data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "\nCalculate to 3 decimal places the following expressions:\n\n1. \n\n$$\n\\frac{2 \\sqrt{3}}{\\sqrt{3}-\\sqrt{2}}\n$$\n\n2. \n\n$$\n\\frac{(3+\\sqrt{3})(1+\\sqrt{5})}{(5+\\sqrt{5})(1+\\sqrt{3})}\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.775"
} | {
"split": "train",
"index": 38449,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number halfway between $\\frac{1}{6}$ and $\\frac{1}{4}$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{24}"
} | {
"split": "train",
"index": 38450,
"solution": "To find the number halfway between two numbers, we calculate the average of the two numbers. The formula for the average of two numbers $a$ and $b$ is:\n\\[\n\\text{Average} = \\frac{a + b}{2}\n\\]\n\nGiven the numbers $\\frac{1}{6}$ and $\\frac{1}{4}$, we first fi... |
deepscaler | [
{
"role": "user",
"content": "In triangle \\(ABC\\), angle bisectors \\(AA_{1}\\), \\(BB_{1}\\), and \\(CC_{1}\\) are drawn. \\(L\\) is the intersection point of segments \\(B_{1}C_{1}\\) and \\(AA_{1}\\), \\(K\\) is the intersection point of segments \\(B_{1}A_{1}\\) and \\(CC_{1}\\). Find the ratio \\(LM:... | math | {
"style": "rule",
"ground_truth": "11/12"
} | {
"split": "train",
"index": 38451,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the following diagram, $AB=50$. Find $AX$.\n\n[asy]\nimport markers;\n\nreal t=.67;\npair A=(0,0);\npair B=(3,-2);\npair C=(1.5,1.5);\npair X=t*A+(1-t)*B;\n\ndraw(C--A--B--C--X);\n\nlabel(\"$A$\",A,SW);\nlabel(\"$B$\",B,E);\nlabel(\"$C$\",C,N);\nlabel(\"$X$\",X,SW);\n\nmark... | math | {
"style": "rule",
"ground_truth": "\\frac{50}3"
} | {
"split": "train",
"index": 38452,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $-\\frac{\\pi}{2} < x < 0$, $\\sin x + \\cos x = \\frac{1}{5}$.\n\n(1) Find the value of $\\sin x - \\cos x$.\n\n(2) Find the value of $\\tan x$. 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": 38453,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Menkara has a $4 \\times 6$ index card. If she shortens the length of one side of this card by $1$ inch, the card would have area $18$ square inches. What would the area of the card be in square inches if instead she shortens the length of the other side by $1$ inch? Let's thin... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 38454,
"solution": "1. **Identify the dimensions of the index card**: Menkara has a $4 \\times 6$ index card. The initial dimensions are $4$ inches and $6$ inches.\n\n2. **Calculate the initial area**: \n \\[\n \\text{Area} = \\text{Length} \\times \\text{Width} = 4 \\times 6 = 24... |
deepscaler | [
{
"role": "user",
"content": "Given that\n\n\\[\n\\frac{1}{x} + \\frac{1}{y} = 4, \\quad x + y = 5,\n\\]\n\ncompute \\(x^2 + y^2\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{35}{2}"
} | {
"split": "train",
"index": 38455,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Which of the followings gives the product of the real roots of the equation $x^4+3x^3+5x^2 + 21x -14=0$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 38456,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, the parametric equation of curve $C$ is $\\begin{cases} & x=3\\cos \\alpha \\\\ & y=\\sin \\alpha \\end{cases}$ ($\\alpha$ is the parameter), and in the polar coordinate system with the origin as the pole and the positive $x$-axis as th... | math | {
"style": "rule",
"ground_truth": "\\dfrac {18 \\sqrt {2}}{5}"
} | {
"split": "train",
"index": 38457,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A particular fortune cookie lists four two-digit positive integers as your lucky numbers. The first three are 57, 13, and 72, but you got sauce on the last one and can't read it. If the sum of the digits of all four numbers equals $\\frac{1}{5}$ of the sum of all four numbers, ... | math | {
"style": "rule",
"ground_truth": "38"
} | {
"split": "train",
"index": 38458,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two circles touch each other internally at point K. The chord \\(A B\\) of the larger circle is tangent to the smaller circle at point \\(L\\), and \\(A L = 10\\). Find \\(B L\\) if \\(A K: B K = 2: 5\\). Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 38459,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, each of the five boxes is to contain a number. Each number in a shaded box must be the average of the number in the box to the left of it and the number in the box to the right of it. Given the numbers:\n\n| 8 | | | 26 | $x$ |\n\nWhat is the value of $x$ ? Let... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 38460,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "18.14 People are participating in a round-robin Japanese chess tournament. Each person plays against 13 others, with no draws in the matches. Find the maximum number of \"circular triples\" (where each of the three participants wins against one and loses to another) in the tour... | math | {
