive integer $n$ , the integers $P(n)$ and $P(P(n))$ are relatively prime to $n$ . If $P(3) = 89$ , what is the value of $P(10)$ ? Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "859"
} | {
"split": "train",
"index": 39858,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The acronym AMC is shown in the rectangular grid below with grid lines spaced $1$ unit apart. In units, what is the sum of the lengths of the line segments that form the acronym AMC$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13 + 4\\sqrt{2}"
} | {
"split": "train",
"index": 39859,
"solution": "To solve this problem, we need to calculate the total length of the line segments that form the acronym \"AMC\" on the given grid. We will count the number of straight line segments and slanted line segments separately, and then compute their total lengths.\n\n1. *... |
deepscaler | [
{
"role": "user",
"content": "Two balls are randomly chosen from a box containing 20 balls numbered from 1 to 20. Calculate the probability that the sum of the numbers on the two balls is divisible by 3. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{32}{95}"
} | {
"split": "train",
"index": 39860,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose in a right triangle LMN, where angle M is the right angle, $\\cos N = \\frac{4}{5}$ and length LM is given by 20 units. What is the length of LN? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 39861,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The areas of two squares are in the ratio $25:36$. What is the ratio of their perimeters? Express your answer in the form $a:b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5:6"
} | {
"split": "train",
"index": 39862,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of $x$-intercepts on the graph of $y = \\sin \\frac{2}{x}$ (evaluated in terms of radians) in the interval $(0.0002, 0.002).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2865"
} | {
"split": "train",
"index": 39863,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the volume in cubic inches of a right, rectangular prism where the areas of the side, front, and bottom faces are 20 square inches, 12 square inches, and 8 square inches, respectively. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8\\sqrt{30}"
} | {
"split": "train",
"index": 39864,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jeremy made a Venn diagram showing the number of students in his class who own types of pets. There are 32 students in his class. In addition to the information in the Venn diagram, Jeremy knows half of the students have a dog, $\\frac{3}{8}$ have a cat, six have some other pet... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 39865,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose \\( x \\), \\( y \\), and \\( z \\) are positive numbers satisfying:\n\\[\nx^2 \\cdot y = 2, \\\\\ny^2 \\cdot z = 4, \\text{ and} \\\\\nz^2 / x = 5.\n\\]\nFind \\( x \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5^{1/7}"
} | {
"split": "train",
"index": 39866,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Adding two dots above the decimal 0.142857 makes it a repeating decimal. The 2020th digit after the decimal point is 5. What is the repeating cycle? $\\quad$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "142857"
} | {
"split": "train",
"index": 39867,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Cameron writes down the smallest positive multiple of 30 that is a perfect square, the smallest positive multiple of 30 that is a perfect cube, and all the multiples of 30 between them. How many integers are in Cameron's list? Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "871"
} | {
"split": "train",
"index": 39868,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Kristoff is planning to transport a number of indivisible ice blocks with positive integer weights from the north mountain to Arendelle. He knows that when he reaches Arendelle, Princess Anna and Queen Elsa will name an ordered pair $(p, q)$ of nonnegative integers satisfying $... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 39869,
"solution": "The answer is 18. First, we will show that Kristoff must carry at least 18 ice blocks. Let $$0<x_{1} \\leq x_{2} \\leq \\cdots \\leq x_{n}$$ be the weights of ice blocks he carries which satisfy the condition that for any $p, q \\in \\mathbb{Z}_{\\geq 0}$ such that... |
deepscaler | [
{
"role": "user",
"content": "The four hydrogen atoms in the methane molecule $\\mathrm{CH}_{4}$ are located at the vertices of a regular tetrahedron with edge length 1. The carbon atom $C$ is located at the center of the tetrahedron $C_{0}$. Let the four hydrogen atoms be $H_{1}, H_{2}, H_{3}, H_{4}$. Then... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 39870,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A secret agent is trying to decipher a passcode. So far, he has obtained the following information:\n\n- It is a four-digit number.\n- It is not divisible by seven.\n- The digit in the tens place is the sum of the digit in the units place and the digit in the hundreds place.\n-... | math | {
