data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "\nIf organisms do not die but only divide, then the population will certainly never die out. \n\nThe conditions are satisfied by the function whose graph is highlighted in the image.\n\n$$\nx(p)=\\left\\{\\begin{array}{l}\n1, \\text { if } 0 \\leq p \\leq \\frac{1}{2} \\\\\n\\f... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 500,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number $x$ satisfies $5x^2 + 4 = 3x + 9$. Find the value of $(10x - 3)^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "109"
} | {
"split": "train",
"index": 501,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive three-digit integers with each digit greater than 4 are divisible by 6? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 502,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The base of the pyramid is a right triangle with a hypotenuse equal to 6 and an acute angle of $15^{\\circ}$. All lateral edges are inclined to the plane of the base at an angle of $45^{\\circ}$. Find the volume of the pyramid. Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "4.5"
} | {
"split": "train",
"index": 503,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the fictional country of Novaguard, they use the same twelve-letter Rotokas alphabet for their license plates, which are also five letters long. However, for a special series of plates, the following rules apply: \n- The plate must start with either P or T.\n- The plate must... | math | {
"style": "rule",
"ground_truth": "1440"
} | {
"split": "train",
"index": 504,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be randomly selected numbers from the set $\\{1,2,3,4\\}$. Given the line $l:y=ax+b$ and the circle $O:x^{2}+y^{2}=1$, the probability that the line $l$ intersects the circle $O$ is ______. The expected number of intersection points between the line $l$ and the ... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{4}"
} | {
"split": "train",
"index": 505,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Vasya and Petya simultaneously started running from the starting point of a circular track in opposite directions at constant speeds. At some point, they met. Vasya completed a full lap and, continuing to run in the same direction, reached the point of their first meeting at th... | math | {
"style": "rule",
"ground_truth": "\\frac{1+\\sqrt{5}}{2}"
} | {
"split": "train",
"index": 506,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the adjoining figure, two circles with radii $8$ and $6$ are drawn with their centers $12$ units apart. At $P$, one of the points of intersection, a line is drawn in such a way that the chords $QP$ and $PR$ have equal length. Find the square of the length of $QP$.\n[asy]size... | math | {
"style": "rule",
"ground_truth": "130"
} | {
"split": "train",
"index": 507,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A three-digit natural number $abc$ is termed a \"convex number\" if and only if the digits $a$, $b$, and $c$ (representing the hundreds, tens, and units place, respectively) satisfy $a < b$ and $c < b$. Given that $a$, $b$, and $c$ belong to the set $\\{5, 6, 7, 8, 9\\}$ and ar... | math | {
"style": "rule",
"ground_truth": "\\frac {1}{3}"
} | {
"split": "train",
"index": 508,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the greatest common divisor of 12 and 20. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 509,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, establish a polar coordinate system with the origin $O$ as the pole and the positive semi-axis of the $x$-axis as the polar axis, using the same unit of length in both coordinate systems. Given that circle $C$ has a center at point ($2$... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 510,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)$ be a polynomial of degree 2006 with real coefficients, and let its roots be $r_1,$ $r_2,$ $\\dots,$ $r_{2006}.$ There are exactly 1006 distinct values among\n\\[|r_1|, |r_2|, \\dots, |r_{2006}|.\\]What is the minimum number of real roots that $f(x)$ can have? Let's ... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 511,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the line \\( y = x - 1 \\) intersects the ellipse \\( \\frac{x^{2}}{a^{2}} + \\frac{y^{2}}{a^{2} - 1} = 1 \\) (where \\( a > 1 \\)) at points \\( A \\) and \\( B \\). If the circle with diameter \\( AB \\) passes through the left focus of the ellipse, find the value of \\... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6} + \\sqrt{2}}{2}"
} | {
"split": "train",
"index": 512,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose there are 15 dogs including Rex and Daisy. We need to divide them into three groups of sizes 6, 5, and 4. How many ways can we form the groups such that Rex is in the 6-dog group and Daisy is in the 4-dog group? Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "72072"