"style": "rule",
"ground_truth": "112"
} | {
"split": "train",
"index": 38461,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ such that $$\\underbrace{2^{2^{2^{2}}}}_{n 2^{\\prime} s}>\\underbrace{((\\cdots((100!)!)!\\cdots)!)!}_{100 \\text { factorials }}$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "104"
} | {
"split": "train",
"index": 38462,
"solution": "Note that $2^{2^{2^{2}}}>100^{2}$. We claim that $a>b^{2} \\Longrightarrow 2^{a}>(b!)^{2}$, for $b>2$. This is because $$2^{a}>b^{2 b} \\Longleftrightarrow a>2 b \\log _{2}(b)$$ and $\\log _{2}(b)<b^{2} / 2$ for $b>2$. Then since $b^{b}>b$ ! this bound works. Then ... |
deepscaler | [
{
"role": "user",
"content": "A certain organism begins as three cells. Each cell splits and becomes two cells at the end of two days. At the end of another two days, every cell of the organism splits and becomes two cells. This process lasts for a total of 8 days, and no cells die during this time. How man... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 38463,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the vertex of angle $\\alpha$ coincides with the origin $O$, its initial side coincides with the non-negative semi-axis of the $x$-axis, and its terminal side passes through point $P(-\\frac{3}{5}, -\\frac{4}{5})$.\n\n(1) Find the value of $\\sin(\\alpha + \\pi)$;\n\... | math | {
"style": "rule",
"ground_truth": "\\frac{16}{65}"
} | {
"split": "train",
"index": 38464,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nA construction company built a cottage village consisting of three identical streets. The houses in the village are identical according to the plan and are up for sale at the same price. The business plan of the company, which includes expenses for all stages of construction,... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 38465,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that line $l$ passes through point $P(-1,2)$ with a slope angle of $\\frac{2\\pi}{3}$, and the circle's equation is $\\rho=2\\cos (\\theta+\\frac{\\pi}{3})$:\n(1) Find the parametric equation of line $l$;\n(2) Let line $l$ intersect the circle at points $M$ and $N$, find ... | math | {
"style": "rule",
"ground_truth": "6+2\\sqrt{3}"
} | {
"split": "train",
"index": 38466,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The letters \\( A, J, H, S, M, E \\) and the numbers \\( 1, 9, 8, 9 \\) are \"rotated\" as follows:\n\\begin{tabular}{rrr} \nAJHSME & 1989 & \\\\\n1. JHSMEA & 9891 & (1st rotation) \\\\\n2. HSMEAJ & 8919 & (2nd rotation) \\\\\n3. SMEAJH & 9198 & (3rd rotation) \\\\\n..... & &\n... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 38467,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The stem-and-leaf plot shows the number of minutes and seconds of one ride on each of the 21 top-rated water slides in the world. In the stem-and-leaf plot, $1 \\ 45$ represents 1 minute, 45 seconds, which is equivalent to 105 seconds. What is the median of this data set? Expre... | math | {
"style": "rule",
"ground_truth": "135"
} | {
"split": "train",
"index": 38468,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x,y,$ and $z$ be real numbers satisfying the system of equations \\begin{align*} xy + 4z &= 60 \\\\ yz + 4x &= 60 \\\\ zx + 4y &= 60. \\end{align*} Let $S$ be the set of possible values of $x.$ Find the sum of the squares of the elements of $S.$ Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "273"
} | {
"split": "train",
"index": 38469,
"solution": "We first subtract the second equation from the first, noting that they both equal $60$. \\begin{align*} xy+4z-yz-4x&=0 \\\\ 4(z-x)-y(z-x)&=0 \\\\ (z-x)(4-y)&=0 \\end{align*}\n Case 1: Let $y=4$.\n The first and third equations simplify to: \\begin{align*} x+z&=15 \... |
deepscaler | [
{
"role": "user",
"content": "If $x = 2$ and $y = 5$, then what is the value of $\\frac{x^4+2y^2}{6}$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 38470,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the rectangular coordinate system (xOy), a pole coordinate system is established with O as the pole and the positive semi-axis of x as the polar axis. The shortest distance between a point on the curve C: ρcosθ - ρsinθ = 1 and a point on the curve M: x = -2 + cosφ, y = 1 + s... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}-1"
} | {
"split": "train",
"index": 38471,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The cost of 1 piece of gum is 1 cent. What is the cost of 1000 pieces of gum, in dollars? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10.00"
} | {
"split": "train",
"index": 38472,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Which are more: three-digit numbers where all digits have the same parity (all even or all odd), or three-digit numbers where adjacent digits have different parity? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "225"
