data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Calculate:<br/>$(1)(\\sqrt{\\frac{1}{3}})^{2}+\\sqrt{0．{3}^{2}}-\\sqrt{\\frac{1}{9}}$;<br/>$(2)(\\sqrt{6}-\\sqrt{\\frac{1}{2}})-(\\sqrt{24}+2\\sqrt{\\frac{2}{3}})$;<br/>$(3)(\\frac{\\sqrt{32}}{3}-4\\sqrt{\\frac{1}{2}}+3\\sqrt{27})÷2\\sqrt{2}$;<br/>$(4)(\\sqrt{3}+\\sqrt{2}-1)(\\... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"split": "train",
"index": 18949,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many natural-number factors does $\\textit{N}$ have if $\\textit{N} = 2^3 \\cdot 3^2 \\cdot 5^1$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 18950,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two circles $C\\_1$: $x^{2}+y^{2}=1$ and $C\\_2$: $(x-2)^{2}+(y-4)^{2}=1$, a moving point $P(a,b)$ passes through and forms tangent lines $PM$ and $PN$ to circles $C\\_1$ and $C\\_2$ respectively with $M$ and $N$ being the points of tangency. If $PM=PN$, find the minimum ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{34}"
} | {
"split": "train",
"index": 18951,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diagonals of a convex quadrilateral \\(ABCD\\) intersect at point \\(E\\). Given that \\(AB = AD\\) and \\(CA\\) is the bisector of angle \\(C\\), with \\(\\angle BAD = 140^\\circ\\) and \\(\\angle BEA = 110^\\circ\\), find angle \\(CDB\\). Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 18952,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive integer is *happy* if:\n\n1. All its digits are different and not $0$ ,\n2. One of its digits is equal to the sum of the other digits.\n\nFor example, 253 is a *happy* number. How many *happy* numbers are there? Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 18953,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four girls and eight boys came for a class photograph. Children approach the photographer in pairs and take a joint photo. Among how many minimum photos must there necessarily be either a photo of two boys, a photo of two girls, or two photos with the same children? Let's think... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 18954,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Carlos took $70\\%$ of a whole pie. Maria took one third of the remainder. What portion of the whole pie was left? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20\\%"
} | {
"split": "train",
"index": 18955,
"solution": "1. **Calculate the remaining pie after Carlos's share:**\n Carlos took $70\\%$ of the pie. Therefore, the remaining part of the pie is:\n \\[\n 100\\% - 70\\% = 30\\%\n \\]\n\n2. **Determine Maria's share of the remaining pie:**\n Maria took one third of ... |
deepscaler | [
{
"role": "user",
"content": "Add $518_{12} + 276_{12}$. Express your answer in base 12, using $A$ for $10$ and $B$ for $11$ if necessary. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "792_{12}"
} | {
"split": "train",
"index": 18956,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that Josie jogs parallel to a canal along which a boat is moving at a constant speed in the same direction and counts 130 steps to reach the front of the boat from behind it, and 70 steps from the front to the back, find the length of the boat in terms of Josie's steps. L... | math | {
"style": "rule",
"ground_truth": "91"
} | {
"split": "train",
"index": 18957,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Julio cuts off the four corners, or vertices, of a regular tetrahedron. How many vertices does the remaining shape have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 18958,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCDEF$ be a regular hexagon with side 1. Point $X, Y$ are on sides $CD$ and $DE$ respectively, such that the perimeter of $DXY$ is $2$. Determine $\\angle XAY$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30^\\circ"
} | {
"split": "train",
"index": 18959,
"solution": "\nTo solve for \\(\\angle XAY\\), we first establish the geometry of the problem. We have a regular hexagon \\(ABCDEF\\) with side length 1. Since it is regular, each interior angle of the hexagon is \\(120^\\circ\\).\n\nPoints \\(X\\) and \\(Y\\) are located on si... |
deepscaler | [
{
"role": "user",
"content": "The Red Sox play the Yankees in a best-of-seven series that ends as soon as one team wins four games. Suppose that the probability that the Red Sox win Game $n$ is $\\frac{n-1}{6}$. What is the probability that the Red Sox will win the series? Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 18960,
"solution": "Note that if we imagine that the series always continues to seven games even after one team has won four, this will never change the winner of the series. Notice also that the probability that the Red Sox will win Game $n$ is precisely the probability that the Yank... |
deepscaler | [
{
"role": "user",
