data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "How many distinct arrangements of the letters in the word \"balloon\" are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1260"
} | {
"split": "train",
"index": 600,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, triangle \\(ABC\\) is isosceles, with \\(AB = AC\\). If \\(\\angle ABC = 50^\\circ\\) and \\(\\angle DAC = 60^\\circ\\), the value of \\(x\\) is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 601,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ $c$ be three distinct positive real numbers such that $a,$ $b,$ $c$ form a geometric sequence, and\n\\[\\log_c a, \\ \\log_b c, \\ \\log_a b\\]form an arithmetic sequence. Find the common difference of the arithmetic sequence. Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 602,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many ways can I choose 2 cooks on a backpacking trip with 8 people if any of the 8 people may be cooks? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 603,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine how many regions of space are divided by:\na) The six planes of a cube's faces.\nb) The four planes of a tetrahedron's faces. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 604,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In right triangle $ABC$ with $\\angle C=90^{\\circ}$ and $AC=4$, find the value of $\\overrightarrow{AB} \\cdot \\overrightarrow{AC}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 605,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\sin \\frac {x}{2}\\cos \\frac {x}{2}+\\cos ^{2} \\frac {x}{2}-1$.\n$(1)$ Find the smallest positive period of the function $f(x)$ and the interval where it is monotonically decreasing;\n$(2)$ Find the minimum value of the function $f(x)$ on the interv... | math | {
"style": "rule",
"ground_truth": "- \\frac { \\sqrt {2}+1}{2}"
} | {
"split": "train",
"index": 606,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively, and it is given that $c\\cos B + b\\cos C = 3a\\cos B$.\n\n$(1)$ Find the value of $\\cos B$; \n\n$(2)$ If $\\overrightarrow{BA} \\cdot \\overrightarrow{BC} = 2$, find th... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"split": "train",
"index": 607,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given positive integer $n$ and $r$ pairwise distinct primes $p_1,p_2,\\cdots,p_r.$ Initially, there are $(n+1)^r$ numbers written on the blackboard: $p_1^{i_1}p_2^{i_2}\\cdots p_r^{i_r} (0 \\le i_1,i_2,\\cdots,i_r \\le n).$\n\nAlice and Bob play a game by making a move by turns... | math | {
"style": "rule",
"ground_truth": "M^{\\lfloor \\frac{n}{2} \\rfloor}"
} | {
"split": "train",
"index": 608,
"solution": "\nGiven positive integer \\( n \\) and \\( r \\) pairwise distinct primes \\( p_1, p_2, \\cdots, p_r \\). Initially, there are \\( (n+1)^r \\) numbers written on the blackboard: \\( p_1^{i_1} p_2^{i_2} \\cdots p_r^{i_r} \\) where \\( 0 \\le i_1, i_2, \\cdots, i_r \\l... |
deepscaler | [
{
"role": "user",
"content": "Let \\( n \\geq 2 \\) be a fixed integer. Find the least constant \\( C \\) such that the inequality\n\\[ \\sum_{i<j} x_{i} x_{j}\\left(x_{i}^{2}+x_{j}^{2}\\right) \\leq C\\left(\\sum_{i} x_{i}\\right)^{4} \\]\nholds for every \\( x_{1}, \\ldots, x_{n} \\geq 0 \\) (the sum on t... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"split": "train",
"index": 609,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A fly trapped inside a rectangular prism with dimensions $1$ meter, $2$ meters, and $3$ meters decides to tour the corners of the box. It starts from the corner $(0,0,0)$ and ends at the corner $(0,0,3)$, visiting each of the other corners exactly once. Determine the maximum po... | math | {
"style": "rule",
"ground_truth": "\\sqrt{14} + 6 + \\sqrt{13} + \\sqrt{5}"
} | {
"split": "train",
"index": 610,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all integers $k$ such that $\\binom{23}{4} + \\binom{23}{5} = \\binom{24}{k}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 611,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence 1, 1+2, 2+3+4, 3+4+5+6, ..., the value of the 8th term in this sequence is: ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 612,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Xanthia can read 100 pages per hour and Molly can read 50 pages per hour. If they each read the same book, and the book has 225 pages, how many more minutes than Xanthia would it take for Molly to finish reading the book? Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "135"
} | {
"split": "train",