} | {
"split": "train",
"index": 38473,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_{n}\\}$ with the sum of the first $n$ terms as $S_{n}$, where the common difference $d\\neq 0$, and $S_{3}+S_{5}=50$, $a_{1}$, $a_{4}$, $a_{13}$ form a geometric sequence.<br/>$(1)$ Find the general formula for the sequence $\\{a_{n}\\}$;<br/>... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{27}"
} | {
"split": "train",
"index": 38474,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the rectangular coordinate system xOy, the parametric equation of line l is $$\\begin{cases} \\overset{x=t}{y=1+t}\\end{cases}$$ (t is the parameter), line m is parallel to line l and passes through the coordinate origin, and the parametric equation of circle C is $$\\begin{... | math | {
"style": "rule",
"ground_truth": "2+ \\sqrt {2}"
} | {
"split": "train",
"index": 38475,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the inequality \n\\[\n((x+y)^2+4)((x+y)^2-2)\\geq A\\cdot (x-y)^2\n\\]\nis hold for every real numbers $x,y$ such that $xy=1$ , what is the largest value of $A$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 38476,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three tenths plus four thousandths is equal to Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.304"
} | {
"split": "train",
"index": 38477,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate: $$\\frac {\\cos 2^\\circ}{\\sin 47^\\circ} + \\frac {\\cos 88^\\circ}{\\sin 133^\\circ}$$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 38478,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Except for the first two terms, each term of the sequence $2000, y, 2000 - y,\\ldots$ is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encountered. What positive integer $y$ produces a sequence of m... | math | {
"style": "rule",
"ground_truth": "1236"
} | {
"split": "train",
"index": 38479,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = ax^3 + (a-1)x^2 + 27(a-2)x + b$, its graph is symmetric about the origin. Determine the monotonicity of $f(x)$ on the interval $[-4, 5]$ and find the maximum and minimum values of $f(x)$ on this interval. Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "-54"
} | {
"split": "train",
"index": 38480,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Seven teams play a soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a $50\\%$ chance of winning each game it plays, and the outcomes of the games are independent. In each game, the winner is awarded a point and the loser get... | math | {
"style": "rule",
"ground_truth": "831"
} | {
"split": "train",
"index": 38481,
"solution": "After the first game, there are $10$ games we care about-- those involving $A$ or $B$. There are $3$ cases of these $10$ games: $A$ wins more than $B$, $B$ wins more than $A$, or $A$ and $B$ win the same number of games. Also, there are $2^{10} = 1024$ total outcom... |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ is isosceles with $AC = BC$ and $\\angle ACB = 106^\\circ.$ Point $M$ is in the interior of the triangle so that $\\angle MAC = 7^\\circ$ and $\\angle MCA = 23^\\circ.$ Find the number of degrees in $\\angle CMB.$ Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "83"
} | {
"split": "train",
"index": 38482,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many values of $x$, $-10 < x < 50$, satisfy $\\cos^2 x + 3\\sin^2 x = 1.5?$ (Note: $x$ is measured in radians.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 38483,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular tetrahedron has the numbers $1,2,3,4$ written on its four faces. Four such identical regular tetrahedrons are thrown simultaneously onto a table. What is the probability that the product of the numbers on the four faces in contact with the table is divisible by $4$? L... | math | {
"style": "rule",
"ground_truth": "\\frac{13}{16}"
} | {
"split": "train",
"index": 38484,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Peter, Emma, and Kyler played chess with each other. Peter won 4 games and lost 2 games. Emma won 3 games and lost 3 games. If Kyler lost 3 games, how many games did he win? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 38485,
"solution": "1. **Identify the total number of games each player played**:\n - Peter played 6 games in total (4 wins + 2 losses).\n - Emma played 6 games in total (3 wins + 3 losses).\n - Kyler's total games are not directly given, but we know he lost 3 games.\n\n2. **Det... |
deepscaler | [
{
"role": "user",
"content": "Find the inclination angle of the line $\\sqrt {2}x+ \\sqrt {6}y+1=0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5\\pi}{6}"
} | {
"split": "train",