} | {
"split": "train",
"index": 513,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a_1, a_2, \\ldots$ be a sequence determined by the rule $a_n = \\frac{a_{n-1}}{2}$ if $a_{n-1}$ is even and $a_n = 3a_{n-1} + 1$ if $a_{n-1}$ is odd. For how many positive integers $a_1 \\le 3000$ is it true that $a_1$ is less than each of $a_2$, $a_3$, $a_4$, and $a_5$? L... | math | {
"style": "rule",
"ground_truth": "750"
} | {
"split": "train",
"index": 514,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCDEFGH$ be a cube with each edge of length $s$. A right square pyramid is placed on top of the cube such that its base aligns perfectly with the top face $EFGH$ of the cube, and its apex $P$ is directly above $E$ at a height $s$. Calculate $\\sin \\angle FAP$. Let's thin... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 515,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An infinite geometric series has a first term of $540$ and a sum of $4500$. What is its common ratio, and what is the second term of the series? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "475.2"
} | {
"split": "train",
"index": 516,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The average age of seven children is 8 years old. Each child is a different age, and there is a difference of three years in the ages of any two consecutive children. In years, how old is the oldest child? Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 517,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\triangle ABC$ have side lengths $AB = 12$, $AC = 16$, and $BC = 20$. Inside $\\angle BAC$, two circles are positioned, each tangent to rays $\\overline{AB}$ and $\\overline{AC}$, and the segment $\\overline{BC}$. Compute the distance between the centers of these two circ... | math | {
"style": "rule",
"ground_truth": "20\\sqrt{2}"
} | {
"split": "train",
"index": 518,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At a crossroads, if vehicles are not allowed to turn back, calculate the total number of possible driving routes. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 519,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two fair coins are simultaneously flipped. This is done repeatedly until at least one of the coins comes up heads, at which point the process stops. What is the probability that the other coin also came up heads on this last flip? Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "1/3"
} | {
"split": "train",
"index": 520,
"solution": "$1 / 3$. Let the desired probability be $p$. There is a $1 / 4$ chance that both coins will come up heads on the first toss. Otherwise, both can come up heads simultaneously only if both are tails on the first toss, and then the process restarts as if from the beginn... |
deepscaler | [
{
"role": "user",
"content": "Calculate the limit of the function:\n\n$$\n\\lim _{x \\rightarrow 0}(1-\\ln (1+\\sqrt[3]{x}))^{\\frac{x}{\\sin ^{4} \\sqrt[3]{x}}}\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "e^{-1}"
} | {
"split": "train",
"index": 521,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given positive integers \\(a_{1}, a_{2}, \\ldots, a_{2023}\\) such that \\(a_{k}=\\sum_{i=1}^{2023}\\left|a_{k}-a_{i}\\right|\\) for all \\(1 \\leq k \\leq 2023\\), find the minimum possible value of \\(a_{1}+a_{2}+\\cdots+a_{2023}\\). Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "2046264"
} | {
"split": "train",
"index": 522,
"solution": "Without loss of generality, let \\(a_{1} \\leq a_{2} \\leq \\cdots \\leq a_{2023}\\). Then, note that \\(a_{k+1}-a_{k}=\\sum_{i=1}^{2023}\\left|a_{k+1}-a_{i}\\right|-\\left|a_{k}-a_{i}\\right|=k(a_{k+1}-a_{k})-(2023-k)(a_{k+1}-a_{k})=(2k-2023)(a_{k+1}-a_{k})\\). Thus... |
deepscaler | [
{
"role": "user",
"content": "Flea Kuzya can make jumps along a line \\(L\\). The starting point for the jumps is point \\(A\\) on line \\(L\\), the length of each jump is \\(h\\), and the direction of each jump is chosen randomly and equally likely. Find the probability that after making between two and fi... | math | {
"style": "rule",
"ground_truth": "5/8"
} | {
"split": "train",
"index": 523,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For any positive integer \\( k \\), let \\( f_{1}(k) \\) be the square of the sum of the digits of \\( k \\) when written in decimal notation. For \\( n > 1 \\), let \\( f_{n}(k) = f_{1}\\left(f_{n-1}(k)\\right) \\). What is \\( f_{1992}\\left(2^{1991}\\right) \\)? Let's think ... | math | {
"style": "rule",
"ground_truth": "256"
} | {
"split": "train",
"index": 524,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the biggest positive integer $n$ such that $n$ is $167$ times the amount of it's positive divisors. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2004"