"content": "Two water particles fall freely in succession from a $300 \\mathrm{~m}$ high cliff. The first one has already fallen $\\frac{1}{1000} \\mathrm{~mm}$ when the second one starts to fall.\n\nHow far apart will the two particles be at the moment when the first particle reaches the ... | math | {
"style": "rule",
"ground_truth": "34.6"
} | {
"split": "train",
"index": 18961,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two boards, one 5 inches wide and the other 7 inches wide, are nailed together to form an X. The angle at which they cross is 45 degrees. If this structure is painted and the boards are later separated, what is the area of the unpainted region on the five-inch board? Assume the... | math | {
"style": "rule",
"ground_truth": "35\\sqrt{2}"
} | {
"split": "train",
"index": 18962,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Exactly three of the interior angles of a convex polygon are obtuse. What is the maximum number of sides of such a polygon? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 18963,
"solution": "1. **Sum of Interior Angles**: The sum of the interior angles of an $n$-sided polygon is given by the formula:\n \\[\n 180(n-2) = 180n - 360\n \\]\n This formula arises from the fact that a polygon can be divided into $(n-2)$ triangles, each contributing $1... |
deepscaler | [
{
"role": "user",
"content": "Find the largest prime divisor of 11! + 12! Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 18964,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $$\\frac{1}{2}\\times4\\times\\frac{1}{8}\\times16\\times\\frac{1}{32}\\times64\\times\\frac{1}{128}\\times256\\times\\frac{1}{512}\\times1024?$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 18965,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The volume of the box is 360 cubic units where $a, b,$ and $c$ are integers with $1<c<b<a$. What is the largest possible value of $b$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 18966,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the largest solution to the equation \\[\\log_{10x^2} 10 + \\log_{100x^3} 10 = -2.\\]Find the value of $\\frac{1}{x^{12}},$ writing your answer in decimal representation. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10000000"
} | {
"split": "train",
"index": 18967,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nThe numbers \\( x \\) and \\( y \\) satisfy the equation:\n\n$$\n\\sqrt{x y}+\\sqrt{(1-x)(1-y)}=\\sqrt{7 x(1-y)}+\\frac{\\sqrt{y(1-x)}}{\\sqrt{7}}\n$$\n\nFind the maximum value of the expression \\( x + 7y \\). Justify your answer. Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "57/8"
} | {
"split": "train",
"index": 18968,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the projection of the vector $\\begin{pmatrix} 4 \\\\ -4 \\\\ -1 \\end{pmatrix}$ onto the line\n\\[2x = -3y = z.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 6/7 \\\\ -4/7 \\\\ 12/7 \\end{pmatrix}"
} | {
"split": "train",
"index": 18969,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\clubsuit(x)$ denote the sum of the digits of the positive integer $x$. For example, $\\clubsuit(8)=8$ and $\\clubsuit(123)=1+2+3=6$. For how many two-digit values of $x$ is $\\clubsuit(\\clubsuit(x))=3$? Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 18970,
"solution": "1. **Define the function and its application**: Let $\\clubsuit(x)$ denote the sum of the digits of the positive integer $x$. For example, $\\clubsuit(8) = 8$ and $\\clubsuit(123) = 1 + 2 + 3 = 6$.\n\n2. **Determine possible values of $y = \\clubsuit(x)$**: Since $... |
deepscaler | [
{
"role": "user",
"content": "Given $\\sqrt{20} \\approx 4.472, \\sqrt{2} \\approx 1.414$, find $-\\sqrt{0.2} \\approx$____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-0.4472"
} | {
"split": "train",
"index": 18971,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( a \\) be a nonzero real number. In the Cartesian coordinate plane \\( xOy \\), the focal distance of the conic section \\( x^2 + a y^2 + a^2 = 0 \\) is 4. Find the value of \\( a \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1 - \\sqrt{17}}{2}"
} | {
"split": "train",
"index": 18972,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a new arcade game, the \"monster\" is the shaded region of a semicircle with radius 2 cm as shown in the diagram. The mouth, which is an unshaded piece within the semicircle, subtends a central angle of 90°. Compute the perimeter of the shaded region.\nA) $\\pi + 3$ cm\nB) $... | math | {
"style": "rule",
"ground_truth": "\\pi + 4"
} | {
"split": "train",
"index": 18973,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Franklin has four bags, numbered 1 through 4. Initially, the first bag contains fifteen balls, numbered 1 through 15 , and the other bags are empty. Franklin randomly pulls a pair of balls out of the first bag, throws away the ball with the lower number, and moves the ball with... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 18974,