"index": 613,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest integer $n$ such that each subset of $\\{1,2,\\ldots, 2004\\}$ with $n$ elements has two distinct elements $a$ and $b$ for which $a^2-b^2$ is a multiple of $2004$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1003"
} | {
"split": "train",
"index": 614,
"solution": "\nTo solve the problem of finding the smallest integer \\( n \\) such that each subset of \\(\\{1, 2, \\ldots, 2004\\}\\) with \\( n \\) elements has two distinct elements \\( a \\) and \\( b \\) for which \\( a^2 - b^2 \\) is a multiple of \\( 2004 \\), we start by ... |
deepscaler | [
{
"role": "user",
"content": "Let $f(x) = \\displaystyle \\frac{1}{ax+b}$ where $a$ and $b$ are nonzero constants. Find all solutions to $f^{-1}(x) = 0$. Express your answer in terms of $a$ and/or $b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac1b"
} | {
"split": "train",
"index": 615,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f$ be a linear function for which $f(6)-f(2)=12$. What is $f(12)-f(2)?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 616,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five integers have an average of 69. The middle integer (the median) is 83. The most frequently occurring integer (the mode) is 85. The range of the five integers is 70. What is the second smallest of the five integers? Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "77"
} | {
"split": "train",
"index": 617,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of triples $(x,y,z)$ of real numbers that satisfy\n\\begin{align*}\nx &= 2018 - 2019 \\operatorname{sign}(y + z), \\\\\ny &= 2018 - 2019 \\operatorname{sign}(x + z), \\\\\nz &= 2018 - 2019 \\operatorname{sign}(x + y).\n\\end{align*}Note: For a real number $a,$\n... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 618,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the area of the portion of the circle defined by the equation $x^2 + 6x + y^2 = 50$ that lies below the $x$-axis and to the left of the line $y = x - 3$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{59\\pi}{4}"
} | {
"split": "train",
"index": 619,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two circles of radius $r$ are externally tangent to each other and internally tangent to the ellipse $x^2 + 4y^2 = 5$. Find $r$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{15}}{4}"
} | {
"split": "train",
"index": 620,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of all of the digits of the integers from 1 to 2008 is to be calculated. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28054"
} | {
"split": "train",
"index": 621,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $\\sqrt{36+64}-\\sqrt{25-16}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 622,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On an island, there are knights, liars, and followers; each one knows who is who among them. All 2018 island inhabitants were lined up and each was asked to answer \"Yes\" or \"No\" to the question: \"Are there more knights than liars on the island?\" The inhabitants answered o... | math | {
"style": "rule",
"ground_truth": "1009"
} | {
"split": "train",
"index": 623,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different positive, six-digit integers can be formed using the digits 2, 2, 5, 5, 9 and 9? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 624,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A convex polyhedron $P$ has $26$ vertices, $60$ edges, and $36$ faces, $24$ of which are triangular and $12$ of which are quadrilaterals. A space diagonal is a line segment connecting two non-adjacent vertices that do not belong to the same face. How many space diagonals does $... | math | {
"style": "rule",
"ground_truth": "241"
} | {
"split": "train",
"index": 625,
"solution": "Every pair of vertices of the polyhedron determines either an edge, a face diagonal or a space diagonal. We have ${26 \\choose 2} = \\frac{26\\cdot25}2 = 325$ total line segments determined by the vertices. Of these, $60$ are edges. Each triangular face has $0$ face ... |
deepscaler | [
{
"role": "user",
"content": "How many distinct lines pass through the point $(0, 2016)$ and intersect the parabola $y = x^2$ at two lattice points? (A lattice point is a point whose coordinates are integers.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 626,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Serezha and Misha, while walking in the park, came across a meadow surrounded by linden trees. Serezha walked around the meadow, counting the trees. Misha did the same but started from a different tree (although he walked in the same direction). The tree that was 20th for Serez... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 627,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many sequences of $0$s and $1$s of length $19$ are there that begin with a $0$, end with a $0$, contain no two consecutive $0$s, and contain no three consecutive $1$s? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "65"