"index": 38486,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( A \\), \\( B \\), and \\( C \\) be pairwise independent events with equal probabilities, and \\( A \\cap B \\cap C = \\varnothing \\). Find the maximum possible value for the probability \\( \\mathrm{P}(A) \\). Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 38487,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be a triangle with $AB=5, BC=4$ and $AC=3$. Let $\\mathcal{P}$ and $\\mathcal{Q}$ be squares inside $ABC$ with disjoint interiors such that they both have one side lying on $AB$. Also, the two squares each have an edge lying on a common line perpendicular to $AB$, and... | math | {
"style": "rule",
"ground_truth": "\\frac{144}{49}"
} | {
"split": "train",
"index": 38488,
"solution": "Let the side lengths of $\\mathcal{P}$ and $\\mathcal{Q}$ be $a$ and $b$, respectively. Label two of the vertices of $\\mathcal{P}$ as $D$ and $E$ so that $D$ lies on $AB$ and $E$ lies on $AC$, and so that $DE$ is perpendicular to $AB$. The triangle $ADE$ is simila... |
deepscaler | [
{
"role": "user",
"content": "A gardener plants three maple trees, four oaks, and five birch trees in a row. He plants them in random order, each arrangement being equally likely. Let $\\frac m n$ in lowest terms be the probability that no two birch trees are next to one another. Find $m+n$.\n Let's think s... | math | {
"style": "rule",
"ground_truth": "106"
} | {
"split": "train",
"index": 38489,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define a T-grid to be a $3\\times3$ matrix which satisfies the following two properties:\nExactly five of the entries are $1$'s, and the remaining four entries are $0$'s.\nAmong the eight rows, columns, and long diagonals (the long diagonals are $\\{a_{13},a_{22},a_{31}\\}$ and... | math | {
"style": "rule",
"ground_truth": "68"
} | {
"split": "train",
"index": 38490,
"solution": "The T-grid can be considered as a tic-tac-toe board: five $1$'s (or X's) and four $0$'s (or O's).\n There are only $\\dbinom{9}{5} = 126$ ways to fill the board with five $1$'s and four $0$'s. Now we just need to subtract the number of bad grids. Bad grids are ones... |
deepscaler | [
{
"role": "user",
"content": "Four boxes with ball capacity 3, 5, 7, and 8 are given. Find the number of ways to distribute 19 identical balls into these boxes. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 38491,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the graph on $1000$ vertices $v_1, v_2, ...v_{1000}$ such that for all $1 \\le i < j \\le 1000$ , $v_i$ is connected to $v_j$ if and only if $i$ divides $j$ . Determine the minimum number of colors that must be used to color the vertices of this graph such... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 38492,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mindy made three purchases for $1.98, $5.04, and $9.89. What was her total, to the nearest dollar? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 38493,
"solution": "1. **Identify the individual amounts**: Mindy made purchases of $\\textdollar 1.98$, $\\textdollar 5.04$, and $\\textdollar 9.89$.\n\n2. **Round each amount to the nearest dollar**:\n - $\\textdollar 1.98$ rounds to $\\textdollar 2$ because it is closer to $\\tex... |
deepscaler | [
{
"role": "user",
"content": "Given the sums of the first n terms of two arithmetic sequences $\\{a_n\\}$ and $\\{b_n\\}$ denoted as $S_n$ and $T_n$, respectively, if $\\frac {S_{n}}{T_{n}} = \\frac {2n}{3n+1}$, calculate the value of $\\frac {a_{6}}{b_{6}}$. Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "\\frac {11}{17}"
} | {
"split": "train",
"index": 38494,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nCalculate the definite integral:\n\n$$\n\\int_{0}^{\\frac{\\pi}{2}} \\frac{\\sin ^{2} x \\, dx}{(1+\\cos x+\\sin x)^{2}}\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2} - \\frac{1}{2} \\ln 2"
} | {
"split": "train",
"index": 38495,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$(1)$ Calculate: $\\sqrt{3}\\tan 45^{\\circ}-\\left(2023-\\pi \\right)^{0}+|2\\sqrt{3}-2|+(\\frac{1}{4})^{-1}-\\sqrt{27}$; <br/>$(2)$ Simplify first, then evaluate: $\\frac{{{x^2}-x}}{{{x^2}+2x+1}}\\div (\\frac{2}{{x+1}}-\\frac{1}{x})$, simplify it first, then choose an integer... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"split": "train",
"index": 38496,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers $1,2,\\dots,9$ are randomly placed into the $9$ squares of a $3 \\times 3$ grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd? Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{14}"
} | {
"split": "train",
"index": 38497,
"solution": "\nTo solve this problem, we need to ensure that the sum of the numbers in each row and each column is odd. We know that the sum of three numbers is odd if and only if either all three numbers are odd or exactly one of them is odd (and the other two are even).\n\n##... |