"solution": "Pretend there is a 16 th ball numbered 16. This process is equivalent to randomly drawing a tournament bracket for the 16 balls, and playing a tournament where the higher ranked ball always wins. The probability that a ball is left in a bag at the end is the probab... |
deepscaler | [
{
"role": "user",
"content": "In a drawer Sandy has $5$ pairs of socks, each pair a different color. On Monday Sandy selects two individual socks at random from the $10$ socks in the drawer. On Tuesday Sandy selects $2$ of the remaining $8$ socks at random and on Wednesday two of the remaining $6$ socks at ... | math | {
"style": "rule",
"ground_truth": "341"
} | {
"split": "train",
"index": 18975,
"solution": "The key is to count backwards. First, choose the pair which you pick on Wednesday in $5$ ways. Then there are four pairs of socks for you to pick a pair of on Tuesday, and you don't want to pick a pair. Since there are $4$ pairs, the number of ways to do this is $\... |
deepscaler | [
{
"role": "user",
"content": "Tim wants to invest some money in a bank which compounds quarterly with an annual interest rate of $7\\%$. To the nearest dollar, how much money should he invest if he wants a total of $\\$60,\\!000$ at the end of $5$ years? Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "\\$42409"
} | {
"split": "train",
"index": 18976,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of arrangements of 4 beads (2 red, 2 green, 2 blue) in a circle such that the two red beads are not adjacent. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 18977,
"solution": "We divide this problem into cases based on the relative position of the two red beads: - They are adjacent. Then, there are 4 possible placements of the green and blue beads: GGBB, GBBG, GBGB, BGGB. - They are 1 bead apart. Then, there are two choices for the bead ... |
deepscaler | [
{
"role": "user",
"content": "Given ten 0's and ten 1's, how many 0-1 binary sequences can be formed such that no three or more consecutive 0's are together? For example, 01001001010011101011 is such a sequence, but the sequence 01001000101001110111 does not satisfy this condition. Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "24068"
} | {
"split": "train",
"index": 18978,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At a certain university, the division of mathematical sciences consists of the departments of mathematics, statistics, and computer science. There are two male and two female professors in each department. A committee of six professors is to contain three men and three women an... | math | {
"style": "rule",
"ground_truth": "88"
} | {
"split": "train",
"index": 18979,
"solution": "Use generating functions. For each department, there is 1 way to pick 2 males, 4 ways to pick one of each, and 1 way to pick 2 females. Since there are three departments in total, and we wish for three males and three females, the answer will be equal to the coeffi... |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be a square. Let $E, F, G$ and $H$ be the centers, respectively, of equilateral triangles with bases $\\overline{AB}, \\overline{BC}, \\overline{CD},$ and $\\overline{DA},$ each exterior to the square. What is the ratio of the area of square $EFGH$ to the area of squ... | math | {
"style": "rule",
"ground_truth": "\\frac{2+\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 18980,
"solution": "1. **Assign Coordinates to Square $ABCD$**:\n Assume the side length of square $ABCD$ is $s$. Without loss of generality, let $s = 6$ for simplicity. Place $ABCD$ in the coordinate plane with $A = (0, 0)$, $B = (6, 0)$, $C = (6, 6)$, and $D = (0, 6)$.\n\n2. **Loc... |
deepscaler | [
{
"role": "user",
"content": "Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(2008,0),(2008,2009),$ and $(0,2009)$. What is the probability that $x > 2y$? Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "\\frac{502}{2009}"
} | {
"split": "train",
"index": 18981,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the direction vectors of two skew lines $l_{1}$ and $l_{2}$ are $\\overrightarrow{a}=\\left(0,-2,-1\\right)$ and $\\overrightarrow{b}=\\left(2,0,4\\right)$, calculate the cosine value of the angle between the two skew lines $l_{1}$ and $l_{2}$. Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"split": "train",
"index": 18982,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among the positive integers less than $10^{4}$, how many positive integers $n$ are there such that $2^{n} - n^{2}$ is divisible by 7? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2857"
} | {
"split": "train",
"index": 18983,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the random variable $\\xi$ follows a normal distribution $N(4, 6^2)$, and $P(\\xi \\leq 5) = 0.89$, find the probability $P(\\xi \\leq 3)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.11"
} | {
"split": "train",
"index": 18984,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Zach has twelve identical-looking chocolate eggs. Exactly three of the eggs contain a special prize inside. Zach randomly gives three of the twelve eggs to each of Vince, Wendy, Xin, and Yolanda. What is the probability that only one child will receive an egg that contains a sp... | math | {