} | {
"split": "train",
"index": 628,
"solution": "To solve this problem, we need to determine the number of valid sequences of length 19 that satisfy the given conditions. We will use a recursive approach to count these sequences.\n\n#### Step 1: Define the Problem\nLet $f(n)$ be the number of valid sequences of len... |
deepscaler | [
{
"role": "user",
"content": "A bag contains 8 red balls, a number of white balls, and no other balls. If $\\frac{5}{6}$ of the balls in the bag are white, then how many white balls are in the bag? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 629,
"solution": "Since $\\frac{5}{6}$ of the balls are white and the remainder of the balls are red, then $\\frac{1}{6}$ of the balls are red. Since the 8 red balls represent $\\frac{1}{6}$ of the total number of balls and $\\frac{5}{6} = 5 \\cdot \\frac{1}{6}$, then the number of wh... |
deepscaler | [
{
"role": "user",
"content": "If $a\\equiv 18\\pmod{42}$ and $b\\equiv 73\\pmod{42}$, then for what integer $n$ in the set $\\{100,101,102,\\ldots,140,141\\}$ is it true that $$a-b\\equiv n\\pmod{42}~?$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "113"
} | {
"split": "train",
"index": 630,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Farmer John has 5 cows, 4 pigs, and 7 horses. How many ways can he pair up the animals so that every pair consists of animals of different species? Assume that all animals are distinguishable from each other. Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "100800"
} | {
"split": "train",
"index": 631,
"solution": "Since there are 9 cow and pigs combined and 7 horses, there must be a pair with 1 cow and 1 pig, and all the other pairs must contain a horse. There are $4 \\times 5$ ways of selecting the cow-pig pair, and 7 ! ways to select the partners for the horses. It follows t... |
deepscaler | [
{
"role": "user",
"content": "What is the smallest possible area, in square units, of a right triangle with two sides measuring $7$ units and $8$ units? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7\\sqrt{15}}{2}"
} | {
"split": "train",
"index": 632,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be nonnegative real numbers such that\n\\[\\sin (ax + b) = \\sin 15x\\]for all integers $x.$ Find the smallest possible value of $a.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 633,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let point P be a fixed point inside a circle ⊙O with a radius of 5, and OP=4. The sum of all possible integer values of the chord lengths passing through point P is. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 634,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For $x > 0$, the area of the triangle with vertices $(0, 0), (x, 2x)$, and $(x, 0)$ is 64 square units. What is the value of $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 635,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two adjacent faces of a tetrahedron are equilateral triangles with a side length of 1 and form a dihedral angle of 45 degrees. The tetrahedron rotates around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto a plane that co... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{4}"
} | {
"split": "train",
"index": 636,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The product of the digits of 4321 is 24. How many distinct four-digit positive integers are such that the product of their digits equals 18? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 637,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Cyclic pentagon $ABCDE$ has side lengths $AB=BC=5, CD=DE=12$, and $AE=14$. Determine the radius of its circumcircle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{225\\sqrt{11}}{88}"
} | {
"split": "train",
"index": 638,
"solution": "Let $C^{\\prime}$ be the point on minor arc $BCD$ such that $BC^{\\prime}=12$ and $C^{\\prime}D=5$, and write $AC^{\\prime}=BD=C^{\\prime}E=x, AD=y$, and $BD=z$. Ptolemy applied to quadrilaterals $ABC^{\\prime}D, BC^{\\prime}DE$, and $ABDE$ gives $$\\begin{aligned} &... |
deepscaler | [
{
"role": "user",
