data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Consider a string of $n$ $7$'s, $7777\\cdots77,$ into which $+$ signs are inserted to produce an arithmetic expression. For example, $7+77+777+7+7=875$ could be obtained from eight $7$'s in this way. For how many values of $n$ is it possible to insert $+$ signs so that the resu... | math | {
"style": "rule",
"ground_truth": "108"
} | {
"index": 6900,
"solution": "To simplify, replace all the $7$’s with $1$’s. Because the sum is congruent to $n \\pmod 9$ and \\[1000 \\equiv 1 \\pmod 9 \\implies n \\equiv 1 \\pmod 9\\] Also, $n \\leq 1000$. There are \\[\\left \\lfloor \\frac{1000}{9} \\right \\rfloor + 1 = 112 \\textrm{ positive integers that sa... |
DeepScaleR | [
{
"role": "user",
"content": "Consider a string of $n$ $7$s, $7777\\cdots77,$ into which $+$ signs are inserted to produce an arithmetic expression. How many values of $n$ are possible if the inserted $+$ signs create a sum of $7350$ using groups of $7$s, $77$s, $777$s, and possibly $7777$s?"
}
] | math | {
"style": "rule",
"ground_truth": "117"
} | {
"index": 6901,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a string of $n$ $8$'s, $8888\\cdots88$, into which $+$ signs are inserted to produce an arithmetic expression. For how many values of $n$ is it possible to insert $+$ signs so that the resulting expression has value $8000$?"
}
] | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"index": 6902,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a terminal with fifteen gates arranged in a straight line with exactly $90$ feet between adjacent gates. A passenger's departure gate is assigned at random. Later, the gate is changed to another randomly chosen gate. Calculate the probability that the passenger walks $... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"index": 6903,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a three-person game involving the following three types of fair six-sided dice. - Dice of type $A$ have faces labelled $2,2,4,4,9,9$. - Dice of type $B$ have faces labelled $1,1,6,6,8,8$. - Dice of type $C$ have faces labelled $3,3,5,5,7,7$. All three players simultane... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{9}"
} | {
"index": 6904,
"solution": "Short version: third player doesn't matter; against 1 opponent, by symmetry, you'd both play the same strategy. Type A beats B, B beats C, and C beats A all with probability $5 / 9$. It can be determined that choosing each die with probability $1 / 3$ is the best strategy. Then, whatev... |
DeepScaleR | [
{
"role": "user",
"content": "Consider a trapezoid field with base lengths of 120 meters and 180 meters and non-parallel sides each measuring 130 meters. The angles adjacent to the longer base are $60^\\circ$. At harvest, the crops at any point in the field are brought to the nearest point on the field's pe... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 6905,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a trapezoidal field where it's planted uniformly with wheat. The trapezoid has the following measurements: side $AB$ is 150 m, base $AD$ (the longest side) is 300 m, and the other base $BC$ is 150 m. The angle at $A$ is $75^\\circ$, and the angle at $B$ is $105^\\circ$... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 6906,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a triangle $DEF$ where the angles of the triangle satisfy\n\\[ \\cos 3D + \\cos 3E + \\cos 3F = 1. \\]\nTwo sides of this triangle have lengths 12 and 14. Find the maximum possible length of the third side."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{127}"
} | {
"index": 6907,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a triangle with vertices at $(2, 1)$, $(4, 7)$, and $(7, 3)$. This triangle is reflected about the line $y=4$. Find the area of the union of the original and the reflected triangles."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 6908,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a triangle with vertices at points \\( (0, 0), (30, 0), \\) and \\( (18, 26) \\). The vertices of its midpoint triangle are the midpoints of its sides. A triangular pyramid is formed by folding the original triangle along the sides of its midpoint triangle. What is the... | math | {
"style": "rule",
"ground_truth": "3380"
} | {
"index": 6909,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a two-digit number in base 12 represented by $AB_{12}$, where $A$ and $B$ are duodecimal digits (0 to 11), and $A \\neq B$. When this number is reversed to $BA_{12}$, under what condition is a particular prime number a necessary factor of the difference $AB_{12} - BA_{... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 6910,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a unit cube in a coordinate system with vertices $A(0,0,0)$, $A'(1,1,1)$, and other vertices placed accordingly. A regular octahedron has vertices placed at fractions $\\frac{1}{3}$ and $\\frac{2}{3}$ along the segments connecting $A$ with $A'$'s adjacent vertices and ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{3}"