deepscaler | [
{
"role": "user",
"content": "Two circles of radius 2 are centered at $(2,0)$ and at $(0,2)$. What is the area of the intersection of the interiors of the two circles? Express your answer in fully expanded form in terms of $\\pi$. Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "2\\pi-4"
} | {
"split": "train",
"index": 38498,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest possible number of whole 2-by-3 non-overlapping rectangles needed to cover a square region exactly, without extra over-hangs and without gaps? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 38499,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $\\displaystyle\\frac{2+2i}{-3+4i}$. Express your answer as a complex number in the form $a+bi$, where $a$ and $b$ are real numbers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{25} - \\frac{14}{25}i"
} | {
"split": "train",
"index": 38500,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Wang Lei and her older sister walk from home to the gym to play badminton. It is known that the older sister walks 20 meters more per minute than Wang Lei. After 25 minutes, the older sister reaches the gym, and then realizes she forgot the racket. She immediately returns along... | math | {
"style": "rule",
"ground_truth": "1500"
} | {
"split": "train",
"index": 38501,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x$, $y\\in \\mathbb{R}$, vectors $\\overrightarrow{a}=(2,x)$, $\\overrightarrow{b}=(y,1)$, $\\overrightarrow{c}=(3,-3)$, and $\\overrightarrow{a}⊥\\overrightarrow{b}$, $\\overrightarrow{b}∥\\overrightarrow{c}$. \n$(1)$ Find $|\\overrightarrow{a}+\\overrightarrow{b}|$; \n... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"split": "train",
"index": 38502,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Dan is walking down the left side of a street in New York City and must cross to the right side at one of 10 crosswalks he will pass. Each time he arrives at a crosswalk, however, he must wait $t$ seconds, where $t$ is selected uniformly at random from the real interval $[0,60]... | math | {
"style": "rule",
"ground_truth": "60\\left(1-\\left(\\frac{1}{10}\\right)^{\\frac{1}{9}}\\right)"
} | {
"split": "train",
"index": 38503,
"solution": "With probability $\\left(1-\\frac{k}{60}\\right)^{9}$, Dan reaches the last crosswalk without crossing at any previous site, in which case the expected value of his wait time is 30 seconds. Otherwise, with probability $1-\\left(1-\\frac{k}{60}\\right)^{9}$, Dan cro... |
deepscaler | [
{
"role": "user",
"content": "Let $m$ be the smallest integer whose cube root is of the form $n+r$, where $n$ is a positive integer and $r$ is a positive real number less than $1/1000$. Find $n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 38504,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The distance from point $\\left(1,0\\right)$ to the line $3x+4y-2+\\lambda \\left(2x+y+2\\right)=0$, where $\\lambda \\in R$, needs to be determined. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{13}"
} | {
"split": "train",
"index": 38505,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{a},$ $\\mathbf{b},$ $\\mathbf{c}$ be vectors such that $\\|\\mathbf{a}\\| = \\|\\mathbf{b}\\| = 1$ and $\\|\\mathbf{c}\\| = 2.$ Find the maximum value of\n\\[\\|\\mathbf{a} - 2 \\mathbf{b}\\|^2 + \\|\\mathbf{b} - 2 \\mathbf{c}\\|^2 + \\|\\mathbf{c} - 2 \\mathbf{a... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 38506,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the board, six numbers are written in a row. It is known that each number, starting from the third, is equal to the product of the two preceding numbers, and the fifth number is equal to 108. Find the product of all six numbers in this row. Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "136048896"
} | {
"split": "train",
"index": 38507,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the function $f(x) = \\frac{bx}{\\ln x} - ax$, where $e$ is the base of the natural logarithm.\n\n(I) If the tangent line to the graph of the function $f(x)$ at the point $(e^2, f(e^2))$ is $3x + 4y - e^2 = 0$, find the values of the real numbers $a$ and $b$.\n\n(II) When $... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2} - \\frac{1}{4e^2}"
} | {
"split": "train",
"index": 38508,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the definite integral:\n\n$$\n\\int_{0}^{\\frac{2\\pi}{3}} \\frac{\\cos^2 x \\, dx}{(1 + \\cos x + \\sin x)^2}\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2} - \\ln 2"
} | {
"split": "train",
"index": 38509,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(Full score: 8 points) \nDuring the 2010 Shanghai World Expo, there were as many as 11 types of admission tickets. Among them, the price for a \"specified day regular ticket\" was 200 yuan per ticket, and the price for a \"specified day concession ticket\" was 120 yuan per tic... | math | {