"content": "Given real numbers $a$ and $b$ satisfying $a^{2}b^{2}+2ab+2a+1=0$, calculate the minimum value of $ab\\left(ab+2\\right)+\\left(b+1\\right)^{2}+2a$. 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": 639,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $y = ax^{2} + x - b$ where $a \\in \\mathbb{R}$ and $b \\in \\mathbb{R}$. \n$(1)$ If $b = 1$ and the set $\\{x | y = 0\\}$ has exactly one element, find the set of possible values for the real number $a$. \n$(2)$ Solve the inequality with respect to $x$: $y < (a-1)x^{2} +... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 640,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In isosceles triangle $\\triangle ABC$, $A$ is located at the origin and $B$ is located at $(20,0)$. Point $C$ is in the first quadrant with $AC = BC$ and angle $BAC = 75^{\\circ}$. If triangle $ABC$ is rotated counterclockwise about point $A$ until the image of $C$ lies on the... | math | {
"style": "rule",
"ground_truth": "875"
} | {
"split": "train",
"index": 641,
"solution": "Call the points of the intersections of the triangles $D$, $E$, and $F$ as noted in the diagram (the points are different from those in the diagram for solution 1). $\\overline{AD}$ bisects $\\angle EDE'$.\n Through HL congruency, we can find that $\\triangle AED$ is... |
deepscaler | [
{
"role": "user",
"content": "Given a sequence of positive terms $\\{a\\_n\\}$, where $a\\_2=6$, and $\\frac{1}{a\\_1+1}$, $\\frac{1}{a\\_2+2}$, $\\frac{1}{a\\_3+3}$ form an arithmetic sequence, find the minimum value of $a\\_1a\\_3$. Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "19+8\\sqrt{3}"
} | {
"split": "train",
"index": 642,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a nonnegative integer $n$, let $r_7(n)$ denote the remainder when $n$ is divided by $7.$ Determine the $15^{\\text{th}}$ entry in an ordered list of all nonnegative integers $n$ that satisfy $$r_7(3n)\\le 3.$$ Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 643,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways are there to insert +'s between the digits of 111111111111111 (fifteen 1's) so that the result will be a multiple of 30? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2002"
} | {
"split": "train",
"index": 644,
"solution": "Note that because there are 15 1's, no matter how we insert +'s, the result will always be a multiple of 3. Therefore, it suffices to consider adding +'s to get a multiple of 10. By looking at the units digit, we need the number of summands to be a multiple of 10. Be... |
deepscaler | [
{
"role": "user",
"content": "Is it possible to represent the number $1986$ as the sum of squares of $6$ odd integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\text{No}"
} | {
"split": "train",
"index": 645,
"solution": "\nTo determine if it is possible to express the number \\( 1986 \\) as the sum of squares of six odd integers, we can analyze the properties of numbers represented as sums of squares of odd integers.\n\n### Step 1: Analyzing Odd Integers\n\nFirst, recall that the squ... |
deepscaler | [
{
"role": "user",
"content": "Let $X = \\{-5,-4,-3,-2,-1,0,1,2,3,4,5\\}$ and $S = \\{(a,b)\\in X\\times X:x^2+ax+b \\text{ and }x^3+bx+a \\text{ have at least a common real zero .}\\}$ How many elements are there in $S$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 646,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangles $\\triangle DEF$ and $\\triangle D'E'F'$ are positioned in the coordinate plane with vertices $D(0,0)$, $E(0,10)$, $F(14,0)$, $D'(20,20)$, $E'(30,20)$, $F'(20,8)$. Determine the angle of rotation $n$ degrees clockwise around the point $(p,q)$ where $0<n<180$, that tra... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 647,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all positive integers $\\nu$ for which $\\mathop{\\text{lcm}}[\\nu,20]=60$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "126"
} | {
"split": "train",
"index": 648,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point $Q$ is chosen within $\\triangle DEF$ such that lines drawn through $Q$, parallel to the sides of $\\triangle DEF$, divide it into three smaller triangles with areas $9$, $16$, and $25$ respectively. Determine the area of $\\triangle DEF$. Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 649,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If you want to form a rectangular prism with a surface area of 52, what is the minimum number of small cubes with edge length 1 needed? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 650,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We drew the face diagonals of a cube with an edge length of one unit and drew a sphere centered around the cube's center. The sphere intersected the diagonals at the vertices of a convex polyhedron, all of whose faces are regular. What was the radius of the sphere? Let's think ... | math | {
"style": "rule",
"ground_truth": "0.579"
} | {
"split": "train",