} | {
"split": "train",
"index": 525,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the rectangular coordinate system $xOy$, the parametric equation of line $l$ is $\\begin{cases} x=t \\\\ y=t+1 \\end{cases}$ (where $t$ is the parameter), and the parametric equation of curve $C$ is $\\begin{cases} x=2+2\\cos \\phi \\\\ y=2\\sin \\phi \\end{cases}$ (where $\... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"split": "train",
"index": 526,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{a},$ $\\mathbf{b},$ $\\mathbf{c}$ be vectors such that $\\|\\mathbf{a}\\| = \\|\\mathbf{b}\\| = 1,$ $\\|\\mathbf{c}\\| = 2,$ and\n\\[\\mathbf{a} \\times (\\mathbf{a} \\times \\mathbf{c}) + \\mathbf{b} = \\mathbf{0}.\\]Find the smallest possible angle between $\\ma... | math | {
"style": "rule",
"ground_truth": "30^\\circ"
} | {
"split": "train",
"index": 527,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{a},$ $\\mathbf{b},$ $\\mathbf{c}$ be vectors, and let $D$ be the determinant of the matrix whose column vectors are $\\mathbf{a},$ $\\mathbf{b},$ and $\\mathbf{c}.$ Then the determinant of the matrix whose column vectors are $\\mathbf{a} \\times \\mathbf{b},$ $\\... | math | {
"style": "rule",
"ground_truth": "(1,2)"
} | {
"split": "train",
"index": 528,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S=\\{1,2, \\ldots, 2008\\}$. For any nonempty subset $A \\subset S$, define $m(A)$ to be the median of $A$ (when $A$ has an even number of elements, $m(A)$ is the average of the middle two elements). Determine the average of $m(A)$, when $A$ is taken over all nonempty subs... | math | {
"style": "rule",
"ground_truth": "\\frac{2009}{2}"
} | {
"split": "train",
"index": 529,
"solution": "For any subset $A$, we can define the \"reflected subset\" $A^{\\prime}=\\{i \\mid 2009-i \\in A\\}$. Then $m(A)=2009-m\\left(A^{\\prime}\\right)$. Note that as $A$ is taken over all nonempty subsets of $S, A^{\\prime}$ goes through all the nonempty subsets of $S$ as... |
deepscaler | [
{
"role": "user",
"content": "If $a \\text{ Y } b$ is defined as $a \\text{ Y } b = a^2 - 2ab + b^2$, what is the value of $3 \\text{ Y } 2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 530,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many multiples of 4 are between 100 and 350? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "62"
} | {
"split": "train",
"index": 531,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the area, in square units, of the interior region formed by the lines $y = 2x - 4, y = -3x +16$ and the $y$-axis? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 532,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x$ be a real number such that $x^{3}+4 x=8$. Determine the value of $x^{7}+64 x^{2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "128"
} | {
"split": "train",
"index": 533,
"solution": "For any integer $n \\geq 0$, the given implies $x^{n+3}=-4 x^{n+1}+8 x^{n}$, so we can rewrite any such power of $x$ in terms of lower powers. Carrying out this process iteratively gives $$\\begin{aligned} x^{7} & =-4 x^{5}+8 x^{4} \\\\ & =8 x^{4}+16 x^{3}-32 x^{2} \... |
deepscaler | [
{
"role": "user",
"content": "The line $l_1$: $x+my+6=0$ is parallel to the line $l_2$: $(m-2)x+3y+2m=0$. Find the value of $m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 534,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Lisa drew graphs of all functions of the form \\( y = ax + b \\), where \\( a \\) and \\( b \\) take all natural values from 1 to 100. How many of these graphs pass through the point \\((3, 333)\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 535,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways are there to put 7 balls in 2 boxes if the balls are distinguishable but the boxes are not? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 536,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rural School USA has 105 students enrolled. There are 60 boys and 45 girls. If $\\frac{1}{10}$ of the boys and $\\frac{1}{3}$ of the girls are absent on one day, what percent of the total student population is absent? Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "20 \\%"
} | {
"split": "train",
"index": 537,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many two-digit numbers have digits whose sum is a perfect square? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 538,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cylindrical barrel with radius $4$ feet and height $10$ feet is full of water. A solid cube with side length $8$ feet is set into the barrel so that the diagonal of the cube is vertical. The volume of water thus displaced is $v$ cubic feet. Find $v^2$.\n[asy] import three; im... | math | {
"style": "rule",
"ground_truth": "384"