} | {
"index": 6911,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a unit square $ABCD$ whose bottom left vertex is at the origin. A circle $\\omega$ with radius $\\frac{1}{3}$ is inscribed such that it touches the square's bottom side at point $M$. If $\\overline{AM}$ intersects $\\omega$ at a point $P$ different from $M$, where $A$ ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 6912,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a unit square $WXYZ$ with midpoints $M_1$, $M_2$, $M_3$, and $M_4$ on sides $WZ$, $XY$, $YZ$, and $XW$ respectively. Let $R_1$ be a point on side $WZ$ such that $WR_1 = \\frac{1}{4}$. A light ray starts from $R_1$ and reflects off at point $S_1$ (which is the intersect... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{24}"
} | {
"index": 6913,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a wardrobe that consists of $6$ red shirts, $7$ green shirts, $8$ blue shirts, $9$ pairs of pants, $10$ green hats, $10$ red hats, and $10$ blue hats. Additionally, you have $5$ ties in each color: green, red, and blue. Every item is distinct. How many outfits can you ... | math | {
"style": "rule",
"ground_truth": "18900"
} | {
"index": 6914,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider all 1000-element subsets of the set $\\{1, 2, 3, ... , 2015\\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$.\nHint\n Use th... | math | {
"style": "rule",
"ground_truth": "431"
} | {
"index": 6915,
"solution": "Let $p$ be the size of the large set and $q$ be the size of the subset (i.e. in this problem, $p = 2015$ and $q = 1000$). We can easily find the answers for smaller values of $p$ and $q$:\n For $p = 2$ and $q = 2$, the answer is $1$.\n For $p = 3$ and $q = 2$, the answer is $\\frac43$.... |
DeepScaleR | [
{
"role": "user",
"content": "Consider all 1000-element subsets of the set $\\{1, 2, 3, \\dots , 2015\\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$."
}
] | math | {
"style": "rule",
"ground_truth": "2016"
} | {
"index": 6916,
"solution": "\nTo solve this problem, we need to find the arithmetic mean of the least elements of all 1000-element subsets of the set $\\{1, 2, 3, \\ldots, 2015\\}$. We will use combinatorial arguments to derive the solution.\n\n#### Step 1: Counting subsets with a fixed least element\nLet $i$ be ... |
DeepScaleR | [
{
"role": "user",
"content": "Consider all 4-digit palindromes that can be written as $\\overline{abba}$, where $a$ is non-zero and $b$ ranges from 1 to 9. Calculate the sum of the digits of the sum of all such palindromes."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 6917,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider all angles measured in degrees. Calculate the product $\\prod_{k=1}^{22} \\csc^2(4k-3)^\\circ = m^n$, where $m$ and $n$ are integers greater than 1. Find the value of $m+n$."
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 6918,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider all four-digit numbers (including leading zeros) from $0000$ to $9999$. A number is considered balanced if the sum of its two leftmost digits equals the sum of its two rightmost digits. Calculate the total number of such balanced four-digit numbers."
}
] | math | {
"style": "rule",
"ground_truth": "670"
} | {
"index": 6919,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider all functions $f: \\mathbb{Z} \\rightarrow \\mathbb{Z}$ satisfying $$f(f(x)+2 x+20)=15$$ Call an integer $n$ good if $f(n)$ can take any integer value. In other words, if we fix $n$, for any integer $m$, there exists a function $f$ such that $f(n)=m$. Find the sum of a... | math | {
"style": "rule",
"ground_truth": "-35"
} | {
"index": 6920,
"solution": "For almost all integers $x, f(x) \\neq-x-20$. If $f(x)=-x-20$, then $$f(-x-20+2 x+20)=15 \\Longrightarrow-x-20=15 \\Longrightarrow x=-35$$ Now it suffices to prove that the $f(-35)$ can take any value. $f(-35)=15$ in the function $f(x) \\equiv 15$. Otherwise, set $f(-35)=c$, and $f(x)=... |
DeepScaleR | [
{
"role": "user",
"content": "Consider all non-empty subsets of the set \\( S = \\{1, 2, \\cdots, 10\\} \\). A subset is called a \"good subset\" if the number of even numbers in the subset is not less than the number of odd numbers. How many \"good subsets\" are there?"
}
] | math | {
"style": "rule",
"ground_truth": "637"
} | {
"index": 6921,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider all ordered pairs $(m, n)$ of positive integers satisfying $59 m - 68 n = mn$ . Find the sum of all the possible values of $n$ in these ordered pairs."
