data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Define the function $f(x) = 2x - 5$. For what value of $x$ is $f(x)$ equal to $f^{-1}(x)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 700,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $F\\_1$ and $F\\_2$ are the left and right foci of the hyperbola $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1 (a > 0, b > 0)$, and there exists a point $P$ on the hyperbola such that $\\angle F\\_1PF\\_2 = 60^{\\circ}$ and $|OP| = 3b$ (where $O$ is the origin), find the ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{42}}{6}"
} | {
"split": "train",
"index": 701,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Xinyi is a beautiful city with abundant tourism resources. The Malingshan Scenic Area is a $4A$-rated tourist attraction in our city. The scenic area has developed a souvenir that costs $30$ yuan to produce and is sold. The selling price is set not to be lower than the cost and... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 702,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x = (2 + \\sqrt{2})^6,$ let $n = \\lfloor x \\rfloor,$ and let $f = x - n.$ Find\n\\[\nx(1 - f).\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 703,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve for $x$: $\\left(\\frac{1}{4}\\right)^{2x+8} = (16)^{2x+5}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 704,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ed and Sue bike at equal and constant rates. Similarly, they jog at equal and constant rates, and they swim at equal and constant rates. Ed covers $74$ kilometers after biking for $2$ hours, jogging for $3$ hours, and swimming for $4$ hours, while Sue covers $91$ kilometers aft... | math | {
"style": "rule",
"ground_truth": "314"
} | {
"split": "train",
"index": 705,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 12 ordered pairs of integers $(x,y)$ that satisfy $x^2 + y^2 = 25$. What is the greatest possible sum $x+y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 706,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain teacher received $10$, $6$, $8$, $5$, $6$ letters from Monday to Friday, then the variance of this data set is $s^{2}=$____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3.2"
} | {
"split": "train",
"index": 707,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the nearest integer to $(2+\\sqrt3)^4$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "194"
} | {
"split": "train",
"index": 708,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $1$; $4$; $\\ldots$ and $9$; $16$; $\\ldots$ be two arithmetic progressions. The set $S$ is the union of the first $2004$ terms of each sequence. How many distinct numbers are in $S$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3722"
} | {
"split": "train",
"index": 709,
"solution": "1. **Identify the sequences and their general terms**:\n - The first sequence is $1, 4, 7, \\ldots$ which is an arithmetic progression with the first term $a_1 = 1$ and common difference $d_1 = 3$. The $k$-th term of this sequence can be expressed as $a_k = 3k + 1$... |
deepscaler | [
{
"role": "user",
"content": "Mrs. Crabapple now teaches two different classes of British Literature. Her first class has 12 students and meets three times a week, while her second class has 9 students and meets twice a week. How many different sequences of crabapple recipients are possible in a week for bo... | math | {
"style": "rule",
"ground_truth": "139,968"
} | {
"split": "train",
"index": 710,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ a,\\ b$ be real constants. Find the minimum value of the definite integral:\r\n\r $ I(a,\\ b)\\equal{}\\int_0^{\\pi} (1\\minus{}a\\sin x \\minus{}b\\sin 2x)^2 dx.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\pi - \\frac{8}{\\pi}"
} | {
"split": "train",
"index": 711,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of ordered pairs $(x,y)$ of real numbers such that\n\\[16^{x^2 + y} + 16^{x + y^2} = 1.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 712,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let natural numbers \\( k \\) and \\( n \\) be coprime, where \\( n \\geq 5 \\) and \\( k < \\frac{n}{2} \\). A regular \\((n ; k)\\)-star is defined as a closed broken line formed by replacing every \\( k \\) consecutive sides of a regular \\( n \\)-gon with a diagonal connect... | math | {
"style": "rule",
"ground_truth": "48432"
} | {
"split": "train",