} | {
"split": "train",
"index": 539,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ in $\\triangle ABC$, respectively, and $\\sqrt{3}c\\sin A = a\\cos C$.\n\n$(I)$ Find the value of $C$;\n\n$(II)$ If $c=2a$ and $b=2\\sqrt{3}$, find the area of $\\triangle ABC$. Let's think step by ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{15} - \\sqrt{3}}{2}"
} | {
"split": "train",
"index": 540,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many integers $n$ with $1 \\le n \\le 2012$ is the product\n\\[\n \\prod_{k=0}^{n-1} \\left( \\left( 1 + e^{2 \\pi i k / n} \\right)^n + 1 \\right) \n\\]equal to zero? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "335"
} | {
"split": "train",
"index": 541,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an equilateral triangle $ABC$ with side length $1$, let $\\overrightarrow{BC} = \\overrightarrow{a}$, $\\overrightarrow{AC} = \\overrightarrow{b}$, and $\\overrightarrow{AB} = \\overrightarrow{c}$. Evaluate the value of $\\overrightarrow{a} \\cdot \\overrightarrow{b} + \\ove... | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{2}"
} | {
"split": "train",
"index": 542,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The population of a small town is $480$. The graph indicates the number of females and males in the town, but the vertical scale-values are omitted. How many males live in the town? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "160"
} | {
"split": "train",
"index": 543,
"solution": "1. **Understanding the Problem**: The total population of the town is given as $480$. The graph, which is not shown here, apparently divides the population into three equal parts, each represented by a square. Each square represents the same number of people, denoted... |
deepscaler | [
{
"role": "user",
"content": "Alicia earns 20 dollars per hour, of which $1.45\\%$ is deducted to pay local taxes. How many cents per hour of Alicia's wages are used to pay local taxes? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 544,
"solution": "1. **Convert Alicia's hourly wage to cents**: Alicia earns $20$ dollars per hour. Since there are $100$ cents in a dollar, her hourly wage in cents is:\n \\[\n 20 \\text{ dollars} \\times 100 \\text{ cents/dollar} = 2000 \\text{ cents}\n \\]\n\n2. **Calculate t... |
deepscaler | [
{
"role": "user",
"content": "Let $f(n) = \\frac{x_1 + x_2 + \\cdots + x_n}{n}$, where $n$ is a positive integer. If $x_k = (-1)^k, k = 1, 2, \\cdots, n$, the set of possible values of $f(n)$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\{0, -\\frac{1}{n}\\}"
} | {
"split": "train",
"index": 545,
"solution": "To find the set of possible values of $f(n)$, we first need to evaluate the sum $x_1 + x_2 + \\cdots + x_n$ where $x_k = (-1)^k$ for $k = 1, 2, \\ldots, n$.\n\n1. **Expression for $x_k$:**\n - $x_k = (-1)^k$ means that $x_k$ alternates between $-1$ and $1$ starting... |
deepscaler | [
{
"role": "user",
"content": "Let $S_{n}$ be the sum of the first $n$ terms of an arithmetic sequence $\\{a_{n}\\}$ with distinct terms, given that $a_{3}a_{8}=3a_{11}$, $S_{3}=9$.\n1. Find the general term formula for the sequence $\\{a_{n}\\}$.\n2. If $b_{n}= \\frac {1}{ \\sqrt {a_{n}}+ \\sqrt {a_{n+1}}}$... | math | {
"style": "rule",
"ground_truth": "4 \\sqrt {2}"
} | {
"split": "train",
"index": 546,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that positive real numbers $a$ and $b$ satisfy $a \\gt b$ and $ab=\\frac{1}{2}$, find the minimum value of $\\frac{4{a}^{2}+{b}^{2}+3}{2a-b}$. 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": 547,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{D}$ be a matrix representing a dilation with scale factor $k > 0,$ and let $\\mathbf{R}$ be a matrix representing a rotation about the origin by an angle of $\\theta$ counter-clockwise. If\n\\[\\mathbf{R} \\mathbf{D} = \\begin{pmatrix} 8 & -4 \\\\ 4 & 8 \\end{pma... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 548,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $m$ is a positive integer, and given that $\\mathop{\\text{lcm}}[40, m] = 120$ and $\\mathop{\\text{lcm}}[m, 45] = 180$, what is $m$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 549,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If \\( p \\) and \\( q \\) are positive integers, \\(\\max (p, q)\\) is the maximum of \\( p \\) and \\( q \\) and \\(\\min (p, q)\\) is the minimum of \\( p \\) and \\( q \\). For example, \\(\\max (30,40)=40\\) and \\(\\min (30,40)=30\\). Also, \\(\\max (30,30)=30\\) and \\(\... | math | {
"style": "rule",
"ground_truth": "4100"
} | {
"split": "train",
"index": 550,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of all numbers of the form $2k + 1$, where $k$ takes on integral values from $1$ to $n$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "n(n+2)"