"style": "rule",
"ground_truth": "4583"
} | {
"split": "train",
"index": 39871,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number $a+\\sqrt{b}$ and its radical conjugate have a sum of $-4$ and a product of $1$. Find $a+b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 39872,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The water tank in the diagram is in the shape of an inverted right circular cone. The radius of its base is 20 feet, and its height is 100 feet. The water in the tank fills 50% of the tank's total capacity. Find the height of the water in the tank, which can be expressed in the... | math | {
"style": "rule",
"ground_truth": "52"
} | {
"split": "train",
"index": 39873,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A high school second-year class has a total of 1000 students. To investigate the vision condition of the students in this grade, if 50 samples are selected using the systematic sampling method, and all students are randomly numbered from 000, 001, 002, …, to 999, if it is deter... | math | {
"style": "rule",
"ground_truth": "101"
} | {
"split": "train",
"index": 39874,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mayuki walks once around a track shaped with straight sides and semicircular ends at a constant speed daily. The track has a width of \\(4\\) meters, and it takes her \\(24\\) seconds longer to walk around the outside edge than the inside edge. Determine Mayuki's speed in meter... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"split": "train",
"index": 39875,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the two trisection points on the minor axis of an ellipse and its two foci form a square, find the eccentricity of the ellipse. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{10}}{10}"
} | {
"split": "train",
"index": 39876,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two rectangles, one $8 \\times 10$ and the other $12 \\times 9$, are overlaid as shown in the picture. The area of the black part is 37. What is the area of the gray part? If necessary, round the answer to 0.01 or write the answer as a common fraction. Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "65"
} | {
"split": "train",
"index": 39877,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bored student walks down a hall that contains a row of closed lockers, numbered $1$ to $1024$. He opens the locker numbered 1, and then alternates between skipping and opening each locker thereafter. When he reaches the end of the hall, the student turns around and starts bac... | math | {
"style": "rule",
"ground_truth": "342"
} | {
"split": "train",
"index": 39878,
"solution": "List all the numbers from $1$ through $1024$, then do the process yourself!!! It will take about 25 minutes (if you don't start to see the pattern), but that's okay, eventually, you will get $\\boxed{342}$.\n (Note: If you try to do this, first look through all the... |
deepscaler | [
{
"role": "user",
"content": "The measures of the interior angles of a particular triangle are in a 5:6:7 ratio. What is the measure, in degrees, of the smallest interior angle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 39879,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a set of four distinct lines in a plane, there are exactly $N$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $N$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 39880,
"solution": "1. **Understanding the Problem:**\n The problem asks for the sum of all possible values of $N$, where $N$ is the number of distinct points that lie on two or more of four distinct lines in a plane.\n\n2. **Maximum Number of Intersections:**\n The maximum number... |
deepscaler | [
{
"role": "user",
"content": "What is \\( \\frac{1}{4} \\) more than 32.5? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32.75"
} | {
"split": "train",
"index": 39881,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $\\sqrt[3]{1+8} \\cdot \\sqrt[3]{1+\\sqrt[3]{8}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 39882,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "You have six blocks in a row, labeled 1 through 6, each with weight 1. Call two blocks $x \\leq y$ connected when, for all $x \\leq z \\leq y$, block $z$ has not been removed. While there is still at least one block remaining, you choose a remaining block uniformly at random an... | math | {
"style": "rule",
"ground_truth": "\\frac{163}{10}"
} | {
"split": "train",
"index": 39883,