"index": 713,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Paul needs to save 40 files onto flash drives, each with 2.0 MB space. 4 of the files take up 1.2 MB each, 16 of the files take up 0.9 MB each, and the rest take up 0.6 MB each. Determine the smallest number of flash drives needed to store all 40 files. Let's think step by step... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 714,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For certain real numbers $a$, $b$, and $c$, the polynomial \\[g(x) = x^3 + ax^2 + x + 10\\]has three distinct roots, and each root of $g(x)$ is also a root of the polynomial \\[f(x) = x^4 + x^3 + bx^2 + 100x + c.\\]What is $f(1)$? Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "-7007"
} | {
"split": "train",
"index": 715,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $f(x)$ is a rational function such that $3f\\left(\\dfrac{1}{x}\\right)+\\dfrac{2f(x)}{x}=x^2$ for $x\\neq 0$. Find $f(-2)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{67}{20}"
} | {
"split": "train",
"index": 716,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a triangle with sides of lengths $a,$ $b,$ and $c,$\n\\[(a + b + c)(a + b - c) = 3ab.\\]Find the angle opposite the side of length $c,$ in degrees. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60^\\circ"
} | {
"split": "train",
"index": 717,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a number matrix as shown, the three numbers in each row are in arithmetic progression, and the three numbers in each column are also in arithmetic progression. Given that \\( a_{22} = 2 \\), find the sum of all 9 numbers in the matrix. Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 718,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 3 screw-in light bulbs and 5 bayonet light bulbs in the box, and light bulbs are randomly drawn without replacement until the 5th light bulb is drawn to have all the screw-in light bulbs. Calculate the probability of this event. Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{28}"
} | {
"split": "train",
"index": 719,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the median and the mean of the set \\(\\{m, m+2, m+4, m+11, m+18\\}\\) where the median of the set is \\(16\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 720,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose we need to divide 12 dogs into three groups, where one group contains 4 dogs, another contains 6 dogs, and the last contains 2 dogs. How many ways can we form the groups so that Rover is in the 4-dog group and Spot is in the 6-dog group? Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "2520"
} | {
"split": "train",
"index": 721,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right triangle with legs $10$ and $8$ is similar to another right triangle with corresponding legs $x$ and $5$, respectively. What is the value of $x$? Express your answer as a decimal to the nearest hundredth. Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "6.25"
} | {
"split": "train",
"index": 722,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify the expression $\\dfrac {\\cos 40 ^{\\circ} }{\\cos 25 ^{\\circ} \\sqrt {1-\\sin 40 ^{\\circ} }}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 723,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all positive integers $n$ such that the unit segments of an $n \\times n$ grid of unit squares can be partitioned into groups of three such that the segments of each group share a common vertex. Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "n \\equiv 0,2(\\bmod 6)"
} | {
"split": "train",
"index": 724,
"solution": "Answer: $n \\equiv 0,2(\\bmod 6)$ We first prove that $n \\equiv 0,2(\\bmod 6)$ is necessary for there to be such a partitioning. We break this down into proving that $n$ has to be even and that $n \\equiv 0,2(\\bmod 3)$. The only way a segment on a side of the squar... |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathcal{P}$ be the parabola in the plane determined by the equation $y = x^2.$ Suppose a circle $\\mathcal{C}$ intersects $\\mathcal{P}$ at four distinct points. If three of these points are $(-28,784),$ $(-2,4),$ and $(13,169),$ find the sum of the distances from the ... | math | {
"style": "rule",
"ground_truth": "1247"
} | {
"split": "train",
"index": 725,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the parabola $C: x^{2}=2py\\left(p \\gt 0\\right)$ with focus $F$, and the minimum distance between $F$ and a point on the circle $M: x^{2}+\\left(y+4\\right)^{2}=1$ is $4$.<br/>$(1)$ Find $p$;<br/>$(2)$ If point $P$ lies on $M$, $PA$ and $PB$ are two tangents to $C$ with... | math | {
"style": "rule",
"ground_truth": "20\\sqrt{5}"
} | {
"split": "train",