}
] | math | {
"style": "rule",
"ground_truth": "237"
} | {
"index": 6922,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider all ordered pairs of integers $(a, b)$ such that $1 \\leq a \\leq b \\leq 100$ and $$\\frac{(a+b)(a+b+1)}{a b}$$ is an integer. Among these pairs, find the one with largest value of $b$. If multiple pairs have this maximal value of $b$, choose the one with largest $a$.... | math | {
"style": "rule",
"ground_truth": "(35,90)"
} | {
"index": 6923,
"solution": "Firstly note that $\\frac{(a+b)(a+b+1)}{a b}=2+\\frac{a^{2}+b^{2}+a+b}{a b}$. Let $c$ be this fraction so that $(a+b)(a+b+1)=a b(c+2)$ for some integers $a, b, c$. Suppose $(a, b)$ with $a \\geq b$ is a solution for some $c$. Consider the quadratic $$x^{2}-(b c-1) x+b^{2}+b=0$$ It has ... |
DeepScaleR | [
{
"role": "user",
"content": "Consider all permutations of the numbers $1, 2, \\cdots, 8$ as eight-digit numbers. How many of these eight-digit numbers are multiples of 11?"
}
] | math | {
"style": "rule",
"ground_truth": "4608"
} | {
"index": 6924,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider all polynomials of the form\n\\[x^7 + b_6 x^6 + b_5 x^5 + \\dots + b_2 x^2 + b_1 x + b_0,\\]\nwhere \\( b_i \\in \\{0,1\\} \\) for all \\( 0 \\le i \\le 6 \\). Find the number of such polynomials that have exactly two different integer roots, -1 and 0."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 6925,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider all polynomials of the form\n\\[x^9 + a_8 x^8 + a_7 x^7 + \\dots + a_2 x^2 + a_1 x + a_0,\\]where $a_i \\in \\{0,1\\}$ for all $0 \\le i \\le 8.$ Find the number of such polynomials that have exactly two different integer roots."
}
] | math | {
"style": "rule",
"ground_truth": "56"
} | {
"index": 6926,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider all possible broken lines that travel along the sides of the cells and connect two opposite corners of a square sheet of grid paper with dimensions $100 \\times 100$ by the shortest path. What is the minimum number of such broken lines that need to be taken so that the... | math | {
"style": "rule",
"ground_truth": "101"
} | {
"index": 6927,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider all possible quadratic polynomials $x^2 + px + q$ with a positive discriminant, where the coefficients $p$ and $q$ are integers divisible by 5. Find the largest natural number $n$ such that for any polynomial with the described properties, the sum of the hundredth powe... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 6928,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider all prisms whose base is a convex 2015-gon.\n\nWhat is the maximum number of edges of such a prism that can be intersected by a plane not passing through its vertices?"
}
] | math | {
"style": "rule",
"ground_truth": "2017"
} | {
"index": 6929,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider all quadrilaterals $ABCD$ such that $AB=14$, $BC=9$, $CD=7$, and $DA=12$. What is the radius of the largest possible circle that fits inside or on the boundary of such a quadrilateral?\n$\\textbf{(A)}\\ \\sqrt{15} \\qquad \\textbf{(B)}\\ \\sqrt{21} \\qquad \\textbf{(C)... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{6}"
} | {
"index": 6930,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider all questions on this year's contest that ask for a single real-valued answer (excluding this one). Let \\(M\\) be the median of these answers. Estimate \\(M\\)."
}
] | math | {
"style": "rule",
"ground_truth": "18.5285921"
} | {
"index": 6931,
"solution": "Looking back to the answers of previous problems in the round (or other rounds) can give you to a rough estimate."
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider all the positive integers $N$ with the property that all of the divisors of $N$ can be written as $p-2$ for some prime number $p$ . Then, there exists an integer $m$ such that $m$ is the maximum possible number of divisors of all\nnumbers $N$ with such pro... | math | {
"style": "rule",
"ground_truth": "135"
} | {
"index": 6932,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider all the subsets of $\\{1,2,3, \\ldots, 2018,2019\\}$ having exactly 100 elements. For each subset, take the greatest element. Find the average of all these greatest elements."