"index": 651,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Factor the expression $x(x+2)+(x+2)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(x+1)(x+2)"
} | {
"split": "train",
"index": 652,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 4 distinct balls and 4 distinct boxes. All the balls need to be placed into the boxes.\n(1) If exactly one box remains empty, how many different arrangements are possible?\n(2) If exactly two boxes remain empty, how many different arrangements are possible? Let's thin... | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 653,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A polynomial $f \\in \\mathbb{Z}[x]$ is called splitty if and only if for every prime $p$, there exist polynomials $g_{p}, h_{p} \\in \\mathbb{Z}[x]$ with $\\operatorname{deg} g_{p}, \\operatorname{deg} h_{p}<\\operatorname{deg} f$ and all coefficients of $f-g_{p} h_{p}$ are di... | math | {
"style": "rule",
"ground_truth": "693"
} | {
"split": "train",
"index": 654,
"solution": "We claim that $x^{4}+a x^{2}+b$ is splitty if and only if either $b$ or $a^{2}-4 b$ is a perfect square. (The latter means that the polynomial splits into $(x^{2}-r)(x^{2}-s)$ ). Assuming the characterization, one can easily extract the answer. For $a=16$ and $b=n$, ... |
deepscaler | [
{
"role": "user",
"content": "I have 7 books, three of which are identical copies of the same novel, and the others are distinct. If a particular book among these must always be placed at the start of the shelf, in how many ways can I arrange the rest of the books? Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 655,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Using the digits $1, 2, 3, 4$, 24 unique four-digit numbers can be formed without repeating any digit. If these 24 four-digit numbers are arranged in ascending order, find the sum of the two middle numbers. Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "4844"
} | {
"split": "train",
"index": 656,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two distinct points $A, B$ and line $\\ell$ that is not perpendicular to $A B$, what is the maximum possible number of points $P$ on $\\ell$ such that $A B P$ is an isosceles triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 657,
"solution": "In an isosceles triangle, one vertex lies on the perpendicular bisector of the opposite side. Thus, either $P$ is the intersection of $A B$ and $\\ell$, or $P$ lies on the circle centered at $A$ with radius $A B$, or $P$ lies on the circle centered at $B$ with radius... |
deepscaler | [
{
"role": "user",
"content": "Find the remainder when\\[\\binom{\\binom{3}{2}}{2} + \\binom{\\binom{4}{2}}{2} + \\dots + \\binom{\\binom{40}{2}}{2}\\]is divided by $1000$.\n ~ pi_is_3.14 Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 658,
"solution": "Doing simple algebra calculation will give the following equation: \\begin{align*} \\binom{\\binom{n}{2}}{2} = \\frac{\\frac{n(n-1)}{2} \\cdot (\\frac{n(n-1)}{2}-1)}{2} \\\\ = \\frac{n(n-1)(n^2-n-2)}{8} \\\\ = \\frac{(n+1)n(n-1)(n-2)}{8} \\\\ = \\frac{(n+1)!}{8\\cdot... |
deepscaler | [
{
"role": "user",
"content": "Five fair six-sided dice are rolled. What is the probability that at least three of the five dice show the same value? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{113}{648}"
} | {
"split": "train",
"index": 659,
"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": 660,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The product of two whole numbers is 24. The smallest possible sum of these two numbers is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 661,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the lateral side \\( CD \\) of the trapezoid \\( ABCD (AD \\parallel BC) \\), point \\( M \\) is marked. From vertex \\( A \\), a perpendicular \\( AH \\) is dropped onto segment \\( BM \\). It turns out that \\( AD = HD \\). Find the length of segment \\( AD \\) if it is kn... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 662,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate $46_8 - 27_8$ and express your answer in base 8. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17_8"
} | {
"split": "train",
"index": 663,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $ { \\sum_{1 \\le i< j \\le 10}(i+j)}_{i+j=odd} $ $ - { \\sum_{1 \\le i< j \\le 10}(i+j)}_{i+j=even} $ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "55"
} | {
"split": "train",