"index": 726,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive real number $x$ is such that \\[\n\\sqrt[3]{1-x^4} + \\sqrt[3]{1+x^4} = 1.\n\\]Find $x^8.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{28}{27}"
} | {
"split": "train",
"index": 727,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Seven cards numbered $1$ through $7$ are to be lined up in a row. Find the number of arrangements of these seven cards where one of the cards can be removed leaving the remaining six cards in either ascending or descending order. Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 728,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the odd function f(x) defined on R satisfies f(x+2) + f(2-x) = 0, and when x ∈ (-2, 0), f(x) = log<sub>2</sub>(x+3) + a. If f(9) = 2f(7) + 1, then the value of the real number a is ( ). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{4}{3}"
} | {
"split": "train",
"index": 729,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a$ and $b$ are positive integers that can each be written as a sum of two squares, then $a b$ is also a sum of two squares. Find the smallest positive integer $c$ such that $c=a b$, where $a=x^{3}+y^{3}$ and $b=x^{3}+y^{3}$ each have solutions in integers $(x, y)$, but $c=x... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 730,
"solution": "We can't have $c=1=1^{3}+0^{3}$ or $c=2=1^{3}+1^{3}$, and if $c=3$, then $a$ or $b= \\pm 3$ which is not a sum of two cubes (otherwise, flipping signs of $x$ and $y$ if necessary, we would get either a sum of two nonnegative cubes to equal 3, which clearly does not h... |
deepscaler | [
{
"role": "user",
"content": "Find the inverse of the matrix\n\\[\\begin{pmatrix} 2 & 3 \\\\ -1 & 7 \\end{pmatrix}.\\]If the inverse does not exist, then enter the zero matrix. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 7/17 & -3/17 \\\\ 1/17 & 2/17 \\end{pmatrix}"
} | {
"split": "train",
"index": 731,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different-looking arrangements are possible when four balls are selected at random from six identical red balls and three identical green balls and then arranged in a line? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 732,
"solution": "Since 4 balls are chosen from 6 red balls and 3 green balls, then the 4 balls could include: - 4 red balls, or - 3 red balls and 1 green ball, or - 2 red balls and 2 green balls, or - 1 red ball and 3 green balls. There is only 1 different-looking way to arrange 4 re... |
deepscaler | [
{
"role": "user",
"content": "There are integers $a, b,$ and $c,$ each greater than $1,$ such that\n\\[\\sqrt[a]{N\\sqrt[b]{N\\sqrt[c]{N}}} = \\sqrt[36]{N^{25}}\\]\nfor all $N \\neq 1$. What is $b$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 733,
"solution": "1. **Expression Simplification**:\n Start by simplifying the left-hand side of the equation:\n \\[\n \\sqrt[a]{N\\sqrt[b]{N\\sqrt[c]{N}}} = N^{\\frac{1}{a} + \\frac{1}{ab} + \\frac{1}{abc}}\n \\]\n This simplification comes from the properties of exponents,... |
deepscaler | [
{
"role": "user",
"content": "A square with side length $x$ is inscribed in a right triangle with sides of length $3$, $4$, and $5$ so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length $y$ is inscribed in another right triangle with sides of lengt... | math | {
"style": "rule",
"ground_truth": "\\frac{37}{35}"
} | {
"split": "train",
"index": 734,
"solution": "#### Analyzing the first right triangle:\nConsider a right triangle $ABC$ with sides $3$, $4$, and $5$, where $5$ is the hypotenuse. Let a square be inscribed such that one vertex of the square coincides with the right-angle vertex $C$ of the triangle. Let the side l... |
deepscaler | [
{
"role": "user",
"content": "The cars Niva and Toyota are driving on a ring track of a test site, a quarter of which is on a dirt road and the remaining part is on asphalt. The speed of the Niva on the dirt road is 80 km/h, and on asphalt it is 90 km/h. The speed of the Toyota on the dirt road is 40 km/h, ... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 735,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $a=(2\\cos \\alpha,\\sin ^{2}\\alpha)$ and $b=(2\\sin \\alpha,t)$, where $\\alpha\\in\\left( 0,\\frac{\\pi}{2} \\right)$ and $t$ is a real number.\n\n$(1)$ If $a-b=\\left( \\frac{2}{5},0 \\right)$, find the value of $t$;\n\n$(2)$ If $t=1$ and $a\\cdot b=1$, find t... | math | {
"style": "rule",
"ground_truth": "\\frac{23}{7}"
} | {
"split": "train",
"index": 736,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $α∈({\\frac{π}{2},π})$, $sinα=\\frac{3}{5}$, find $\\tan 2\\alpha$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{24}{7}"