}
] | math | {
"style": "rule",
"ground_truth": "2000"
} | {
"index": 6933,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider all triangles $ABC$ satisfying in the following conditions: $AB = AC$, $D$ is a point on $AC$ for which $BD \\perp AC$, $AC$ and $CD$ are integers, and $BD^{2} = 57$. Among all such triangles, the smallest possible value of $AC$ is"
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 6934,
"solution": "Given that $ABC$ is an isosceles triangle with $AB = AC$, and $D$ is a point on $\\overline{AC}$ such that $\\overline{BD} \\perp \\overline{AC}$. We know $BD^2 = 57$, and $AC$ and $CD$ are integers. We need to find the smallest possible value of $AC$.\n\n1. **Using the Pythagorean The... |
DeepScaleR | [
{
"role": "user",
"content": "Consider an $8 \\times 8$ grid of squares. A rook is placed in the lower left corner, and every minute it moves to a square in the same row or column with equal probability (the rook must move; i.e. it cannot stay in the same square). What is the expected number of minutes unti... | math | {
"style": "rule",
"ground_truth": "70"
} | {
"index": 6935,
"solution": "Let the expected number of minutes it will take the rook to reach the upper right corner from the top or right edges be $E_{e}$, and let the expected number of minutes it will take the rook to reach the upper right corner from any other square be $E_{c}$. Note that this is justified be... |
DeepScaleR | [
{
"role": "user",
"content": "Consider an $m$-by-$n$ grid of unit squares, indexed by $(i,j)$ with $1 \\leq i \\leq m$ and $1 \\leq j \\leq n$. There are $(m-1)(n-1)$ coins, which are initially placed in the squares $(i,j)$ with $1 \\leq i \\leq m-1$ and $1 \\leq j \\leq n-1$. If a coin occupies the square ... | math | {
"style": "rule",
"ground_truth": "\\binom{m+n-2}{m-1}"
} | {
"index": 6936,
"solution": "The number of such configurations is $\\binom{m+n-2}{m-1}$. Initially the unoccupied squares form a path from $(1,n)$ to $(m,1)$ consisting of $m-1$ horizontal steps and $n-1$ vertical steps, and every move preserves this property. This yields an injective map from the set of reachable... |
DeepScaleR | [
{
"role": "user",
"content": "Consider an $n$ -by- $n$ board of unit squares for some odd positive integer $n$ . We say that a collection $C$ of identical dominoes is a maximal grid-aligned configuration on the board if $C$ consists of $(n^2-1)/2$ dominoes where each domino covers exactly two neighboring sq... | math | {
"style": "rule",
"ground_truth": "\\[\n\\left(\\frac{n+1}{2}\\right)^2\n\\]"
} | {
"index": 6937,
"solution": "We claim the answer is $(\\frac{n+1}{2})^2$ .\nFirst, consider a checkerboard tiling of the board with 4 colors: R, G, B, Y. Number each column from $1$ to $n$ from left to right and each row from $1$ to $n$ from top to bottom. We color a tile R if its row and column are odd, a tile G ... |
DeepScaleR | [
{
"role": "user",
"content": "Consider an alphabet of 2 letters. A word is any finite combination of letters. We will call a word unpronounceable if it contains more than two identical letters in a row. How many unpronounceable words of 7 letters are there?"
}
] | math | {
"style": "rule",
"ground_truth": "86"
} | {
"index": 6938,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider an alphabet of 2 letters. A word is any finite combination of letters. We will call a word unpronounceable if it contains more than two identical letters in a row. How many unpronounceable words of 7 letters exist? Points for the problem: 8."
}
] | math | {
"style": "rule",
"ground_truth": "86"
} | {
"index": 6939,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider an arithmetic sequence $\\{a\\_n\\}$ with the sum of its first $n$ terms denoted as $S\\_n$. When $k \\geqslant 2$, if $S\\_{k-1}=8$, $S\\_k=0$, and $S\\_{k+1}=-10$, what is the maximum value of $S\\_n$?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 6940,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider an arithmetic sequence $\\{a_n\\}$ with the sum of its first $n$ terms denoted as $S_n$. Given that $a_1=9$, $a_2$ is an integer, and $S_n \\leq S_5$, find the sum of the first 9 terms of the sequence $\\{\\frac{1}{a_n a_{n+1}}\\}$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{9}"
} | {
"index": 6941,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider an arithmetic sequence where the first four terms are $x+2y$, $x-2y$, $2xy$, and $x/y$. Determine the fifth term of the sequence."