"index": 664,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Fisherman Vasya caught several fish. He placed the three largest fish, which constitute 35% of the total weight of the catch, in the refrigerator. He gave the three smallest fish, which constitute 5/13 of the weight of the remaining fish, to the cat. Vasya ate all the rest of t... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 665,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The coefficient of $x^{2}$ in the expansion of $\\left( \\frac {3}{x}+x\\right)\\left(2- \\sqrt {x}\\right)^{6}$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "243"
} | {
"split": "train",
"index": 666,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the line $l$: $2mx - y - 8m - 3 = 0$ and the circle $C$: $x^2 + y^2 - 6x + 12y + 20 = 0$, find the shortest length of the chord that line $l$ cuts on circle $C$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{15}"
} | {
"split": "train",
"index": 667,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the first 2011 terms of a geometric sequence is 200. The sum of the first 4022 terms is 380. Find the sum of the first 6033 terms. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "542"
} | {
"split": "train",
"index": 668,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A manufacturer of airplane parts makes a certain engine that has a probability $p$ of failing on any given flight. There are two planes that can be made with this sort of engine, one that has 3 engines and one that has 5. A plane crashes if more than half its engines fail. For ... | math | {
"style": "rule",
"ground_truth": "0, \\frac{1}{2}, 1"
} | {
"split": "train",
"index": 669,
"solution": "They have the same probability of failing if $\\binom{5}{2} p^{3}(1-p)^{2}+\\binom{5}{1} p^{4}(1-p)+p^{5}=\\binom{3}{1} p^{2}(1-p)+p^{3}$, which is true iff $p^{2}\\left(6 p^{3}-15 p^{2}+12 p-3\\right)=0$. This is clearly true for $p=0$. We know it is true for $p=1$,... |
deepscaler | [
{
"role": "user",
"content": "Suppose $x-3$ and $y+3$ are multiples of $7$.\n\nWhat is the smallest positive integer, $n,$ for which $x^2+xy+y^2+n$ is a multiple of $7$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 670,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sequence $\\{a_{n}\\}$ is a geometric sequence with a common ratio $q\\neq 1$, $a_{1}=3$, $3a_{1}$, $2a_{2}$, $a_{3}$ form an arithmetic sequence, and the terms of the sequence $\\{a_{n}\\}$ are arranged in a certain order as $a_{1}$, $a_{1}$, $a_{2}$, $a_{1}$, $... | math | {
"style": "rule",
"ground_truth": "1641"
} | {
"split": "train",
"index": 671,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the projection of the vector $\\begin{pmatrix} 3 \\\\ 0 \\\\ -2 \\end{pmatrix}$ onto the line\n\\[\\frac{x}{2} = y = \\frac{z}{-1}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 8/3 \\\\ 4/3 \\\\ -4/3 \\end{pmatrix}"
} | {
"split": "train",
"index": 672,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cube has edges of length 1 cm and has a dot marked in the centre of the top face. The cube is sitting on a flat table. The cube is rolled, without lifting or slipping, in one direction so that at least two of its vertices are always touching the table. The cube is rolled u... | math | {
"style": "rule",
"ground_truth": "\\dfrac{1+\\sqrt{5}}{2}"
} | {
"split": "train",
"index": 673,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be a positive integer. E. Chen and E. Chen play a game on the $n^2$ points of an $n \\times n$ lattice grid. They alternately mark points on the grid such that no player marks a point that is on or inside a non-degenerate triangle formed by three marked points. Ea... | math | {
"style": "rule",
"ground_truth": "1007"
} | {
"split": "train",
"index": 674,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an increasing sequence $\\{a_{n}\\}$ where all terms are positive integers, the sum of the first $n$ terms is $S_{n}$. If $a_{1}=3$ and $S_{n}=2023$, calculate the value of $a_{n}$ when $n$ takes its maximum value. Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "73"
} | {
"split": "train",
"index": 675,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways are there to allocate a group of 8 friends among a basketball team, a soccer team, a track team, and the option of not participating in any sports? Each team, including the non-participant group, could have anywhere from 0 to 8 members. Assume the friends are dist... | math | {
"style": "rule",
"ground_truth": "65536"
} | {
"split": "train",
"index": 676,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the maximum value of the expression \\( x + y \\) if \\( (2 \\sin x - 1)(2 \\cos y - \\sqrt{3}) = 0 \\), \\( x \\in [0, \\frac{3\\pi}{2}] \\), \\( y \\in [\\pi, 2\\pi] \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8\\pi}{3}"
} | {
"split": "train",