"solution": "Note that the total cost is the total number of ordered pairs $(x, y)$ with $1 \\leq x, y \\leq 6$ such that $x$ and $y$ are connected right before $x$ gets removed. The probability that blocks $x$ and $y$ are connected just before block $x$ is removed is simply $\... |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is given that $a = b\\cos C + c\\sin B$.\n(1) Find angle $B$.\n(2) If $b = 4$, find the maximum area of triangle $ABC$. Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{2} + 4"
} | {
"split": "train",
"index": 39884,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number of four-digit even numbers formed without repeating digits from the numbers $2$, $0$, $1$, $7$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 39885,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $α$ is an angle in the second quadrant and $\\sin α= \\frac {3}{5}$, find $\\sin 2α$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac{24}{25}"
} | {
"split": "train",
"index": 39886,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many 12 step paths are there from point $A$ to point $C$ which pass through point $B$ on a grid, where $A$ is at the top left corner, $B$ is 5 steps to the right and 2 steps down from $A$, and $C$ is 7 steps to the right and 4 steps down from $A$? Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "126"
} | {
"split": "train",
"index": 39887,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\pi$ be a permutation of the numbers from 2 through 2012. Find the largest possible value of $\\log _{2} \\pi(2) \\cdot \\log _{3} \\pi(3) \\cdots \\log _{2012} \\pi(2012)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 39888,
"solution": "Note that $$\\begin{aligned} \\prod_{i=2}^{2012} \\log _{i} \\pi(i) & =\\prod_{i=2}^{2012} \\frac{\\log \\pi(i)}{\\log i} \\\\ & =\\frac{\\prod_{i=2}^{2012} \\log \\pi(i)}{\\prod_{i=2}^{2012} \\log i} \\\\ & =1 \\end{aligned}$$ where the last equality holds since $... |
deepscaler | [
{
"role": "user",
"content": "In the cells of a 9 × 9 square, there are non-negative numbers. The sum of the numbers in any two adjacent rows is at least 20, and the sum of the numbers in any two adjacent columns does not exceed 16. What can be the sum of the numbers in the entire table? Let's think step by... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 39889,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A B C$ be a triangle with $A B=9, B C=10$, and $C A=17$. Let $B^{\\prime}$ be the reflection of the point $B$ over the line $C A$. Let $G$ be the centroid of triangle $A B C$, and let $G^{\\prime}$ be the centroid of triangle $A B^{\\prime} C$. Determine the length of segm... | math | {
"style": "rule",
"ground_truth": "\\frac{48}{17}"
} | {
"split": "train",
"index": 39890,
"solution": "Let $M$ be the midpoint of $A C$. For any triangle, we know that the centroid is located $2 / 3$ of the way from the vertex, so we have $M G / M B=M G^{\\prime} / M B^{\\prime}=1 / 3$, and it follows that $M G G^{\\prime} \\sim M B B^{\\prime}$. Thus, $G G^{\\prime... |
deepscaler | [
{
"role": "user",
"content": "Integers $x$ and $y$ with $x>y>0$ satisfy $x+y+xy=101$. What is $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 39891,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the parallelogram with vertices $(10,45)$, $(10,114)$, $(28,153)$, and $(28,84)$. A line through the origin cuts this figure into two congruent polygons. The slope of the line is $m/n,$ where $m_{}$ and $n_{}$ are relatively prime positive integers. Find $m+n$. Let's t... | math | {
"style": "rule",
"ground_truth": "118"
} | {
"split": "train",
"index": 39892,
"solution": "You can clearly see that a line that cuts a parallelogram into two congruent pieces must go through the center of the parallelogram. Taking the midpoint of $(10,45)$, and $(28,153)$ gives $(19,99)$, which is the center of the parallelogram. Thus the slope of the li... |
deepscaler | [
{
"role": "user",
"content": "A line parallel to leg \\(AC\\) of right triangle \\(ABC\\) intersects leg \\(BC\\) at point \\(K\\) and the hypotenuse \\(AB\\) at point \\(N\\). On leg \\(AC\\), a point \\(M\\) is chosen such that \\(MK = MN\\). Find the ratio \\(\\frac{AM}{MC}\\) if \\(\\frac{BK}{BC} = 14\\... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 39893,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A quadrilateral in the plane has vertices $(1, 3)$, $(1, 1)$, $(2, 1)$ and $(2006, 2007)$. How many square units is the area of the quadrilateral? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3008"
} | {
"split": "train",
"index": 39894,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the set of all positive integer divisors of $100,000.$ How many numbers are the product of two distinct elements of $S?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "117"
} | {
"split": "train",
"index": 39895,
"solution": "1. **Identify the prime factorization of 100,000**: \n The prime factorization of $100,000$ is $100,000 = 10^5 = (2 \\cdot 5)^5 = 2^5 \\cdot 5^5$.\n\n2. **Determine the form of elements in set $S$**:\n The elements of $S$, the set of all positive integer diviso... |
deepscaler | [
{