} | {
"split": "train",
"index": 737,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many nonnegative solutions are there to the equation $x^2 = -4x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 738,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width is 2:1. What is the area of the rectangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 739,
"solution": "1. **Identify the relationship between the circle and the rectangle**: The circle is inscribed in the rectangle, which means the diameter of the circle is equal to the height (shorter side) of the rectangle.\n\n2. **Calculate the diameter of the circle**: Given that ... |
deepscaler | [
{
"role": "user",
"content": "Round 8899.50241201 to the nearest whole number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8900"
} | {
"split": "train",
"index": 740,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate: $6 - 8\\left(9 - 4^2\\right) \\div 2 - 3.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 741,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The intersection of two squares with perimeter $8$ is a rectangle with diagonal length $1$ . Given that the distance between the centers of the two squares is $2$ , the perimeter of the rectangle can be expressed as $P$ . Find $10P$ . Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 742,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four of the five vertices of a polygon shaped as a right trapezoid are (5, 11), (16, 11), (16, -2), and (5, -2); it includes a semicircle with a diameter along the bottom base, centered at (10.5, -2) and radius 5.5. Determine the total area bounded by the lower base of the trap... | math | {
"style": "rule",
"ground_truth": "15.125\\pi"
} | {
"split": "train",
"index": 743,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the coefficient of $x^6$ in the expansion of $(x^2-a)(x+\\frac{1}{x})^{10}$ is 30, then find the value of $\\int_{0}^{a} (3x^2+1) \\, dx$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 744,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\frac{1}{3}x^3-ax^2+(a^2-1)x+b$, where $(a,b \\in \\mathbb{R})$\n(I) If $x=1$ is an extreme point of $f(x)$, find the value of $a$;\n(II) If the equation of the tangent line to the graph of $y=f(x)$ at the point $(1, f(1))$ is $x+y-3=0$, find the maxim... | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 745,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What value of $x$ will give the minimum value for $x^2 + 11x - 5$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{11}{2}"
} | {
"split": "train",
"index": 746,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Subset \\( S \\subseteq \\{1, 2, 3, \\ldots, 1000\\} \\) is such that if \\( m \\) and \\( n \\) are distinct elements of \\( S \\), then \\( m + n \\) does not belong to \\( S \\). What is the largest possible number of elements in \\( S \\)? Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "501"
} | {
"split": "train",
"index": 747,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Inside of the square $ABCD$ the point $P$ is given such that $|PA|:|PB|:|PC|=1:2:3$ . Find $\\angle APB$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "135"
} | {
"split": "train",
"index": 748,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The percentage of seventh-grade students participating in the gymnastics section is between 2.9% and 3.1%. Determine the smallest possible number of students in this class. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 749,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Every 1 kilogram of soybeans can produce 0.8 kilograms of soybean oil. With 20 kilograms of soybeans, you can produce \\_\\_\\_\\_\\_\\_ kilograms of soybean oil. To obtain 20 kilograms of soybean oil, you need \\_\\_\\_\\_\\_\\_ kilograms of soybeans. Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 750,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Below is the graph of $y = a \\csc bx$ for some positive constants $a$ and $b.$ Find $a.$\n\n[asy]import TrigMacros;\n\nsize(500);\n\nreal g(real x)\n{\n\treturn 2*csc(x/3);\n}\n\ndraw(graph(g,-6*pi + 0.01, -3*pi - 0.01),red);\ndraw(graph(g,-3*pi + 0.01,-0.01),red);\ndraw(grap... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 751,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Starting with the display \"1,\" calculate the fewest number of keystrokes needed to reach \"400\". Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 752,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A parallelogram $ABCD$ is inscribed in the ellipse $\\frac{x^{2}}{4}+y^{2}=1$, where the slope of the line $AB$ is $k_{1}=1$. Determine the slope of the line $AD$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{4}"