} | {
"split": "train",
"index": 551,
"solution": "1. **Identify the sequence**: The problem asks for the sum of numbers of the form $2k + 1$ where $k$ ranges from $1$ to $n$. This forms a sequence of odd numbers starting from $3$ (when $k=1$, $2k+1=3$) up to $2n+1$ (when $k=n$, $2k+1=2n+1$).\n\n2. **Write out the se... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = x^3 - 3x^2 - 9x + 1$,\n(1) Determine the monotonicity of the function on the interval $[-4, 4]$.\n(2) Calculate the function's local maximum and minimum values as well as the absolute maximum and minimum values on the interval $[-4, 4]$. Let's think s... | math | {
"style": "rule",
"ground_truth": "-75"
} | {
"split": "train",
"index": 552,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P(x)=x^2+bx+c$ . Suppose $P(P(1))=P(P(-2))=0$ and $P(1)\\neq P(-2)$ . Then $P(0)=$ 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": 553,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a question: \"If the value of the algebraic expression $5a+3b$ is $-4$, then what is the value of the algebraic expression $2\\left(a+b\\right)+4\\left(2a+b\\right)$?\" Tang, who loves to use his brain, solved the problem as follows: \nOriginal expression $=2a+2b+8a+4... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 554,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In rectangle $ABCD$, $AB=6$ and $AD=8$. Point $M$ is the midpoint of $\\overline{AD}$. What is the area of $\\triangle AMC$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 555,
"solution": "\n1. **Understanding the Problem:**\n - We have a rectangle $ABCD$ with $AB = 6$ and $AD = 8$.\n - $M$ is the midpoint of $\\overline{AD}$, so $AM = MD = \\frac{AD}{2} = \\frac{8}{2} = 4$.\n - We need to find the area of $\\triangle AMC$.\n\n2. **Using the Tria... |
deepscaler | [
{
"role": "user",
"content": "Given the inequality about $x$, $|x-1|+|x+a|\\leqslant 8$, the minimum value of $a$ is ________. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-9"
} | {
"split": "train",
"index": 556,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In Pascal's Triangle, each number is the sum of the number just above it and to the left and the number just above it and to the right. So the middle number in Row 2 is $2$ because $1+1=2.$ What is the sum of the numbers in Row 8 of Pascal's Triangle?\n\n\\begin{tabular}{rccccc... | math | {
"style": "rule",
"ground_truth": "256"
} | {
"split": "train",
"index": 557,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A diagonal of a polygon is a segment joining two nonconsecutive vertices of the polygon. How many diagonals does a regular octagon have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 558,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Art, Bella, Carla, and Dan each bake cookies of different shapes with the same thickness. Art's cookies are rectangles: 4 inches by 6 inches, while Bella's cookies are equilateral triangles with a side length of 5 inches. Carla's cookies are squares with a side length of 4 inch... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 559,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest real $C$ such that for all pairwise distinct positive real $a_{1}, a_{2}, \\ldots, a_{2019}$ the following inequality holds $$\\frac{a_{1}}{\\left|a_{2}-a_{3}\\right|}+\\frac{a_{2}}{\\left|a_{3}-a_{4}\\right|}+\\ldots+\\frac{a_{2018}}{\\left|a_{2019}-a_{1}\\rig... | math | {
"style": "rule",
"ground_truth": "1010"
} | {
"split": "train",
"index": 560,
"solution": "Without loss of generality we assume that $\\min \\left(a_{1}, a_{2}, \\ldots, a_{2019}\\right)=a_{1}$. Note that if $a, b, c$ $(b \\neq c)$ are positive, then $\\frac{a}{|b-c|}>\\min \\left(\\frac{a}{b}, \\frac{a}{c}\\right)$. Hence $$S=\\frac{a_{1}}{\\left|a_{2}-a_... |
deepscaler | [
{
"role": "user",
"content": "Compute the number of ordered triples of integers $(a,b,c)$ between $1$ and $12$ , inclusive, such that, if $$ q=a+\\frac{1}{b}-\\frac{1}{b+\\frac{1}{c}}, $$ then $q$ is a positive rational number and, when $q$ is written in lowest terms, the numerator is divisible... | math | {
"style": "rule",
"ground_truth": "132"
} | {
"split": "train",
"index": 561,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that Ben spent some amount of money and David spent $0.5 less for each dollar Ben spent, and Ben paid $16.00 more than David, determine the total amount they spent together in the bagel store. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48.00"
} | {
"split": "train",
"index": 562,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the numbers $\\mathbf{1 5 3 , 3 7 0 , 3 7 1 , 4 0 7}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "153, 370, 371, 407"
} | {
"split": "train",
"index": 563,
"solution": "The numbers are $\\mathbf{1 5 3 , 3 7 0 , 3 7 1 , 4 0 7}$."