"role": "user",
"content": "Kelvin the Frog was bored in math class one day, so he wrote all ordered triples $(a, b, c)$ of positive integers such that $a b c=2310$ on a sheet of paper. Find the sum of all the integers he wrote down. In other words, compute $$\\sum_{\\substack{a b c=2310 \\\\ a, b, c \\in... | math | {
"style": "rule",
"ground_truth": "49140"
} | {
"split": "train",
"index": 39896,
"solution": "Note that $2310=2 \\cdot 3 \\cdot 5 \\cdot 7 \\cdot 11$. The given sum clearly equals $3 \\sum_{a b c=2310} a$ by symmetry. The inner sum can be rewritten as $$\\sum_{a \\mid 2310} a \\cdot \\tau\\left(\\frac{2310}{a}\\right)$$ as for any fixed $a$, there are $\\ta... |
deepscaler | [
{
"role": "user",
"content": "A polynomial with integer coefficients is of the form\n\\[12x^3 - 4x^2 + a_1x + 18 = 0.\\]\nDetermine the number of different possible rational roots of this polynomial. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 39897,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the function $f(x) = \\frac{1}{2}(m-2)x^2 + (n-8)x + 1$ with $m \\geq 0$ and $n \\geq 0$ is monotonically decreasing in the interval $\\left[\\frac{1}{2}, 2\\right]$, then the maximum value of $mn$ is __________. Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 39898,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two points are taken on the lateral edge of a pyramid, dividing the edge into three equal parts. Planes parallel to the base are drawn through these points. Find the volume of the section of the pyramid between these planes, given that the volume of the entire pyramid is 1. Let... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{27}"
} | {
"split": "train",
"index": 39899,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse \\(\\frac{x^{2}}{16}+\\frac{y^{2}}{9}=1\\) with three points \\(P\\), \\(Q\\), and \\(R\\) on it, where \\(P\\) and \\(Q\\) are symmetric with respect to the origin. Find the maximum value of \\(|RP| + |RQ|\\). Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 39900,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four elevators in a skyscraper, differing in color (red, blue, green, and yellow), move in different directions at different but constant speeds. An observer timed the events as follows: At the 36th second, the red elevator caught up with the blue one (moving in the same direct... | math | {
"style": "rule",
"ground_truth": "46"
} | {
"split": "train",
"index": 39901,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circular table has 60 chairs around it. There are $N$ people seated at this table in such a way that the next person seated must sit next to someone. 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": "20"
} | {
"split": "train",
"index": 39902,
"solution": "To find the smallest possible value for $N$, we need to ensure that any new person seated must sit next to someone already seated. We will analyze the seating arrangement to determine the minimum $N$ that satisfies this condition.\n\n1. **Understanding the seating ... |
deepscaler | [
{
"role": "user",
"content": "Evaluate $|(12-9i)(8+15i)|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "255"
} | {
"split": "train",
"index": 39903,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$(-1)^{5^{2}} + 1^{2^{5}} = $ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 39904,
"solution": "1. **Evaluate the first term** $(-1)^{5^2}$:\n - Calculate the exponent: $5^2 = 25$.\n - Since $(-1)^{n}$ is $-1$ if $n$ is odd, and $25$ is odd, we have $(-1)^{25} = -1$.\n\n2. **Evaluate the second term** $1^{2^5}$:\n - Calculate the exponent: $2^5 = 32$.\n... |
deepscaler | [
{
"role": "user",
"content": "A dog and a cat simultaneously grab a sausage from different ends with their teeth. If the dog bites off its piece and runs away, the cat will get 300 grams more than the dog. If the cat bites off its piece and runs away, the dog will get 500 grams more than the cat. How much s... | math | {
"style": "rule",
"ground_truth": "400"
} | {
"split": "train",
"index": 39905,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What percent of the positive integers less than or equal to $120$ have no remainders when divided by $6$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16.\\overline{6}\\%"
} | {
"split": "train",
"index": 39906,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "( Elgin Johnston ) Legs $L_1, L_2, L_3, L_4$ of a square table each have length $n$ , where $n$ is a positive integer. For how many ordered 4-tuples $(k_1, k_2, k_3, k_4)$ of nonnegative integers can we cut a piece of length $k_i$ from the end of leg $L_i \\; (i = 1,2,3,4)$... | math | {
"style": "rule",
"ground_truth": "\\binom{n+3}{3}"
} | {
"split": "train",
"index": 39907,
"solution": "\nThe problem involves determining how many ways we can cut legs of a square table such that the table remains stable. Specifically, we aim to find the number of ordered 4-tuples \\((k_1, k_2, k_3, k_4)\\) where the lengths of the cuts \\(k_i\\) are non-negative in... |