} | {
"split": "train",
"index": 753,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nThe pensioners on one of the planets of Alpha Centauri enjoy spending their free time solving numeric puzzles: they choose natural numbers from a given range $[A, B]$ such that the sum of any two chosen numbers is not divisible by a certain number $N$. Last week, the newspape... | math | {
"style": "rule",
"ground_truth": "356"
} | {
"split": "train",
"index": 754,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In trapezoid \\(A B C D\\), the bases \\(A D\\) and \\(B C\\) are 8 and 18, respectively. It is known that the circumscribed circle of triangle \\(A B D\\) is tangent to lines \\(B C\\) and \\(C D\\). Find the perimeter of the trapezoid. Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "56"
} | {
"split": "train",
"index": 755,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "4.3535… is a decimal, which can be abbreviated as , and the repeating cycle is . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 756,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = \\sqrt{3}\\sin x \\cos x - \\sin^2 x$: \n(1) Find the smallest positive period of $f(x)$ and the intervals where the function is increasing; \n(2) When $x \\in [0, \\frac{\\pi}{2}]$, find the maximum and minimum values of $f(x)$. Let's think step by... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 757,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alice and Bob are playing a game with dice. They each roll a die six times and take the sums of the outcomes of their own rolls. The player with the higher sum wins. If both players have the same sum, then nobody wins. Alice's first three rolls are 6, 5, and 6, while Bob's firs... | math | {
"style": "rule",
"ground_truth": "3895"
} | {
"split": "train",
"index": 758,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the area, in square units, of the triangle formed by the $x$ and $y$ intercepts of the curve $y = (x-3)^2 (x+2) (x-1)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 759,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a finite sequence \\(P = \\left(p_{1}, p_{2}, \\cdots, p_{n}\\right)\\), the Caesar sum (named after a mathematician Caesar) is defined as \\(\\frac{s_{1}+s_{2}+\\cdots+s_{n}}{n}\\), where \\(s_{k} = p_{1} + p_{2} + \\cdots + p_{k}\\) for \\(1 \\leq k \\leq n\\). If a seq... | math | {
"style": "rule",
"ground_truth": "991"
} | {
"split": "train",
"index": 760,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many positive integers $n$ less than or equal to 1000 is $$(\\sin t+i\\cos t)^n=\\sin nt+i\\cos nt$$true for all real $t$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "250"
} | {
"split": "train",
"index": 761,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(1) Find the domain of the function $f(x) = \\log(2\\sin 2x - 1)$. \n(2) Calculate: $$\\log_{2}\\cos \\frac{\\pi}{9} + \\log_{2}\\cos \\frac{2\\pi}{9} + \\log_{2}\\cos \\frac{4\\pi}{9}.$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 762,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It is known that the complex number \\( z \\) satisfies \\( |z| = 1 \\). Find the maximum value of \\( u = \\left| z^3 - 3z + 2 \\right| \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{3}"
} | {
"split": "train",
"index": 763,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Express as a common fraction: $\\cfrac{ \\frac{2}{5}+\\frac{3}{4} }{ \\frac{4}{9}+\\frac{1}{6}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{207}{110}"
} | {
"split": "train",
"index": 764,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A set consists of five different odd positive integers, each greater than 2. When these five integers are multiplied together, their product is a five-digit integer of the form $AB0AB$, where $A$ and $B$ are digits with $A \\neq 0$ and $A \\neq B$. (The hundreds digit of the pr... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 765,
"solution": "Solution 1: Let $N=AB0AB$ and let $t$ be the two-digit integer $AB$. We note that $N=1001t$, and that $1001=11 \\cdot 91=11 \\cdot 7 \\cdot 13$. Therefore, $N=t \\cdot 7 \\cdot 11 \\cdot 13$. We want to write $N$ as the product of 5 distinct odd integers, each greate... |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all the integers between -12.1 and 3.3? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-72"
} | {
"split": "train",