"style": "rule",
"ground_truth": "300"
} | {
"split": "train",
"index": 38510,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What are the first three digits to the right of the decimal point in the decimal representation of $\\left(10^{1001}+1\\right)^{3/2}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "743"
} | {
"split": "train",
"index": 38511,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Cátia leaves school every day at the same time and rides her bicycle home. When she pedals at $20 \\mathrm{~km} / \\mathrm{h}$, she arrives home at $4:30$ PM. If she pedals at $10 \\mathrm{~km} / \\mathrm{h}$, she arrives home at $5:15$ PM. At what speed should she pedal to arr... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 38512,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sequence $\\{a_n\\}$ is an arithmetic sequence, and $a_2=-1$, the sequence $\\{b_n\\}$ satisfies $b_n-b_{n-1}=a_n$ ($n\\geqslant 2, n\\in \\mathbb{N}$), and $b_1=b_3=1$ \n(I) Find the value of $a_1$; \n(II) Find the general formula for the sequence $\\{b_n\\}$.... | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 38513,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If 3913 were to be expressed as a sum of distinct powers of 2, what would be the least possible sum of the exponents of these powers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "47"
} | {
"split": "train",
"index": 38514,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the max. value of $ M$,such that for all $ a,b,c>0$:\n $ a^{3}+b^{3}+c^{3}-3abc\\geq M(|a-b|^{3}+|a-c|^{3}+|c-b|^{3})$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{9 + 6\\sqrt{3}}"
} | {
"split": "train",
"index": 38515,
"solution": "\nTo find the maximum value of \\( M \\) such that the inequality\n\n\\[\na^3 + b^3 + c^3 - 3abc \\geq M(|a-b|^3 + |a-c|^3 + |c-b|^3)\n\\]\n\nholds for all \\( a, b, c > 0 \\), we start by analyzing both sides of the inequality.\n\n### Step 1: Understand the Expres... |
deepscaler | [
{
"role": "user",
"content": "Place four balls numbered 1, 2, 3, and 4 into three boxes labeled A, B, and C. \n(1) If none of the boxes are empty and ball number 3 must be in box B, how many different arrangements are there? \n(2) If ball number 1 cannot be in box A and ball number 2 cannot be in box B, h... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 38516,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the coordinate plane, consider points $A = (0, 0)$, $B = (11, 0)$, and $C = (18, 0)$. Line $\\ell_A$ has slope 1 and passes through $A$. Line $\\ell_B$ is vertical and passes through $B$. Line $\\ell_C$ has slope $-1$ and passes through $C$. The three lines $\\ell_A$, $\... | math | {
"style": "rule",
"ground_truth": "85"
} | {
"split": "train",
"index": 38517,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right triangle XYZ has legs XY = YZ = 8 cm. In each step of an iterative process, the triangle is divided into four smaller right triangles by joining the midpoints of the sides. However, for this problem, the area of the shaded triangle in each iteration is reduced by a fact... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 38518,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the Island of Misfortune, there are knights, who always tell the truth, and liars, who always lie. One day, $n$ islanders gathered in a room.\n\nThe first person said: \"Exactly 1 percent of the people present in this room are liars.\"\n\nThe second person said: \"Exactly 2 ... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 38519,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be the 200th smallest positive real solution to the equation $x-\\frac{\\pi}{2}=\\tan x$. Find the greatest integer that does not exceed $\\frac{n}{2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "314"
} | {
"split": "train",
"index": 38520,
"solution": "Drawing the graphs of the functions $y=x-\\frac{\\pi}{2}$ and $y=\\tan x$, we may observe that the graphs intersect exactly once in each of the intervals $\\left(\\frac{(2 k-1) \\pi}{2}, \\frac{(2 k+1) \\pi}{2}\\right)$ for each $k=1,2, \\cdots$. Hence, the 200th i... |
deepscaler | [
{
"role": "user",
"content": "Find the remainder when $5^{2021}$ is divided by $17$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 38521,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Boston weatherman says there is a 75 percent chance of rain for each day of the Labor Day four-day weekend. If it doesn't rain, then the weather will be sunny. Paul and Yuri want it to be sunny one of those days for a World War III preenactment, but if it's sunny for more ... | math | {
"style": "rule",
"ground_truth": "\\frac{27}{64}"
} | {
"split": "train",