"style": "rule",
"ground_truth": "1/55"
} | {
"split": "train",
"index": 18985,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A$ be the number of unordered pairs of ordered pairs of integers between 1 and 6 inclusive, and let $B$ be the number of ordered pairs of unordered pairs of integers between 1 and 6 inclusive. (Repetitions are allowed in both ordered and unordered pairs.) Find $A-B$. Let's... | math | {
"style": "rule",
"ground_truth": "225"
} | {
"split": "train",
"index": 18986,
"solution": "There are $6 \\cdot 6$ ordered pairs of integers between 1 and 6 inclusive and 21 unordered pairs of integers \\left(\\binom{6}{2}=15\\right.$ different pairs and 6 doubles). Then, $A=\\binom{36}{2}+36=666$ and $B=21 \\cdot 21=441$. Therefore $A-B=225$. For general... |
deepscaler | [
{
"role": "user",
"content": "Find the greatest natural number $n$ such that $n\\leq 2008$ and $(1^2+2^2+3^2+\\cdots + n^2)\\left[(n+1)^2+(n+2)^2+(n+3)^2+\\cdots + (2n)^2\\right]$ is a perfect square.\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1921"
} | {
"split": "train",
"index": 18987,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\\dfrac{1}{10} + \\dfrac{9}{100} + \\dfrac{9}{1000} + \\dfrac{7}{10000} = Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.1997"
} | {
"split": "train",
"index": 18988,
"solution": "1. **Convert each fraction to its decimal form**:\n - $\\dfrac{1}{10} = 0.1$\n - $\\dfrac{9}{100} = 0.09$\n - $\\dfrac{9}{1000} = 0.009$\n - $\\dfrac{7}{10000} = 0.0007$\n\n2. **Add the decimals**:\n - Start by adding the largest decimal places first:\n ... |
deepscaler | [
{
"role": "user",
"content": "Integers $x$ and $y$ with $x > y > 0$ satisfy $x + y + xy = 119$. What is $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "39"
} | {
"split": "train",
"index": 18989,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "After shifting the graph of the function $y=\\sin^2x-\\cos^2x$ to the right by $m$ units, the resulting graph is symmetric to the graph of $y=k\\sin x\\cos x$ ($k>0$) with respect to the point $\\left( \\frac{\\pi}{3}, 0 \\right)$. Find the minimum positive value of $k+m$. Let'... | math | {
"style": "rule",
"ground_truth": "2+ \\frac{5\\pi}{12}"
} | {
"split": "train",
"index": 18990,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $P$ is a point on the side $BC$ such that $\\overrightarrow{BP} = \\frac{1}{2}\\overrightarrow{PC}$. Points $M$ and $N$ lie on the line passing through $P$ such that $\\overrightarrow{AM} = \\lambda \\overrightarrow{AB}$ and $\\overrightarrow{AN} = \\mu \\o... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{3}"
} | {
"split": "train",
"index": 18991,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "One can holds $12$ ounces of soda, what is the minimum number of cans needed to provide a gallon ($128$ ounces) of soda? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 18992,
"solution": "To find the minimum number of cans needed to provide at least one gallon (128 ounces) of soda, where each can holds 12 ounces, we need to calculate the smallest integer $n$ such that $12n \\geq 128$.\n\n1. **Calculate the exact number of cans needed if there were n... |
deepscaler | [
{
"role": "user",
"content": "Given the line $l: y = kx + m$ intersects the ellipse $C: \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$ at points $A$ and $P$, and intersects the $x$-axis and $y$-axis at points $N$ and $M$ respectively, with $PM = MN$. Point $Q$ is the reflection of point $P$ across the... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 18993,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sequence of numbers is written on the blackboard: \\(1, 2, 3, \\cdots, 50\\). Each time, the first 4 numbers are erased, and the sum of these 4 erased numbers is written at the end of the sequence, creating a new sequence. This operation is repeated until there are fewer than... | math | {
"style": "rule",
"ground_truth": "755"
} | {
"split": "train",
"index": 18994,
"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 $a = 2c \\cos A$ and $\\sqrt{5} \\sin A = 1$, find:\n1. $\\sin C$\n2. $\\frac{b}{c}$ Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{5} + 5\\sqrt{3}}{5}"
} | {
"split": "train",
"index": 18995,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all complex numbers $z$ such that\n\\[z^2 = -77 - 36i.\\]Enter all complex numbers, separated by commas. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 - 9i, -2 + 9i"
} | {
"split": "train",
"index": 18996,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Are there integers $m$ and $n$ such that\n\\[5m^2 - 6mn + 7n^2 = 1985 \\ ?\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\text{No}"
} | {
"split": "train",
"index": 18997,