"index": 766,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 7 parking spaces in a row in a parking lot, and now 4 cars need to be parked. If 3 empty spaces need to be together, calculate the number of different parking methods. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 767,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find constants $A$, $B$, and $C$, such that\n$$\\frac{-x^2+3x-4}{x^3+x}= \\frac{A}{x} +\\frac{Bx+C}{x^2+1} $$Enter your answer as the ordered triplet $(A,B,C)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(-4,3,3)"
} | {
"split": "train",
"index": 768,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $A$ and $B$ are 10 units apart. Points $B$ and $C$ are 4 units apart. Points $C$ and $D$ are 3 units apart. If $A$ and $D$ are as close as possible, then the number of units between them is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 769,
"solution": "1. **Understanding the Problem:**\n - We are given four points $A$, $B$, $C$, and $D$ with specific distances between them: $AB = 10$, $BC = 4$, and $CD = 3$.\n - We need to find the minimum possible distance between $A$ and $D$.\n\n2. **Applying the Triangle Ine... |
deepscaler | [
{
"role": "user",
"content": "Given the line $y=-x+1$ and the ellipse $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1(a > b > 0)$, they intersect at points $A$ and $B$. $OA \\perp OB$, where $O$ is the origin. If the eccentricity of the ellipse $e \\in [\\frac{1}{2}, \\frac{\\sqrt{3}}{2}]$, find the maximum v... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{10}}{2}"
} | {
"split": "train",
"index": 770,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular box has a total surface area of 166 square inches and the sum of the lengths of all its edges is 64 inches. Find the sum of the lengths in inches of all of its interior diagonals. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12\\sqrt{10}"
} | {
"split": "train",
"index": 771,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\dbinom{16}{5}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4368"
} | {
"split": "train",
"index": 772,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $M$ be the least common multiple of all the integers $10$ through $30,$ inclusive. Let $N$ be the least common multiple of $M,32,33,34,35,36,37,38,39,$ and $40.$ What is the value of $\\frac{N}{M}$? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "74"
} | {
"split": "train",
"index": 773,
"solution": "1. **Calculate $M$:** \n $M$ is the least common multiple (LCM) of all integers from $10$ to $30$. To find $M$, we need to consider the highest powers of all prime numbers that appear in the factorizations of these numbers:\n - $2^4$ comes from $16 = 2^4$.\n -... |
deepscaler | [
{
"role": "user",
"content": "Consider an equilateral triangle and a square both inscribed in a unit circle such that one side of the square is parallel to one side of the triangle. Compute the area of the convex heptagon formed by the vertices of both the triangle and the square. Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "\\frac{3+\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 774,
"solution": "Consider the diagram above. We see that the shape is a square plus 3 triangles. The top and bottom triangles have base $\\sqrt{2}$ and height $\\frac{1}{2}(\\sqrt{3}-\\sqrt{2})$, and the triangle on the side has the same base and height $1-\\frac{\\sqrt{2}}{2}$. Addi... |
deepscaler | [
{
"role": "user",
"content": "Given a regular 2017-sided polygon \\(A_{1} A_{2} \\cdots A_{2017}\\) inscribed in a unit circle \\(\\odot O\\), choose any two different vertices \\(A_{i}, A_{j}\\). Find the probability that \\(\\overrightarrow{O A_{i}} \\cdot \\overrightarrow{O A_{j}} > \\frac{1}{2}\\). Let'... | math | {
"style": "rule",
"ground_truth": "1/3"
} | {
"split": "train",
"index": 775,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many subsets containing three different numbers can be selected from the set \\(\\{ 89,95,99,132, 166,173 \\}\\) so that the sum of the three numbers is even? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 776,
"solution": "1. **Identify the parity of each number in the set**: \n The set given is $\\{89, 95, 99, 132, 166, 173\\}$. \n - Odd numbers: $89, 95, 99, 173$\n - Even numbers: $132, 166$\n\n2. **Understand the condition for the sum to be even**:\n The sum of three numbers... |
deepscaler | [
{
"role": "user",
"content": "Let \\(P_1\\) be a regular \\(r\\)-sided polygon and \\(P_2\\) be a regular \\(s\\)-sided polygon with \\(r \\geq s \\geq 3\\), such that each interior angle of \\(P_1\\) is \\(\\frac{61}{60}\\) as large as each interior angle of \\(P_2\\). What is the largest possible value of... | math | {