}
] | math | {
"style": "rule",
"ground_truth": "-27.7"
} | {
"index": 6942,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider an arithmetic sequence {a\\_n} with a non-zero common difference. Given that a\\_3 = 7 and a\\_1 - 1, a\\_2 - 1, a\\_4 - 1 form a geometric sequence, find the value of a\\_10."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 6943,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider an equilateral triangle $ABC$ with side length $3$. Right triangle $CBD$ is constructed outwardly on side $BC$ of triangle $ABC$ such that $CB = BD$ and $BCD$ is a right angle at $B$. Find $\\sin^2\\left(\\angle CAD\\right)$.\nA) $\\frac{3}{4}$ \nB) $\\frac{1}{4}$ \n... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 6944,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider an equilateral triangle $T$ of side length 12. Matthew cuts $T$ into $N$ smaller equilateral triangles, each of which has side length 1,3, or 8. Compute the minimum possible value of $N$."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 6945,
"solution": "Matthew can cut $T$ into 16 equilateral triangles with side length 3. If he instead included a triangle of side 8, then let him include $a$ triangles of side length 3. He must include $12^{2}-8^{2}-3^{2} a=80-9 a$ triangles of side length 1. Thus $a \\leq 8$, giving that he includes at... |
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."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3+\\sqrt{3}}{2}"
} | {
"index": 6946,
"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 $1-\\frac{\\sqrt{2}}{2}$. Adding their areas, we ... |
DeepScaleR | [
{
"role": "user",
"content": "Consider an equilateral triangle with a side length of 10 cm, where one side of the triangle is also the diameter of a semicircle. Find the sum of the areas of two small shaded regions formed inside the semicircle but outside the triangle. Express your answer in the form \\(a\\... | math | {
"style": "rule",
"ground_truth": "78"
} | {
"index": 6947,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider an equilateral triangular grid $G$ with 20 points on a side, where each row consists of points spaced 1 unit apart. More specifically, there is a single point in the first row, two points in the second row, ..., and 20 points in the last row, for a total of 210 points.... | math | {
"style": "rule",
"ground_truth": "52 \\sqrt{3}"
} | {
"index": 6948,
"solution": "Imagine deforming the triangle lattice such that now it looks like a lattice of 45-45-90 right triangles with legs of length 1. Note that by doing this, the area has multiplied by $\\frac{2}{\\sqrt{3}}$, so we need to readjust out answer on the isosceles triangle lattice by a factor of... |
DeepScaleR | [
{
"role": "user",
"content": "Consider an infinite grid of equilateral triangles. Each edge (that is, each side of a small triangle) is colored one of $N$ colors. The coloring is done in such a way that any path between any two nonadjacent vertices consists of edges with at least two different colors. What ... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 6949,
"solution": "Note that the condition is equivalent to having no edges of the same color sharing a vertex by just considering paths of length two. Consider a hexagon made out of six triangles. Six edges meet at the center, so $N \\geq 6$. To prove $N=6$, simply use two colors for each of the three p... |
DeepScaleR | [
{
"role": "user",
"content": "Consider an infinite grid of unit squares. An $n$-omino is a subset of $n$ squares that is connected. Below are depicted examples of 8 -ominoes. Two $n$-ominoes are considered equivalent if one can be obtained from the other by translations and rotations. What is the number of ... | math | {
"style": "rule",
"ground_truth": "3426576"
} | {
"index": 6950,
"solution": "We claim that there are approximately $\\frac{3^{n-1}}{4} n$-ominoes. First, we define an order on the squares in an $n$-omino, as follows: we order the squares from left to right, and within a column, we order the squares from top to bottom. We construct an $n$-omino by starting with ... |
DeepScaleR | [
{
"role": "user",
"content": "Consider an isosceles right triangle with leg lengths of 1 each. Inscribed in this triangle is a square in such a way that one vertex of the square coincides with the right-angle vertex of the triangle. Another square with side length $y$ is inscribed in an identical isosceles ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 6951,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider an isosceles triangle $T$ with base 10 and height 12. Define a sequence $\\omega_{1}, \\omega_{2}, \\ldots$ of circles such that $\\omega_{1}$ is the incircle of $T$ and $\\omega_{i+1}$ is tangent to $\\omega_{i}$ and both legs of the isosceles triangle for $i>1$. Find... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{9}"
} | {
"index": 6952,
"solution": "The ratio of the radius of $\\omega_{i+1}$ to the radius of $\\omega_{i}$ is $\\frac{4}{9}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider an isosceles triangle $T$ with base 10 and height 12. Define a sequence $\\omega_{1}, \\omega_{2}, \\ldots$ of circles such that $\\omega_{1}$ is the incircle of $T$ and $\\omega_{i+1}$ is tangent to $\\omega_{i}$ and both legs of the isosceles triangle for $i>1$. Find... | math | {
"style": "rule",
"ground_truth": "\\frac{180 \\pi}{13}"
} | {
"index": 6953,
"solution": "Using the notation from the previous solution, the area contained in the $i$th circle is equal to $\\pi r_{i}^{2}$. Since the radii form a geometric sequence, the areas do as well. Specifically, the areas form a sequence with initial term $\\pi \\cdot \\frac{100}{9}$ and common ratio $... |
DeepScaleR | [
{
"role": "user",
"content": "Consider an octagonal lattice where each vertex is evenly spaced and one unit from its nearest neighbor. How many equilateral triangles have all three vertices in this lattice? Every side of the octagon is extended one unit outward with a single point placed at each extension, ... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 6954,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider an ordinary $6$-sided die, with numbers from $1$ to $6$. How many ways can I paint three faces of a die so that the sum of the numbers on the painted faces is not exactly $9$?"