"solution": "\nTo determine whether there are integers \\( m \\) and \\( n \\) such that\n\n\\[\n5m^2 - 6mn + 7n^2 = 1985,\n\\]\n\nwe begin by analyzing the quadratic form. We can rewrite the equation as:\n\n\\[\n5m^2 - 6mn + 7n^2.\n\\]\n\nFirst, let's complete the square with ... |
deepscaler | [
{
"role": "user",
"content": "The consignment shop received for sale cameras, clocks, pens, and receivers totaling 240 rubles. The sum of the prices of the receiver and one clock is 4 rubles more than the sum of the prices of the camera and the pen, and the sum of the prices of one clock and the pen is 24 r... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 18998,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $A B C$ has incircle $\\omega$ which touches $A B$ at $C_{1}, B C$ at $A_{1}$, and $C A$ at $B_{1}$. Let $A_{2}$ be the reflection of $A_{1}$ over the midpoint of $B C$, and define $B_{2}$ and $C_{2}$ similarly. Let $A_{3}$ be the intersection of $A A_{2}$ with $\\omeg... | math | {
"style": "rule",
"ground_truth": "14/65"
} | {
"split": "train",
"index": 18999,
"solution": "Notice that $A_{2}$ is the point of tangency of the excircle opposite $A$ to $B C$. Therefore, by considering the homothety centered at $A$ taking the excircle to the incircle, we notice that $A_{3}$ is the intersection of $\\omega$ and the tangent line parallel to... |
deepscaler | [
{
"role": "user",
"content": "When a student multiplied the number $66$ by the repeating decimal, \\(1.\\overline{ab}\\), where $a$ and $b$ are digits, he did not notice the notation and just multiplied $66$ times $1.ab.$ Later he found that his answer is $0.5$ less than the correct answer. What is the $2$-... | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 19000,
"solution": "1. **Understanding the Problem:**\n We are given a repeating decimal $1.\\overline{ab}$, where $a$ and $b$ are digits, and the student mistakenly multiplied $66$ by $1.ab$ instead of $1.\\overline{ab}$. The error in the calculation resulted in an answer that was ... |
deepscaler | [
{
"role": "user",
"content": "Let M be a subst of {1,2,...,2006} with the following property: For any three elements x,y and z (x<y<z) of M, x+y does not divide z. Determine the largest possible size of M. Justify your claim. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1004"
} | {
"split": "train",
"index": 19001,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\(\\alpha, \\beta \\in \\left(\\frac{3\\pi}{4}, \\pi \\right)\\), \\(\\cos (\\alpha + \\beta) = \\frac{4}{5}\\), and \\(\\sin \\left(\\alpha - \\frac{\\pi}{4}\\right) = \\frac{12}{13}\\), find \\(\\cos \\left(\\beta + \\frac{\\pi}{4}\\right)\\). Let's think step by ... | math | {
"style": "rule",
"ground_truth": "-\\frac{56}{65}"
} | {
"split": "train",
"index": 19002,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A $3$ by $3$ determinant has three entries equal to $2$ , three entries equal to $5$ , and three entries equal to $8$ . Find the maximum possible value of the determinant. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "405"
} | {
"split": "train",
"index": 19003,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define the ☆ operation: observe the following expressions: \n$1$☆$2=1\\times 3+2=5$; \n$4$☆$(-1)=4\\times 3-1=11$; \n$(-5)$☆$3=(-5)\\times 3+3=-12$; \n$(-6)$☆$(-3)=(-6)\\times 3-3=-21\\ldots$ \n$(1)-1$☆$2=$______, $a$☆$b=$______; \n$(2)$ If $a \\lt b$, then $a$☆$b-b$☆$a$ ... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 19004,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $DEF$ has side lengths $DE = 15$, $EF = 36$, and $FD = 39$. Rectangle $WXYZ$ has vertex $W$ on $\\overline{DE}$, vertex $X$ on $\\overline{DF}$, and vertices $Y$ and $Z$ on $\\overline{EF}$. In terms of the side length $WX = \\omega$, the area of $WXYZ$ can be expresse... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 19005,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $f$ and $g$ are polynomials, and that $h(x)=f(g(x))+g(x)$. Find the degree of $g(x)$ given that the degree of $h(x)$ is $6$ and the degree of $f(x)$ is $2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 19006,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of four-digit numbers with distinct digits, formed using the digits 0, 1, 2, ..., 9, such that the absolute difference between the units and hundreds digit is 8. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "154"
} | {
"split": "train",
"index": 19007,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five points $A$, $B$, $C$, $D$, and $O$ lie on a flat field. $A$ is directly north of $O$, $B$ is directly west of $O$, $C$ is directly south of $O$, and $D$ is directly east of $O$. The distance between $C$ and $D$ is 140 m. A hot-air balloon is positioned in the air at $H$... | math | {
"style": "rule",
"ground_truth": "30\\sqrt{11}"
} | {
"split": "train",