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( z \\) be a complex number such that \\( |z| = 1 \\). If the equation \\( z x^{2} + 2 \\bar{z} x + 2 = 0 \\) in terms of \\( x \\) has a real root, find the sum of all such complex numbers \\( z \\). Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{2}"
} | {
"split": "train",
"index": 564,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer $n$ for which $n^2$ is divisible by 18 and $n^3$ is divisible by 640? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 565,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a_1, a_2, \\ldots$ and $b_1, b_2, \\ldots$ be arithmetic progressions such that $a_1 = 25, b_1 = 75$, and $a_{100} + b_{100} = 100$. Find the sum of the first hundred terms of the progression $a_1 + b_1, a_2 + b_2, \\ldots$ Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "10,000"
} | {
"split": "train",
"index": 566,
"solution": "1. **Identify the general form of the terms in the sequences:**\n Given that $a_1 = 25$ and $b_1 = 75$, and both sequences are arithmetic progressions, we can write the $n$-th terms of these sequences as:\n \\[ a_n = a_1 + (n-1)d_a = 25 + (n-1)d_a \\]\n \\[ b_n... |
deepscaler | [
{
"role": "user",
"content": "In acute triangle $\\text{ABC}$, the lengths of the sides opposite to angles $\\text{A}$, $\\text{B}$, and $\\text{C}$ are $a$, $b$, and $c$ respectively, and $a = 2b\\sin{\\text{A}}$.\n\n(II) Find the measure of angle $\\text{B}$;\n\n(III) If $a = 3\\sqrt{3}$ and $c = 5$, find... | math | {
"style": "rule",
"ground_truth": "\\sqrt{7}"
} | {
"split": "train",
"index": 567,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, what is the perimeter of polygon $PQRST$? [asy]\n\nimport olympiad;\n\nsize(6cm); // ADJUST\n\npair p = (0, 6);\npair q = (3, 6);\n\npair r = (3, 3);\npair t = (0, 0);\npair s = (7, 0);\n\ndraw(p--q--r--s--t--cycle);\nlabel(\"$P$\", p, NW);\nlabel(\"$Q$\", q, NE... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 568,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The hyperbola $C: \\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1 \\ (a > 0, b > 0)$ has an asymptote that is 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 such that $|F_1A| = 2|F_2A|$. Find $\\cos \\angle AF... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{5}"
} | {
"split": "train",
"index": 569,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the polar coordinate system and the Cartesian coordinate system xOy, which have the same unit of length, with the origin O as the pole and the positive half-axis of x as the polar axis. The parametric equation of line l is $$\\begin{cases} x=2+ \\frac {1}{2}t \\\\ y= \\frac ... | math | {
"style": "rule",
"ground_truth": "\\frac {8 \\sqrt {7}}{3}"
} | {
"split": "train",
"index": 570,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\alpha$ is an angle in the third quadrant, the function $f(\\alpha)$ is defined as:\n$$f(\\alpha) = \\frac {\\sin(\\alpha - \\frac {\\pi}{2}) \\cdot \\cos( \\frac {3\\pi}{2} + \\alpha) \\cdot \\tan(\\pi - \\alpha)}{\\tan(-\\alpha - \\pi) \\cdot \\sin(-\\alpha - \\p... | math | {
"style": "rule",
"ground_truth": "\\frac{6\\sqrt{2} - 1}{10}"
} | {
"split": "train",
"index": 571,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different integers can be expressed as the sum of three distinct members of the set $\\{1,4,7,10,13,16,19\\}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 572,
"solution": "1. **Identify the set and its properties**: The set given is $\\{1, 4, 7, 10, 13, 16, 19\\}$. This set is an arithmetic sequence where each term increases by 3 from the previous term.\n\n2. **Determine the range of possible sums**: We need to find the sums of three d... |
deepscaler | [
{
"role": "user",
"content": "Calculate: \n\n$$\\frac{\\left(1+\\frac{1}{2}\\right)^{2} \\times\\left(1+\\frac{1}{3}\\right)^{2} \\times\\left(1+\\frac{1}{4}\\right)^{2} \\times\\left(1+\\frac{1}{5}\\right)^{2} \\times \\cdots \\times\\left(1+\\frac{1}{10}\\right)^{2}}{\\left(1-\\frac{1}{2^{2}}\\right) \\ti... | math | {
"style": "rule",
"ground_truth": "55"
} | {
"split": "train",
"index": 573,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( a \\), \\( b \\), and \\( c \\) be the roots of \\( x^3 - x + 2 = 0 \\). Find \\( \\frac{1}{a+2} + \\frac{1}{b+2} + \\frac{1}{c+2} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{4}"
} | {
"split": "train",