"style": "rule",
"ground_truth": "121"
} | {
"split": "train",
"index": 777,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\\(5^{-2 \\log_{0.04}\\left(3-4x^{2}\\right)} + 1.5 \\log_{\\frac{1}{8}} 4^{x} = 0\\) 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": 778,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $78$ is divided into three parts which are proportional to $1, \\frac{1}{3}, \\frac{1}{6},$ the middle part is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17\\frac{1}{3}"
} | {
"split": "train",
"index": 779,
"solution": "1. **Assign Variables to Parts**: Let the parts be proportional to $1$, $\\frac{1}{3}$, and $\\frac{1}{6}$. If the part proportional to $1$ is $x$, then the other parts are $\\frac{1}{3}x$ and $\\frac{1}{6}x$ respectively.\n\n2. **Set Up the Equation**: According to ... |
deepscaler | [
{
"role": "user",
"content": "Determine the smallest integer $k$ such that $k>1$ and $k$ has a remainder of $3$ when divided by any of $11,$ $4,$ and $3.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "135"
} | {
"split": "train",
"index": 780,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If 3400 were 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": "38"
} | {
"split": "train",
"index": 781,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $\\angle A = 90^\\circ$ and $\\tan C = 3$. If $BC = 90$, what is the length of $AB$, and what is the perimeter of triangle ABC? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36\\sqrt{10} + 90"
} | {
"split": "train",
"index": 782,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A pentagon is formed by placing an equilateral triangle atop a square. Each side of the square is equal to the height of the equilateral triangle. What percent of the area of the pentagon is the area of the equilateral triangle? Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "\\frac{3(\\sqrt{3} - 1)}{6} \\times 100\\%"
} | {
"split": "train",
"index": 783,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest integer $k$ such that for all integers $x$ and $y$, if $xy + 1$ is divisible by $k$, then $x + y$ is also divisible by $k$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 784,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest positive integer $m$ such that an $m \\times m$ square can be exactly divided into 7 rectangles with pairwise disjoint interiors, and the lengths of the 14 sides of these 7 rectangles are $1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14$. Let's think step by step... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 785,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A decorative window is made up of a rectangle with semicircles at either end. The ratio of $AD$ to $AB$ is $3:2$. And $AB$ is 30 inches. What is the ratio of the area of the rectangle to the combined area of the semicircles? Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "6:\\pi"
} | {
"split": "train",
"index": 786,
"solution": "1. **Set up the proportion for $AD$ and $AB$:**\n Given the ratio of $AD$ to $AB$ is $3:2$, and $AB = 30$ inches, we can write:\n \\[\n \\frac{AD}{AB} = \\frac{3}{2}\n \\]\n Substituting $AB = 30$ inches into the proportion:\n \\[\n \\frac{AD}{30} = \\f... |
deepscaler | [
{
"role": "user",
"content": "A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last 2 marbles. What is the probability that Cheryl gets 2 marbles of the same c... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"split": "train",
"index": 787,
"solution": "To solve this problem, we need to calculate the probability that Cheryl gets 2 marbles of the same color. We can use the principle of symmetry and consider the probability of any specific outcome for Cheryl, as the draws are random and independent of the order in whi... |
deepscaler | [
{
"role": "user",
"content": "An environmental agency decides to expand its monitoring teams due to new regulations requiring more extensive testing. They estimate needing 120 new employees to monitor water pollution and 105 new employees to monitor air pollution. Additionally, they need 65 new employees ca... | math | {
"style": "rule",
"ground_truth": "160"
} | {
"split": "train",
"index": 788,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\Omega$ be a unit circle and $A$ be a point on $\\Omega$ . An angle $0 < \\theta < 180^\\circ$ is chosen uniformly at random, and $\\Omega$ is rotated $\\theta$ degrees clockwise about $A$ . What is the expected area swept by this rotation? Let's think step by... | math | {