"index": 19008,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Everyone in a class of 30 students takes math and history. Seven students received an A in history and 13 received an A in math, including four that received an A in both courses. How many students did not receive an A in any of these two courses? Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 19009,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the limit of the function:\n\n$$\\lim _{x \\rightarrow 0} \\frac{e^{4 x}-1}{\\sin \\left(\\pi\\left(\\frac{x}{2}+1\\right)\\right)}$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{8}{\\pi}"
} | {
"split": "train",
"index": 19010,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\cos (α+ \\dfrac {π}{6})= \\dfrac {1}{3}$, where $α∈(0\\,,\\,\\, \\dfrac {π}{2})$, find $\\sin α$ and $\\sin (2α+ \\dfrac {5π}{6})$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\dfrac {7}{9}"
} | {
"split": "train",
"index": 19011,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A solitaire game is played as follows. Six distinct pairs of matched tiles are placed in a bag. The player randomly draws tiles one at a time from the bag and retains them, except that matching tiles are put aside as soon as they appear in the player's hand. The game ends if th... | math | {
"style": "rule",
"ground_truth": "394"
} | {
"split": "train",
"index": 19012,
"solution": "Let $P_k$ be the probability of emptying the bag when it has $k$ pairs in it. Let's consider the possible draws for the first three cards:\nCase 1. We draw a pair on the first two cards. The second card is the same as the first with probability $\\frac {1}{2k - 1}$... |
deepscaler | [
{
"role": "user",
"content": "What is the measure of $\\angle X Z Y$ if $M$ is the midpoint of $Y Z$, $\\angle X M Z=30^{\\circ}$, and $\\angle X Y Z=15^{\\circ}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "75^{\\circ}"
} | {
"split": "train",
"index": 19013,
"solution": "Since $\\angle X M Z=30^{\\circ}$, then $\\angle X M Y=180^{\\circ}-\\angle X M Z=180^{\\circ}-30^{\\circ}=150^{\\circ}$. \n Since the angles in $\\triangle X M Y$ add to $180^{\\circ}$, then \n $$ \\angle Y X M=180^{\\circ}-\\angle X Y Z-\\angle X M Y=180^{\\circ}... |
deepscaler | [
{
"role": "user",
"content": "The diagram shows a segment of a circle such that \\( CD \\) is the perpendicular bisector of the chord \\( AB \\). Given that \\( AB = 16 \\) and \\( CD = 4 \\), find the diameter of the circle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 19014,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If you flip a fair coin 1000 times, what is the expected value of the product of the number of heads and the number of tails? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "249750"
} | {
"split": "train",
"index": 19015,
"solution": "We solve the problem for $n$ coins. We want to find $$E(n)=\\sum_{k=0}^{n} \\frac{1}{2^{n}}\\binom{n}{k} k(n-k)$$ We present three methods for evaluating this sum. Method 1: Discard the terms $k=0, k=n$. Since $\\binom{n}{k} k(n-k)=n(n-1)\\binom{n-2}{k-1}$ by the f... |
deepscaler | [
{
"role": "user",
"content": "Three prime numbers are randomly selected without replacement from the first ten prime numbers. What is the probability that the sum of the three selected numbers is odd? Express your answer as a common fraction. Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{10}"
} | {
"split": "train",
"index": 19016,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence $\\{a_n\\}$ satisfying $(\\log_3{a_n}+1=\\log_3{a_{n+1}}\\ (n\\in \\mathbb{N}^*)$, and $(a_2+a_4+a_6=9$, find the value of $(\\log_{\\frac{1}{3}}(a_5+a_7+a_9))$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"split": "train",
"index": 19017,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "**p1.** Is it possible to place six points in the plane and connect them by nonintersecting segments so that each point will be connected with exactly\na) Three other points?\nb) Four other points?**p2.** Martian bank notes can have denomination of $1, 3, 5, 25$ marts. Is it ... | math | {
"style": "rule",
"ground_truth": "1376"
} | {
"split": "train",
"index": 19018,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The entries in a $3 \\times 3$ array include all the digits from $1$ through $9$, arranged so that the entries in every row and column are in increasing order. How many such arrays are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 19019,
"solution": "To solve this problem, we need to consider the constraints given by the increasing order in rows and columns. We will analyze the possible placements of the numbers, especially focusing on the central number and the corners, which are critical due to the increasing... |
deepscaler | [
{
"role": "user",
"content": "In a convex 10-gon \\(A_{1} A_{2} \\ldots A_{10}\\), all sides and all diagonals connecting vertices skipping one (i.e., \\(A_{1} A_{3}, A_{2} A_{4},\\) etc.) are drawn, except for the side \\(A_{1} A_{10}\\) and the diagonals \\(A_{1} A_{9}\\), \\(A_{2} A_{10}\\).\n\nA path fr... | math | {
"style": "rule",
"ground_truth": "55"