"index": 677,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the area of the parallelogram formed by the vectors \\( a \\) and \\( b \\).\n\nGiven:\n\\[ a = 3p - 4q \\]\n\\[ b = p + 3q \\]\n\\[ |p| = 2 \\]\n\\[ |q| = 3 \\]\n\\[ \\text{Angle between } p \\text{ and } q \\text{ is } \\frac{\\pi}{4} \\] Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "39\\sqrt{2}"
} | {
"split": "train",
"index": 678,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate $\\lim_{n \\to \\infty} \\frac{C_n^2}{n^2+1}$. 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": 679,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Natural numbers \\( a, b, c \\) are chosen such that \\( a < b < c \\). It is also known that the system of equations \\( 2x + y = 2033 \\) and \\( y = |x-a| + |x-b| + |x-c| \\) has exactly one solution. Find the minimum possible value of \\( c \\). Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "1017"
} | {
"split": "train",
"index": 680,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a five-team tournament, each team plays one game with every other team. Each team has a $50\\%$ chance of winning any game it plays. (There are no ties.) Let $\\dfrac{m}{n}$ be the probability that the tournament will produce neither an undefeated team nor a winless team, wh... | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 681,
"solution": "We can use complementary counting: finding the probability that at least one team wins all games or at least one team loses all games.\n No more than 1 team can win or lose all games, so at most one team can win all games and at most one team can lose all games.\n No... |
deepscaler | [
{
"role": "user",
"content": "Let $f: \\mathbb{Z} \\rightarrow \\mathbb{Z}$ be a function such that for any integers $x, y$, we have $f\\left(x^{2}-3 y^{2}\\right)+f\\left(x^{2}+y^{2}\\right)=2(x+y) f(x-y)$. Suppose that $f(n)>0$ for all $n>0$ and that $f(2015) \\cdot f(2016)$ is a perfect square. Find the ... | math | {
"style": "rule",
"ground_truth": "246"
} | {
"split": "train",
"index": 682,
"solution": "Plugging in $-y$ in place of $y$ in the equation and comparing the result with the original equation gives $(x-y) f(x+y)=(x+y) f(x-y)$. This shows that whenever $a, b \\in \\mathbb{Z}-\\{0\\}$ with $a \\equiv b(\\bmod 2)$, we have $\\frac{f(a)}{a}=\\frac{f(b)}{b}$ wh... |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be a parallelogram with area $15$. Points $P$ and $Q$ are the projections of $A$ and $C,$ respectively, onto the line $BD;$ and points $R$ and $S$ are the projections of $B$ and $D,$ respectively, onto the line $AC.$ See the figure, which also shows the relative loca... | math | {
"style": "rule",
"ground_truth": "81"
} | {
"split": "train",
"index": 683,
"solution": "1. **Identify the given information and setup:**\n - $ABCD$ is a parallelogram with area $15$.\n - $PQ = 6$ and $RS = 8$ are the lengths of the projections of $A$ and $C$ onto $BD$, and $B$ and $D$ onto $AC$, respectively.\n - We need to find $d^2$, where $d$ i... |
deepscaler | [
{
"role": "user",
"content": "Positive integers $a$, $b$, and $c$ are such that $a<b<c$. Consider the system of equations \n\\[\n2x + y = 2022 \\quad \\text{and} \\quad y = |x-a| + |x-b| + |x-c|\n\\]\nDetermine the minimum value of $c$ such that the system has exactly one solution. Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "1012"
} | {
"split": "train",
"index": 684,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all triples of positive integers $(x,y,z)$ that satisfy the equation \n$$2(x+y+z+2xyz)^2=(2xy+2yz+2zx+1)^2+2023.$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(2, 3, 3)"
} | {
"split": "train",
"index": 685,
"solution": "\nTo solve the given equation for triples \\((x, y, z)\\) of positive integers:\n\n\\[\n2(x + y + z + 2xyz)^2 = (2xy + 2yz + 2zx + 1)^2 + 2023,\n\\]\n\nwe start by analyzing the structure of the equation. The equation can be seen as comparing the square of two polyno... |
deepscaler | [
{
"role": "user",
"content": "Two circles lie outside regular hexagon $ABCDEF$. The first is tangent to $\\overline{AB}$, and the second is tangent to $\\overline{DE}$. Both are tangent to lines $BC$ and $FA$. What is the ratio of the area of the second circle to that of the first circle? Let's think step b... | math | {
"style": "rule",
"ground_truth": "81"
} | {
"split": "train",
"index": 686,
"solution": "1. **Assumption and Setup**: Assume without loss of generality (WLOG) that the regular hexagon $ABCDEF$ has a side length of 1. The first circle is tangent to $\\overline{AB}$ and the second circle is tangent to $\\overline{DE}$. Both circles are tangent to lines $BC... |
deepscaler | [
{
"role": "user",