deepscaler | [
{
"role": "user",
"content": "Inside a square $R_1$ with area 81, an equilateral triangle $T_1$ is inscribed such that each vertex of $T_1$ touches one side of $R_1$. Each midpoint of $T_1’s$ sides is connected to form a smaller triangle $T_2$. The process is repeated with $T_2$ to form $T_3$. Find the area... | math | {
"style": "rule",
"ground_truth": "\\frac{81\\sqrt{3}}{256}"
} | {
"split": "train",
"index": 39908,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In 10 boxes, place ping-pong balls such that the number of balls in each box is at least 11, not equal to 17, not a multiple of 6, and all numbers are distinct. What is the minimum number of ping-pong balls needed? Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "174"
} | {
"split": "train",
"index": 39909,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Abigail, Beatrice, and Carson combine their eggs to sell them at the market. If Abigail has 37 eggs, Beatrice has 49 eggs, and Carson has 14 eggs, and if eggs can only be sold in cartons of 12, how many eggs will be left over if all cartons are sold? Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 39910,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let\n\\[\\mathbf{M} = \\begin{pmatrix} a & b & c \\\\ b & c & a \\\\ c & a & b \\end{pmatrix}\\]be a matrix with complex entries such that $\\mathbf{M}^2 = \\mathbf{I}.$ If $abc = 1,$ then find the possible values of $a^3 + b^3 + c^3.$ Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "2,4"
} | {
"split": "train",
"index": 39911,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is an urn with $N$ slips of paper numbered from 1 to $N$. Two slips are drawn at random. What is the expected value of the ratio of the smaller number to the larger number drawn? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 39912,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The keys of a safe with five locks are cloned and distributed among eight people such that any of five of eight people can open the safe. What is the least total number of keys? $ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 39913,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A three-dimensional rectangular box with dimensions $X$, $Y$, and $Z$ has faces whose surface areas are $24$, $24$, $48$, $48$, $72$, and $72$ square units. What is $X$ + $Y$ + $Z$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 39914,
"solution": "1. **Identify the relationships between dimensions and surface areas**: Given that the surface areas of the faces of the box are $24$, $24$, $48$, $48$, $72$, and $72$ square units, we can infer that each pair of opposite faces has the same area. Let's denote the d... |
deepscaler | [
{
"role": "user",
"content": "Betty has a $3 \\times 4$ grid of dots. She colors each dot either red or maroon. Compute the number of ways Betty can color the grid such that there is no rectangle whose sides are parallel to the grid lines and whose vertices all have the same color. Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "408"
} | {
"split": "train",
"index": 39915,
"solution": "First suppose no 3 by 1 row is all red or all blue. Then each row is either two red and one blue, or two blue and one red. There are 6 possible configurations of such a row, and as long as no row is repeated, there's no monochromatic rectangle This gives $6 \\cdot ... |
deepscaler | [
{
"role": "user",
"content": "What number is a multiple of every integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 39916,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bin has 8 black balls and 7 white balls. 3 of the balls are drawn at random. What is the probability of drawing 2 of one color and 1 of the other color? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{5}"
} | {
"split": "train",
"index": 39917,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the product of $1011_2 \\cdot 101_2$. Express your answer in base 2. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "110111"
} | {
"split": "train",
"index": 39918,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a=2001$. Consider the set $A$ of all pairs of integers $(m,n)$ with $n\\neq0$ such that\n(i) $m<2a$;\n(ii) $2n|(2am-m^2+n^2)$;\n(iii) $n^2-m^2+2mn\\leq2a(n-m)$.\nFor $(m, n)\\in A$, let \\[f(m,n)=\\frac{2am-m^2-mn}{n}.\\]\nDetermine the maximum and minimum values of $f$. L... | math | {
"style": "rule",
"ground_truth": "2 \\text{ and } 3750"
} | {
"split": "train",
"index": 39919,
"solution": "\nLet \\( a = 2001 \\). Consider the set \\( A \\) of all pairs of integers \\((m, n)\\) with \\( n \\neq 0 \\) such that:\n1. \\( m < 2a \\),\n2. \\( 2n \\mid (2am - m^2 + n^2) \\),\n3. \\( n^2 - m^2 + 2mn \\leq 2a(n - m) \\).\n\nFor \\((m, n) \\in A\\), let\n\\[ ... |
deepscaler | [
{
"role": "user",