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 6955,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider an unusual biased coin, with probability $p$ of landing heads, probability $q \\leq p$ of landing tails, and probability \\frac{1}{6}$ of landing on its side (i.e. on neither face). It is known that if this coin is flipped twice, the likelihood that both flips will hav... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 6956,
"solution": "The probability that both flips are the same is $p^{2}+q^{2}+\\frac{1}{36}$. For this to be \\frac{1}{2}$, we must have $$p^{2}+q^{2}+\\frac{1}{36}=p^{2}+\\left(\\frac{5}{6}-p\\right)^{2}+\\frac{1}{36}=\\frac{1}{2}$$ Using the quadratic formula, $p=\\frac{2}{3}$ or \\frac{1}{6}$. Since... |
DeepScaleR | [
{
"role": "user",
"content": "Consider arrangements of the $9$ numbers $1, 2, 3, \\dots, 9$ in a $3 \\times 3$ array. For each such arrangement, let $a_1$, $a_2$, and $a_3$ be the medians of the numbers in rows $1$, $2$, and $3$ respectively, and let $m$ be the median of $\\{a_1, a_2, a_3\\}$. Let $Q$ be th... | math | {
"style": "rule",
"ground_truth": "360"
} | {
"index": 6957,
"solution": "We see that if one of the medians is 5, then there are two remaining numbers greater than 5 and two less than 5, so it follows that $m=5$. There are 3 ways to choose which row to have 5 in, $4\\cdot 4=16$ ways to choose the other two numbers in that row, $3!=6$ ways to arrange the numb... |
DeepScaleR | [
{
"role": "user",
"content": "Consider equations of the form $x^2 + bx + c = 0$. How many such equations have real roots and have coefficients $b$ and $c$ selected from the set of integers $\\{1,2,3, 4, 5,6\\}$?"
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 6958,
"solution": "To determine how many quadratic equations of the form $x^2 + bx + c = 0$ have real roots with $b$ and $c$ chosen from the set $\\{1, 2, 3, 4, 5, 6\\}$, we need to ensure that the discriminant of the quadratic equation is non-negative. The discriminant $\\Delta$ for a quadratic equation... |
DeepScaleR | [
{
"role": "user",
"content": "Consider equilateral triangle $ABC$ with side length $1$ . Suppose that a point $P$ in the plane of the triangle satisfies \\[2AP=3BP=3CP=\\kappa\\] for some constant $\\kappa$ . Compute the sum of all possible values of $\\kappa$ .\n\n*2018 CCA Math Bonanza Lightning Ro... | math | {
"style": "rule",
"ground_truth": "\\frac{18\\sqrt{3}}{5}"
} | {
"index": 6959,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider finding the result when we compute the series $$1^3 + 2^3 + 3^3 + \\dots + 49^3 + 50^3$$ and the series $$(-1)^3 + (-2)^3 + (-3)^3 + \\dots + (-49)^3 + (-50)^3,$$ then subtract the second series' result from the first series' result. What is the sum?"