"content": "Find a nonzero monic polynomial $P(x)$ with integer coefficients and minimal degree such that $P(1-\\sqrt[3]{2}+\\sqrt[3]{4})=0$. (A polynomial is called monic if its leading coefficient is 1.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "x^{3}-3x^{2}+9x-9"
} | {
"split": "train",
"index": 687,
"solution": "Note that $(1-\\sqrt[3]{2}+\\sqrt[3]{4})(1+\\sqrt[3]{2})=3$, so $1-\\sqrt[3]{2}+\\sqrt[3]{4}=\\frac{3}{1+\\sqrt[3]{2}}$. Now, if $f(x)=x^{3}-2$, we have $f(\\sqrt[3]{2})=0$, so if we let $g(x)=f(x-1)=(x-1)^{3}-2=x^{3}-3x^{2}+3x-3$, then $g(1+\\sqrt[3]{2})=f(\\sqrt[3]... |
deepscaler | [
{
"role": "user",
"content": "Danielle Bellatrix Robinson is organizing a poker tournament with 9 people. The tournament will have 4 rounds, and in each round the 9 players are split into 3 groups of 3. During the tournament, each player plays every other player exactly once. How many different ways can Dan... | math | {
"style": "rule",
"ground_truth": "20160"
} | {
"split": "train",
"index": 688,
"solution": "We first split the 9 people up arbitrarily into groups of 3. There are $\\frac{\\binom{9}{3}\\binom{6}{3}\\binom{3}{3}}{3!}=280$ ways of doing this. Without loss of generality, label the people 1 through 9 so that the first round groups are $\\{1,2,3\\},\\{4,5,6\\}$,... |
deepscaler | [
{
"role": "user",
"content": "Given that $3\\sin \\alpha + 4\\cos \\alpha = 5$.\n(1) Find the value of $\\tan \\alpha$;\n(2) Find the value of $\\cot (\\frac{3\\pi}{2} - \\alpha) \\cdot \\sin^2 (\\frac{3\\pi}{2} + \\alpha)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{12}{25}"
} | {
"split": "train",
"index": 689,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest five-digit number whose digits' product equals 120. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "85311"
} | {
"split": "train",
"index": 690,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is $88 \\div 4 \\div 2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 691,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\frac{5+7+9}{3} = \\frac{4020+4021+4022}{M}$, find $M$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1723"
} | {
"split": "train",
"index": 692,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$(1)$ Solve the system of inequalities: $\\left\\{\\begin{array}{l}{2x-1＜5,①}\\\\{\\frac{4-x}{2}≥1,②}\\end{array}\\right.$<br/>$(2)$ Simplify first, then find the value: $\\frac{1}{a-1}÷(\\frac{{a}^{2}+a}{{a}^{2}-1}+\\frac{1}{a-1})$, where $a=(-2023)^0+(\\frac{1}{2})^{-1}⋅$ Let... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 693,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the quadratic equation \\( (5a + 2b)x^2 + ax + b = 0 \\) has a unique solution for \\( x \\), find the value of \\( x \\). 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": 694,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function $g$, defined on the set of ordered pairs of positive integers, satisfies the following properties:\n\\[\ng(x,x) = x, \\quad\ng(x,y) = g(y,x), \\quad\n(x + y) g(x,y) = yg(x, x + y).\n\\]\nCalculate $g(18,63)$. Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "126"
} | {
"split": "train",
"index": 695,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The point $P$ is inside of an equilateral triangle with side length $10$ so that the distance from $P$ to two of the sides are $1$ and $3$. Find the distance from $P$ to the third side. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5\\sqrt{3} - 4"
} | {
"split": "train",
"index": 696,
"solution": "\nGiven an equilateral triangle with side length \\(10\\) and a point \\(P\\) inside the triangle, we are required to find the distance from \\(P\\) to the third side, knowing the distances from \\(P\\) to the other two sides are \\(1\\) and \\(3\\).\n\nFirst, recall... |
deepscaler | [
{
"role": "user",
"content": "Two spheres touch the plane of triangle \\(ABC\\) at points \\(B\\) and \\(C\\) and are located on opposite sides of this plane. The sum of the radii of these spheres is 11, and the distance between their centers is \\(5 \\sqrt{17}\\). The center of a third sphere with a radius... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{19}"
} | {
"split": "train",
"index": 697,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There were cards with numbers from 1 to 20 in a bag. Vlad pulled out 6 cards and asserted that all these cards can be divided into pairs such that the sums of the numbers in each pair were the same. Lena managed to sneak a peek at 5 of Vlad's cards: they had the numbers $2, 4, ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 698,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $\\log_2 (4^2)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 699,
"solution": ""
} |
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