"content": "A circle is tangent to sides \\( AB \\) and \\( AD \\) of rectangle \\( ABCD \\) and intersects side \\( DC \\) at a single point \\( F \\) and side \\( BC \\) at a single point \\( E \\).\n\nFind the area of trapezoid \\( AFCB \\) if \\( AB = 32 \\), \\( AD = 40 \\), and \\( B... | math | {
"style": "rule",
"ground_truth": "1180"
} | {
"split": "train",
"index": 39920,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "50 students from fifth to ninth grade collectively posted 60 photos on Instagram, with each student posting at least one photo. All students in the same grade (parallel) posted an equal number of photos, while students from different grades posted different numbers of photos. H... | math | {
"style": "rule",
"ground_truth": "46"
} | {
"split": "train",
"index": 39921,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Circles of radius $3$ and $6$ are externally tangent to each other and are internally tangent to a circle of radius $9$. The circle of radius $9$ has a chord that is a common external tangent of the other two circles. Find the square of the length of this chord.\n[asy] pointpen... | math | {
"style": "rule",
"ground_truth": "224"
} | {
"split": "train",
"index": 39922,
"solution": "We label the points as following: the centers of the circles of radii $3,6,9$ are $O_3,O_6,O_9$ respectively, and the endpoints of the chord are $P,Q$. Let $A_3,A_6,A_9$ be the feet of the perpendiculars from $O_3,O_6,O_9$ to $\\overline{PQ}$ (so $A_3,A_6$ are the ... |
deepscaler | [
{
"role": "user",
"content": "There are 2008 distinct points on a circle. If you connect two of these points to form a line and then connect another two points (distinct from the first two) to form another line, what is the probability that the two lines intersect inside the circle? Let's think step by step... | math | {
"style": "rule",
"ground_truth": "1/3"
} | {
"split": "train",
"index": 39923,
"solution": "Given four of these points, there are 3 ways in which to connect two of them and then connect the other two, and of these possibilities exactly one will intersect inside the circle. Thus $1 / 3$ of all the ways to connect two lines and then connect two others have ... |
deepscaler | [
{
"role": "user",
"content": "Six chairs sit in a row. Six people randomly seat themselves in the chairs. Each person randomly chooses either to set their feet on the floor, to cross their legs to the right, or to cross their legs to the left. There is only a problem if two people sitting next to each oth... | math | {
"style": "rule",
"ground_truth": "1106"
} | {
"split": "train",
"index": 39924,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the ellipse $\\frac{x^2}{16} + \\frac{y^2}{9} = 1$, find the slope of the line containing a chord that has the point $M(-2,1)$ as its midpoint. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{8}"
} | {
"split": "train",
"index": 39925,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the graph of the power function $y=mx^{\\alpha}$ (where m and $\\alpha \\in \\mathbb{R}$) passes through the point $(8, \\frac{1}{4})$, then $\\alpha$ equals \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{2}{3}"
} | {
"split": "train",
"index": 39926,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define $f(x) = \\frac{3}{27^x + 3}.$ Calculate the sum\n\\[ f\\left(\\frac{1}{2001}\\right) + f\\left(\\frac{2}{2001}\\right) + f\\left(\\frac{3}{2001}\\right) + \\dots + f\\left(\\frac{2000}{2001}\\right). \\] Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"split": "train",
"index": 39927,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the area of a trapezoid with bases 11 and 4 and diagonals 9 and 12. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 39928,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In right triangle $GHI$, we have $\\angle G = 30^\\circ$, $\\angle H = 90^\\circ$, and $HI = 12$. Find $GH$ to the nearest tenth. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20.8"
} | {
"split": "train",
"index": 39929,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an isosceles triangle $ABC$ satisfying $AB=AC$, $\\sqrt{3}BC=2AB$, and point $D$ is on side $BC$ with $AD=BD$, then the value of $\\sin \\angle ADB$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2 \\sqrt{2}}{3}"
} | {
"split": "train",
"index": 39930,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many $4$-digit positive integers (that is, integers between $1000$ and $9999$, inclusive) having only even digits are divisible by $5?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 39931,
"solution": "1. **Identify the range and conditions**: We are looking for 4-digit integers between 1000 and 9999, inclusive, that have only even digits and are divisible by 5.\n\n2. **Digits must be even**: The possible even digits are 0, 2, 4, 6, and 8.\n\n3. **Divisibility by... |
deepscaler | [
{
"role": "user",