"index": 38522,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $XYZ$, points $X'$, $Y'$, and $Z'$ are on sides $YZ$, $XZ$, and $XY$ respectively. Given that $XX'$, $YY'$, and $ZZ'$ are concurrent at point $P$, and that $\\frac{XP}{PX'} + \\frac{YP}{PY'} + \\frac{ZP}{PZ'} = 100$, find $\\frac{XP}{PX'} \\cdot \\frac{YP}{PY'} \\cd... | math | {
"style": "rule",
"ground_truth": "98"
} | {
"split": "train",
"index": 38523,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\log_6 (4x)=2$, find $\\log_x 27$. Express your answer in simplest fractional form. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac32"
} | {
"split": "train",
"index": 38524,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the hyperbola $C$: $mx^{2}+ny^{2}=1(mn < 0)$, one of its asymptotes is tangent to the circle $x^{2}+y^{2}-6x-2y+9=0$. Determine the eccentricity of $C$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac {5}{4}"
} | {
"split": "train",
"index": 38525,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If the area of $\\triangle ABC$ equals $8$, $a=5$, and $\\tan B= -\\frac{4}{3}$, then $\\frac{a+b+c}{\\sin A+\\sin B+\\sin C}=$ \\_\\_\\_\\_\\_\\_. Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "\\frac{5\\sqrt{65}}{4}"
} | {
"split": "train",
"index": 38526,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function\n\\[f(z) = \\frac{(-1 + i \\sqrt{3}) z + (-2 \\sqrt{3} - 18i)}{2}\\]represents a rotation around some complex number $c$. Find $c$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3} - 5i"
} | {
"split": "train",
"index": 38527,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the smallest constant $n$, such that for any positive real numbers $x$, $y$, and $z$,\n\\[\\sqrt{\\frac{x}{y + 2z}} + \\sqrt{\\frac{y}{2x + z}} + \\sqrt{\\frac{z}{x + 2y}} > n.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 38528,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x^2 + 2x + 5$ is a factor of $x^4 + Px^2 + Q,$ find $P + Q.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 38529,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that point $P$ is the intersection point of the lines $l_{1}$: $mx-ny-5m+n=0$ and $l_{2}$: $nx+my-5m-n=0$ ($m$,$n\\in R$, $m^{2}+n^{2}\\neq 0$), and point $Q$ is a moving point on the circle $C$: $\\left(x+1\\right)^{2}+y^{2}=1$, calculate the maximum value of $|PQ|$. Let... | math | {
"style": "rule",
"ground_truth": "6 + 2\\sqrt{2}"
} | {
"split": "train",
"index": 38530,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the sum of the first $n$ terms of an arithmetic sequence $\\{a_n\\}$ be $S_n$, and it satisfies $S_{2016} > 0, S_{2017} < 0$. For any positive integer $n$, it holds that $|a_n| \\geqslant |a_k|$, find the value of $k$. Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "1009"
} | {
"split": "train",
"index": 38531,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(1+11+21+31+41)+(9+19+29+39+49)= Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 38532,
"solution": "1. **Identify the terms in each group and their properties**: \n The problem gives us two sums:\n \\[\n (1+11+21+31+41) \\quad \\text{and} \\quad (9+19+29+39+49)\n \\]\n Each group contains five terms, and the terms in each group are increasing by 10.\n\n... |
deepscaler | [
{
"role": "user",
"content": "Let $T$ be a subset of $\\{1,2,3,\\ldots,2021\\}$ such that no two members of $T$ differ by $5$ or $8$. What is the largest number of elements $T$ can have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "918"
} | {
"split": "train",
"index": 38533,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of\n$$\n(x+1)(x+2006)\\left[\\frac{1}{(x+1)(x+2)}+\\frac{1}{(x+2)(x+3)}+\\ldots+\\frac{1}{(x+2005)(x+2006)}\\right] ?\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2005"
} | {
"split": "train",
"index": 38534,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that $2a\\cos A=c\\cos B+b\\cos C$.\n\n1. Find the value of $\\cos A$.\n2. If $a=1$ and $\\cos^{2}\\frac{B}{2}+\\cos^{2}\\frac{C}{2}=1+\\frac{\\sqrt{3}}{4}$, find the le... | math | {
"style": "rule",
"ground_truth": "\\frac { \\sqrt {3}}{3}"
} | {
"split": "train",
"index": 38535,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $AAA_4$ can be expressed as $33_b$, where $A$ is a digit in base 4 and $b$ is a base greater than 5, what is the smallest possible sum $A+b$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 38536,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ellina has twelve blocks, two each of red ($\\textbf{R}$), blue ($\\textbf{B}$), yellow ($\\textbf{Y}$), green ($\\textbf{G}$), orange ($\\textbf{O}$), and purple ($\\textbf{P}$). Call an arrangement of blocks $\\textit{even}$ if there is an even number of blocks between each p... | math | {
"style": "rule",
"ground_truth": "247"
} | {
"split": "train",
"index": 38537,
"solution": "We can simply use constructive counting. First, let us place the red balls; choose the first slot in $12$ ways, and the second in $6$ ways, because the number is cut in half due to the condition in the problem. This gives $12 \\cdot 6$ ways to place the blue balls.... |