"style": "rule",
"ground_truth": "2\\pi"
} | {
"split": "train",
"index": 789,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ali wants to move from point $A$ to point $B$. He cannot walk inside the black areas but he is free to move in any direction inside the white areas (not only the grid lines but the whole plane). Help Ali to find the shortest path between $A$ and $B$. Only draw the path and writ... | math | {
"style": "rule",
"ground_truth": "7 + 5\\sqrt{2}"
} | {
"split": "train",
"index": 790,
"solution": "\nThe task is to find the shortest path for Ali to move from point \\( A \\) to point \\( B \\), only navigating through the white areas in the given plane. Based on the diagram provided, we will employ geometric considerations to determine the path and length.\n\n##... |
deepscaler | [
{
"role": "user",
"content": "What is the value of $x$ if $x=\\frac{2023^2 - 2023 + 1}{2023}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2022 + \\frac{1}{2023}"
} | {
"split": "train",
"index": 791,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The [Fibonacci sequence](https://artofproblemsolving.com/wiki/index.php/Fibonacci_sequence) $1,1,2,3,5,8,13,21,\\ldots$ starts with two 1s, and each term afterwards is the sum of its two predecessors. Which one of the ten [digits](https://artofproblemsolving.com/wiki/index.php/... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 792,
"solution": "To solve this problem, we need to determine the first occurrence of each digit from 0 to 9 in the units position of the Fibonacci sequence. We can simplify the problem by considering the Fibonacci sequence modulo 10, which will directly give us the units digit of eac... |
deepscaler | [
{
"role": "user",
"content": "The relevant departments want to understand the popularization of knowledge about the prevention of H1N1 influenza in schools, so they designed a questionnaire with 10 questions and conducted a survey in various schools. Two classes, A and B, from a certain middle school were r... | math | {
"style": "rule",
"ground_truth": "\\dfrac{2}{5}"
} | {
"split": "train",
"index": 793,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Josef and Timothy play a game in which Josef picks an integer between 1 and 1000 inclusive and Timothy divides 1000 by that integer and states whether or not the quotient is an integer. How many integers could Josef pick such that Timothy's quotient is an integer? Let's think ... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 794,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A B$ be a segment of length 2 with midpoint $M$. Consider the circle with center $O$ and radius $r$ that is externally tangent to the circles with diameters $A M$ and $B M$ and internally tangent to the circle with diameter $A B$. Determine the value of $r$. Let's think st... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 795,
"solution": "Let $X$ be the midpoint of segment $A M$. Note that $O M \\perp M X$ and that $M X=\\frac{1}{2}$ and $O X=\\frac{1}{2}+r$ and $O M=1-r$. Therefore by the Pythagorean theorem, we have $$O M^{2}+M X^{2}=O X^{2} \\Longrightarrow(1-r)^{2}+\\frac{1}{2^{2}}=\\left(\\frac{1... |
deepscaler | [
{
"role": "user",
"content": "The forecast predicts an 80 percent chance of rain for each day of a three-day festival. If it doesn't rain, there is a 50% chance it will be sunny and a 50% chance it will be cloudy. Mina and John want exactly one sunny day during the festival for their outdoor activities. Wha... | math | {
"style": "rule",
"ground_truth": "0.243"
} | {
"split": "train",
"index": 796,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many cubic centimeters are in the volume of a cone having a diameter of 12 cm and a slant height of 10 cm? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "96 \\pi"
} | {
"split": "train",
"index": 797,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of functions $f : \\mathbb{R} \\to \\mathbb{R}$ such that\n\\[f(x + y) f(x - y) = (f(x) + f(y))^2 - 4x^2 f(y)\\]for all real numbers $x$ and $y.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 798,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$\\frac{(3!)!}{3!} = $ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 799,
"solution": "1. **Calculate $3!$:**\n \\[\n 3! = 3 \\times 2 \\times 1 = 6\n \\]\n\n2. **Calculate $(3!)!$:**\n \\[\n (3!)! = (6)!\n \\]\n\n3. **Expand $(6)!$:**\n \\[\n 6! = 6 \\times 5 \\times 4 \\times 3 \\times 2 \\times 1\n \\]\n\n4. **Simplify the expressi... |
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