"index": 574,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(1) Consider the function $f(x) = |x - \\frac{5}{2}| + |x - a|$, where $x \\in \\mathbb{R}$. If the inequality $f(x) \\geq a$ holds true for all $x \\in \\mathbb{R}$, find the maximum value of the real number $a$.\n(2) Given positive numbers $x$, $y$, and $z$ satisfying $x + 2y... | math | {
"style": "rule",
"ground_truth": "16 + 8\\sqrt{3}"
} | {
"split": "train",
"index": 575,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\\begin{align*}\n2a + 3b + 5c + 7d &= 34 \\\\\n3(d+c) &= b \\\\\n3b + c &= a \\\\\nc - 1 &= d \\\\\n\\end{align*} Given the above system of equations, find $a \\cdot b \\cdot c \\cdot d$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 576,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of \\[\\frac{x^1\\cdot x^2\\cdot x^3\\cdots x^{15}}{x^2\\cdot x^4 \\cdot x^6 \\cdots x^{20}}\\]if $x=2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1024"
} | {
"split": "train",
"index": 577,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nThe different ways to obtain the number of combinations of dice, as discussed in Example 4-15 of Section 4.6, can also be understood using the generating function form of Pólya’s enumeration theorem as follows:\n$$\n\\begin{aligned}\nP= & \\frac{1}{24} \\times\\left[\\left(x_... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 578,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the prime factorization of 1007021035035021007001. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7^{7} \\cdot 11^{7} \\cdot 13^{7}"
} | {
"split": "train",
"index": 579,
"solution": "The number in question is $$\\sum_{i=0}^{7}\\binom{7}{i} 1000^{i}=(1000+1)^{7}=1001^{7}=7^{7} \\cdot 11^{7} \\cdot 13^{7}$$"
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the expression: $2\\log_{2}\\;\\sqrt {2}-\\lg 2-\\lg 5+ \\frac{1}{ 3(\\frac{27}{8})^{2} }$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{9}"
} | {
"split": "train",
"index": 580,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Conduct an experiment by throwing 2 dice, and denote the point P with the coordinates $(x, y)$, where $x$ represents the number shown on the first die, and $y$ represents the number shown on the second die. \n(I) Find the probability that point P lies on the line $y = x$. \n(... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{36}"
} | {
"split": "train",
"index": 581,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$10 \\cdot 52 \\quad 1990-1980+1970-1960+\\cdots-20+10$ equals: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"split": "train",
"index": 582,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "One writes 268 numbers around a circle, such that the sum of 20 consectutive numbers is always equal to 75. The number 3, 4 and 9 are written in positions 17, 83 and 144 respectively. Find the number in position 210. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 583,
"solution": "Given the problem, we have to find the number in position 210 under the constraints provided. We have 268 numbers written in a circle, denoted as \\( a_1, a_2, \\ldots, a_{268} \\), and we know that the sum of any 20 consecutive numbers is 75. \n\nThis implies:\n\\[\... |
deepscaler | [
{
"role": "user",
"content": "In a large box of ribbons, $\\frac{1}{4}$ are yellow, $\\frac{1}{3}$ are purple, $\\frac{1}{6}$ are orange, and the remaining 40 ribbons are black. How many of the ribbons are purple? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "53"
} | {
"split": "train",
"index": 584,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "As shown in the diagram, points \\(E\\), \\(F\\), \\(G\\), and \\(H\\) are located on the four sides of the square \\(ABCD\\) as the trisection points closer to \\(B\\), \\(C\\), \\(D\\), and \\(A\\) respectively. If the side length of the square is 6, what is the area of the s... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 585,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest integer that must divide the product of any $5$ consecutive integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 586,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(1) Convert the following angles between degrees and radians: ① -15°; ② $\\frac {7\\pi }{12}$ \n(2) Given that the terminal side of angle $\\alpha$ passes through point P(2sin30°, -2cos30°), find the sine, cosine, and tangent values of angle $\\alpha$. Let's think step by step... | math | {
"style": "rule",
"ground_truth": "-\\sqrt {3}"
} | {
"split": "train",