} | {
"split": "train",
"index": 19020,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Robert has 4 indistinguishable gold coins and 4 indistinguishable silver coins. Each coin has an engraving of one face on one side, but not on the other. He wants to stack the eight coins on a table into a single stack so that no two adjacent coins are face to face. Find the nu... | math | {
"style": "rule",
"ground_truth": "630"
} | {
"split": "train",
"index": 19021,
"solution": "There are two separate parts to this problem: one is the color (gold vs silver), and the other is the orientation.\n There are ${8\\choose4} = 70$ ways to position the gold coins in the stack of 8 coins, which determines the positions of the silver coins.\n Create ... |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, the measure of $\\angle A$ is $86$ degrees. The measure of $\\angle B$ is $22$ degrees more than three times the measure of $\\angle C$. What is the measure, in degrees, of $\\angle C$? Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "18 \\text{ degrees}"
} | {
"split": "train",
"index": 19022,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a,b,c,d$ are Distinct Real no. such that $a = \\sqrt{4+\\sqrt{5+a}}$ $b = \\sqrt{4-\\sqrt{5+b}}$ $c = \\sqrt{4+\\sqrt{5-c}}$ $d = \\sqrt{4-\\sqrt{5-d}}$ Then $abcd = $ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 19023,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given 5 points \\( A, B, C, D, E \\) on a plane, with no three points being collinear. How many different ways can one connect these points with 4 segments such that each point is an endpoint of at least one segment? Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "135"
} | {
"split": "train",
"index": 19024,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the $xy$-plane, the segment with endpoints $(-3,0)$ and $(27,0)$ is the diameter of a circle. A vertical line $x=k$ intersects the circle at two points, and one of the points has a $y$-coordinate of $12$. Find the value of $k$. Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 19025,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many values of $n$ with $3 \\leq n \\leq 12$ can a Fano table be created? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 19026,
"solution": "First, we calculate the number of pairs that can be formed from the integers from 1 to $n$. One way to form a pair is to choose one number to be the first item of the pair ($n$ choices) and then a different number to be the second item of the pair ($n-1$ choices). ... |
deepscaler | [
{
"role": "user",
"content": "Michael has never taken a foreign language class, but is doing a story on them for the school newspaper. The school offers French and Spanish. Michael has a list of all 25 kids in the school enrolled in at least one foreign language class. He also knows that 18 kids are in th... | math | {
"style": "rule",
"ground_truth": "\\frac{91}{100}"
} | {
"split": "train",
"index": 19027,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, it is known that $\\sin A : \\sin B : \\sin C = 3 : 5 : 7$. The largest interior angle of this triangle is equal to ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\pi}{3}"
} | {
"split": "train",
"index": 19028,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $b\\sin A=a\\sin C$ and $c=1$, find the value of $b$ and the maximum area of $\\triangle ABC$. 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": 19029,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A digital watch displays hours and minutes in a 24-hour format. Calculate the largest possible sum of the digits in the display. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 19030,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four people are sitting at four sides of a table, and they are dividing a 32-card Hungarian deck equally among themselves. If one selected player does not receive any aces, what is the probability that the player sitting opposite them also has no aces among their 8 cards? Let's... | math | {
"style": "rule",
"ground_truth": "130/759"
} | {
"split": "train",
"index": 19031,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain school club has 10 members, and two of them are put on duty each day from Monday to Friday. Given that members A and B must be scheduled on the same day, and members C and D cannot be scheduled together, the total number of different possible schedules is (▲). Choices... | math | {
"style": "rule",
"ground_truth": "5400"
} | {
"split": "train",
"index": 19032,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alvin rides on a flat road at $25$ kilometers per hour, downhill at $35$ kph, and uphill at $10$ kph. Benny rides on a flat road at $35$ kph, downhill at $45$ kph, and uphill at $15$ kph. Alvin goes from town $X$ to town $Y$, a distance of $15$ km all uphill, then from town $Y$... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 19033,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The expression $16x^2-106x-105$ can be written as $(8x + a)(2x + b),$ where $a$ and $b$ are integers. What is $a + 2b$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-23"