deepscaler | [
{
"role": "user",
"content": "Let $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$ be real numbers whose sum is 20. Determine with proof the smallest possible value of \\(\\sum_{1 \\leq i<j \\leq 5}\\left\\lfloor a_{i}+a_{j}\\right\\rfloor\\). Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 38538,
"solution": "We claim that the minimum is 72. This can be achieved by taking $a_{1}=a_{2}=a_{3}=a_{4}=0.4$ and $a_{5}=18.4$. To prove that this is optimal, note that \\(\\sum_{1 \\leq i<j \\leq 5}\\left\\lfloor a_{i}+a_{j}\\right\\rfloor=\\sum_{1 \\leq i<j \\leq 5}\\left(a_{i}+... |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{p}$ and $\\mathbf{q}$ be two three-dimensional unit vectors such that the angle between them is $30^\\circ.$ Find the area of the parallelogram whose diagonals correspond to $\\mathbf{p} + 2 \\mathbf{q}$ and $2 \\mathbf{p} + \\mathbf{q}.$ Let's think step by step... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 38539,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many unordered pairs of edges of a given cube determine a plane? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 38540,
"solution": "1. **Understanding the Problem**: We need to find how many unordered pairs of edges in a cube determine a plane. Two edges determine a plane if they are either parallel or intersecting (not skew).\n\n2. **Total Number of Edges in a Cube**: A cube has 12 edges.\n\n3... |
deepscaler | [
{
"role": "user",
"content": "Expand the following expression: $3(8x^2-2x+1)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24x^2-6x+3"
} | {
"split": "train",
"index": 38541,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=ax^{2}-2x+1$.\n$(1)$ When $a\\neq 0$, discuss the monotonicity of the function $f(x)$;\n$(2)$ If $\\frac {1}{3}\\leqslant a\\leqslant 1$, and the maximum value of $f(x)$ on $[1,3]$ is $M(a)$, the minimum value is $N(a)$, let $g(a)=M(a)-N(a)$, find the e... | math | {
"style": "rule",
"ground_truth": "\\frac {1}{2}"
} | {
"split": "train",
"index": 38542,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In square $ABCE$, $AF=2FE$ and $CD=2DE$. What is the ratio of the area of $\\triangle BFD$ to the area of square $ABCE$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{18}"
} | {
"split": "train",
"index": 38543,
"solution": "1. **Assigning Lengths**: Let the length of $\\overline{FE}$ be $x$. Since $AF = 2FE$, we have $\\overline{AF} = 2x$. Similarly, since $CD = 2DE$, we have $\\overline{ED} = x$ and $\\overline{DC} = 2x$.\n\n2. **Calculating Side Length of Square**: The side length o... |
deepscaler | [
{
"role": "user",
"content": "When $(a-b)^n,n\\ge2,ab\\ne0$, is expanded by the binomial theorem, it is found that when $a=kb$, where $k$ is a positive integer, the sum of the second and third terms is zero. Then $n$ equals: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2k+1"
} | {
"split": "train",
"index": 38544,
"solution": "1. **Substitute $a$ with $kb$:** Given $a = kb$, substitute into $(a-b)^n$:\n \\[\n (a-b)^n = (kb - b)^n = (b(k-1))^n\n \\]\n\n2. **Apply the Binomial Theorem:** The Binomial Theorem states that\n \\[\n (x+y)^n = \\sum_{i=0}^n \\binom{n}{i} x^{n-i} y^i\n ... |
deepscaler | [
{
"role": "user",
"content": "The maximum value of the function $y=\\sin x \\cos x + \\sin x + \\cos x$ is __________. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2} + \\sqrt{2}"
} | {
"split": "train",
"index": 38545,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$48n$ is the smallest positive integer that satisfies the following conditions:\n1. $n$ is a multiple of 75;\n2. $n$ has exactly 75 positive divisors (including 1 and itself).\n\nFind $\\frac{n}{75}$. Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "432"
} | {
"split": "train",
"index": 38546,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The expression $10y^2 - 51y + 21$ can be written as $(Cy - D)(Ey - F)$, where $C, D, E, F$ are integers. What is $CE + C$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 38547,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each vertex of convex pentagon $ABCDE$ is to be assigned a color. There are $6$ colors to choose from, and the ends of each diagonal must have different colors. How many different colorings are possible? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "3120"
} | {
"split": "train",
"index": 38548,
"solution": "To solve this problem, we need to consider the constraints given by the diagonals of the pentagon. Each diagonal connects two vertices, and the vertices at the ends of each diagonal must have different colors. We will analyze the possible colorings by considering d... |
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