"content": "Let \\( G \\) be the centroid of \\(\\triangle ABC\\). Given \\( BG \\perp CG \\) and \\( BC = \\sqrt{2} \\), find the maximum value of \\( AB + AC \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{5}"
} | {
"split": "train",
"index": 39932,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function f(x) = 3^x with a range of M, where x < -1, find the probability that a number x randomly chosen from the interval (-1, 1) belongs to M. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"split": "train",
"index": 39933,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The coefficient of the $x^2$ term in the expansion of $(1+x)(1+ \\sqrt{x})^5$ is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 39934,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many odd integers are there between $\\frac{17}{4}$ and $\\frac{35}{2}?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 39935,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that out of 6 products, 2 are defective and the rest are qualified, calculate the probability of selecting exactly one defective product from these 6 products. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{15}"
} | {
"split": "train",
"index": 39936,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The inverse of \\( f(x) = \\frac{3x-2}{x+4} \\) may be written in the form \\( f^{-1}(x) = \\frac{ax+b}{cx+d} \\), where \\( a, b, c, \\) and \\( d \\) are real numbers. Find \\( \\frac{a}{c} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 39937,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangles $ABC$ and $AEF$ are such that $B$ is the midpoint of $\\overline{EF}.$ Also, $AB = EF = 1,$ $BC = 6,$ $CA = \\sqrt{33},$ and\n\\[\\overrightarrow{AB} \\cdot \\overrightarrow{AE} + \\overrightarrow{AC} \\cdot \\overrightarrow{AF} = 2.\\]Find the cosine of the angle be... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 39938,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence $\\{a_n\\}$, its first term is $7$, and $a_n= \\frac{1}{2}a_{n-1}+3(n\\geqslant 2)$, find the value of $a_6$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{193}{32}"
} | {
"split": "train",
"index": 39939,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If \\( 8 + 6 = n + 8 \\), what is the value of \\( n \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 39940,
"solution": "Since \\( 8+6=n+8 \\), then subtracting 8 from both sides, we obtain \\( 6=n \\) and so \\( n \\) equals 6."
} |
deepscaler | [
{
"role": "user",
"content": "In convex quadrilateral \\(EFGH\\), \\(\\angle E = \\angle G\\), \\(EF = GH = 150\\), and \\(EH \\neq FG\\). The perimeter of \\(EFGH\\) is 580. Find \\(\\cos E\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{14}{15}"
} | {
"split": "train",
"index": 39941,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number of real roots of the equation $\\frac{x}{100} = \\sin x$ is: \n(32nd United States of America Mathematical Olympiad, 1981) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "63"
} | {
"split": "train",
"index": 39942,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f : [0, 1] \\rightarrow \\mathbb{R}$ be a monotonically increasing function such that $$ f\\left(\\frac{x}{3}\\right) = \\frac{f(x)}{2} $$ $$ f(1 0 x) = 2018 - f(x). $$ If $f(1) = 2018$ , find $f\\left(\\dfrac{12}{13}\\right)$ .\n Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "2018"
} | {
"split": "train",
"index": 39943,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function f(x) = 3x^2 + 5x - 2, find the minimum value of the function f(x) in the interval [-2, -1]. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 39944,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many distinct four-digit numbers are divisible by 3 and have 23 as their last two digits? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 39945,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alice is jogging north at a speed of 6 miles per hour, and Tom is starting 3 miles directly south of Alice, jogging north at a speed of 9 miles per hour. Moreover, assume Tom changes his path to head north directly after 10 minutes of eastward travel. How many minutes after thi... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 39946,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose I have 6 shirts, 4 ties, and 3 pairs of pants. If an outfit requires a shirt and pants, and can either have a tie or not have a tie, how many outfits can I make? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 39947,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Kolya, after walking one-fourth of the way from home to school, realized that he forgot his problem book. If he does not go back for it, he will arrive at school 5 minutes before the bell rings, but if he goes back, he will be 1 minute late. How long (in minutes) does it take t... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 39948,
"solution": ""
} |
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