"index": 587,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the minimum possible value of the sum\n\\[\\frac{a}{2b} + \\frac{b}{4c} + \\frac{c}{8a},\\]where $a,$ $b,$ and $c$ are positive real numbers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 588,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Line $m$ has the equation $y = 3x + 5$. Line $n$ has the equation $y = kx - 7$. Lines $m$ and $n$ intersect at the point $(-4, -7)$. What is the value of $k$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 589,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $({x-1})^4({x+2})^5=a_0+a_1x+a_2x^2+⋯+a_9x^9$, find the value of $a_{2}+a_{4}+a_{6}+a_{8}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-24"
} | {
"split": "train",
"index": 590,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the digits in the answer to\n$\\underbrace{9999\\cdots 99}_{94\\text{ nines}} \\times \\underbrace{4444\\cdots 44}_{94\\text{ fours}}$\nwhere a string of $94$ nines is multiplied by a string of $94$ fours. Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "846"
} | {
"split": "train",
"index": 591,
"solution": "1. **Identify the pattern**: We start by observing the multiplication of smaller strings of nines and fours:\n - $9 \\times 4 = 36$, and the sum of the digits is $3 + 6 = 9$.\n - $99 \\times 44 = 4356$, and the sum of the digits is $4 + 3 + 5 + 6 = 18$.\n\n2. **G... |
deepscaler | [
{
"role": "user",
"content": "$ABCDE$ is a regular pentagon inscribed in a circle of radius 1. What is the area of the set of points inside the circle that are farther from $A$ than they are from any other vertex? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{5}"
} | {
"split": "train",
"index": 592,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If\n\\[\\begin{pmatrix} 1 & 2 & a \\\\ 0 & 1 & 4 \\\\ 0 & 0 & 1 \\end{pmatrix}^n = \\begin{pmatrix} 1 & 18 & 2007 \\\\ 0 & 1 & 36 \\\\ 0 & 0 & 1 \\end{pmatrix},\\]then find $a + n.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 593,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In rectangle $WXYZ$, $P$ is a point on $WY$ such that $\\angle WPZ=90^{\\circ}$. $UV$ is perpendicular to $WY$ with $WU=UP$, as shown. $PZ$ intersects $UV$ at $Q$. Point $R$ is on $YZ$ such that $WR$ passes through $Q$. In $\\triangle PQW$, $PW=15$, $WQ=20$ and $QP=25$. Find $V... | math | {
"style": "rule",
"ground_truth": "\\dfrac{20}{3}"
} | {
"split": "train",
"index": 594,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Chester is traveling from Hualien to Lugang, Changhua, to participate in the Hua Luogeng Golden Cup Mathematics Competition. Before setting off, his father checked the car's odometer, which read a palindromic number of 69,696 kilometers (a palindromic number remains the same wh... | math | {
"style": "rule",
"ground_truth": "82.2"
} | {
"split": "train",
"index": 595,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five people are sitting around a round table, with identical coins placed in front of each person. Everyone flips their coin simultaneously. If the coin lands heads up, the person stands up; if it lands tails up, the person remains seated. Determine the probability that no two ... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{32}"
} | {
"split": "train",
"index": 596,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\( f(x)=a \\sin ((x+1) \\pi)+b \\sqrt[3]{x-1}+2 \\), where \\( a \\) and \\( b \\) are real numbers and \\( f(\\lg 5) = 5 \\), find \\( f(\\lg 20) \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 597,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For an integer $m\\geq 1$, we consider partitions of a $2^m\\times 2^m$ chessboard into rectangles consisting of cells of chessboard, in which each of the $2^m$ cells along one diagonal forms a separate rectangle of side length $1$. Determine the smallest possible sum of rectan... | math | {
"style": "rule",
"ground_truth": "2^{m+2}(m+1)"
} | {
"split": "train",
"index": 598,
"solution": "\nTo determine the smallest possible sum of rectangle perimeters when a \\(2^m \\times 2^m\\) chessboard is partitioned into rectangles such that each of the \\(2^m\\) cells along one diagonal is a separate rectangle, we begin by analyzing the conditions and the requ... |
deepscaler | [
{
"role": "user",
"content": "If $\\left\\lfloor n^2/4 \\right\\rfloor - \\lfloor n/2 \\rfloor^2 = 2$, then find all integer values of $n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 599,
"solution": ""
} |
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