} | {
"split": "train",
"index": 19034,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many solutions of the equation $\\tan x = \\tan (\\tan x)$ are on the interval $0 \\le x \\le \\tan^{-1} 942$? (Here $\\tan^{-1}$ means the inverse tangent function, sometimes written $\\arctan$.)\n\nNote: You can assume the result that $\\tan \\theta > \\theta$ for $0 < \... | math | {
"style": "rule",
"ground_truth": "300"
} | {
"split": "train",
"index": 19035,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many of the natural numbers from 1 to 800, inclusive, contain the digit 7 at least once? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "233"
} | {
"split": "train",
"index": 19036,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "2 diagonals of a regular heptagon (a 7-sided polygon) are chosen. What is the probability that they intersect inside the heptagon? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{5}{13}"
} | {
"split": "train",
"index": 19037,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If each side of a regular hexagon consists of 6 toothpicks, and there are 6 sides, calculate the total number of toothpicks used to build the hexagonal grid. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 19038,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the probability that the same number will be facing up on each of five standard six-sided dice that are tossed simultaneously? Express your answer as a simplified common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{1296}"
} | {
"split": "train",
"index": 19039,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the equation \\(\\left|x^{2}-2ax+b\\right|=8\\) has exactly three real roots, and these roots are the side lengths of a right triangle. Find the value of \\(a+b\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "264"
} | {
"split": "train",
"index": 19040,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many such five-digit Shenma numbers exist, where the middle digit is the smallest, the digits increase as they move away from the middle, and all the digits are different? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1512"
} | {
"split": "train",
"index": 19041,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The positive integers \\( x \\) and \\( y \\), for which \\( \\gcd(x, y) = 3 \\), are the coordinates of the vertex of a square centered at the origin with an area of \\( 20 \\cdot \\operatorname{lcm}(x, y) \\). Find the perimeter of the square. Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "24\\sqrt{5}"
} | {
"split": "train",
"index": 19042,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Seven students stand in a row for a photo, among them, students A and B must stand next to each other, and students C and D must not stand next to each other. The total number of different arrangements is. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "960"
} | {
"split": "train",
"index": 19043,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $|\\vec{a}|=|\\vec{b}|=1$ and $\\vec{a} \\perp \\vec{b}$, find the projection of $2\\vec{a}+\\vec{b}$ in the direction of $\\vec{a}+\\vec{b}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{3\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 19044,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Makarla attended two meetings during her $9$-hour work day. The first meeting took $45$ minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 19045,
"solution": "1. **Convert the work day into minutes**: \n Makarla's work day is $9$ hours long. Since there are $60$ minutes in an hour, the total number of minutes in her work day is:\n \\[\n 9 \\times 60 = 540 \\text{ minutes}\n \\]\n\n2. **Calculate the total time sp... |
deepscaler | [
{
"role": "user",
"content": "If $M = 2007 \\div 3$, $N = M \\div 3$, and $X = M - N$, then 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": "446"
} | {
"split": "train",
"index": 19046,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse $\\frac {x^{2}}{a^{2}} + \\frac {y^{2}}{b^{2}} = 1 (a > b > 0)$ with vertex $B$ at the top, vertex $A$ on the right, and right focus $F$. Let $E$ be a point on the lower half of the ellipse such that the tangent at $E$ is parallel to $AB$. If the eccentricity o... | math | {
"style": "rule",
"ground_truth": "\\frac {\\sqrt {2}}{4}"
} | {
"split": "train",
"index": 19047,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diagram shows a polygon made by removing six $2\\times 2$ squares from the sides of an $8\\times 12$ rectangle. Find the perimeter of this polygon.\n Let's think step... | math | {
"style": "rule",
"ground_truth": "52"
} | {
"split": "train",
"index": 19048,
"solution": ""
} |
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