}
] | math | {
"style": "rule",
"ground_truth": "3251250"
} | {
"index": 6960,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider five-dimensional Cartesian space $\\mathbb{R}^{5}=\\left\\{\\left(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\\right) \\mid x_{i} \\in \\mathbb{R}\\right\\}$ and consider the hyperplanes with the following equations: - $x_{i}=x_{j}$ for every $1 \\leq i<j \\leq 5$; - $x_{1}+x_{2... | math | {
"style": "rule",
"ground_truth": "480"
} | {
"index": 6961,
"solution": "Note that given a set of plane equations $P_{i}\\left(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\\right)=0$, for $i=1,2, \\ldots, n$, each region that the planes separate the space into correspond to a $n$-tuple of -1 and 1 , representing the sign of $P_{1}, P_{2}, \\ldots P_{n}$ for all points... |
DeepScaleR | [
{
"role": "user",
"content": "Consider four functions, labelled from (6) to (9). The domain of function (7) is $\\{-6, -5, -4, -3, -2, -1, 0, 1\\}$. Determine the product of the labels of the functions that are invertible. The graphs are as follows: \n- Function (6): $y = x^3 - 3x$\n- Function (7): A set of... | math | {
"style": "rule",
"ground_truth": "504"
} | {
"index": 6962,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider integers \\( \\{1, 2, \\ldots, 10\\} \\). A particle is initially at 1. It moves to an adjacent integer in the next step. What is the expected number of steps it will take to reach 10 for the first time?"
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 6963,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider integers c and d where c consists of 1986 nines, and d consists of 1986 sixes. What is the sum of the digits of the resulting number in base 10 when these numbers are added?"
}
] | math | {
"style": "rule",
"ground_truth": "9931"
} | {
"index": 6964,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider license plates consisting of a sequence of four digits followed by two letters. Assume each arrangement is equally likely for these plates. What is the probability that such a license plate contains at least one palindrome sequence (either the four-digit sequence or th... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{104}"
} | {
"index": 6965,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider numbers of the form $1a1$ , where $a$ is a digit. How many pairs of such numbers are there such that their sum is also a palindrome?\n\n*Note: A palindrome is a number which reads the same from left to right and from right to left. Examples: $353$ , $91719$ .*"
... | math | {
"style": "rule",
"ground_truth": "55"
} | {
"index": 6966,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider numbers of the form \\(10n + 1\\), where \\(n\\) is a positive integer. We shall call such a number 'grime' if it cannot be expressed as the product of two smaller numbers, possibly equal, both of which are of the form \\(10k + 1\\), where \\(k\\) is a positive integer... | math | {
"style": "rule",
"ground_truth": "87"
} | {
"index": 6967,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider one positive even integer and one positive odd integer less than $16$, where the even integer is a multiple of $3$. Compute how many different possible values result when the sum of these two numbers is added to their product."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 6968,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider pairs $(f,g)$ of functions from the set of nonnegative integers to itself such that \n[list]\n[*]$f(0) \\geq f(1) \\geq f(2) \\geq \\dots \\geq f(300) \\geq 0$\n[*]$f(0)+f(1)+f(2)+\\dots+f(300) \\leq 300$\n[*]for any 20 nonnegative integers $n_1, n_2, \\dots, n_{20}$, ... | math | {
"style": "rule",
"ground_truth": "115440"
} | {
"index": 6969,
"solution": "\nConsider pairs \\((f, g)\\) of functions from the set of nonnegative integers to itself such that:\n- \\(f(0) \\geq f(1) \\geq f(2) \\geq \\dots \\geq f(300) \\geq 0\\),\n- \\(f(0) + f(1) + f(2) + \\dots + f(300) \\leq 300\\),\n- for any 20 nonnegative integers \\(n_1, n_2, \\dots, n... |
DeepScaleR | [
{
"role": "user",
"content": "Consider parallelogram $A B C D$ with $A B>B C$. Point $E$ on $\\overline{A B}$ and point $F$ on $\\overline{C D}$ are marked such that there exists a circle $\\omega_{1}$ passing through $A, D, E, F$ and a circle $\\omega_{2}$ passing through $B, C, E, F$. If $\\omega_{1}, \\o... | math | {
"style": "rule",
"ground_truth": "51"
} | {
"index": 6970,
"solution": "We want to find $A D=B C=E F$. So, let $E F$ intersect $B D$ at $O$. It is clear that $\\triangle B O E \\sim \\triangle D O F$. However, we can show by angle chase that $\\triangle B X E \\sim \\triangle D Y F$ : $$\\angle B E G=\\angle A D G=\\angle C B H=\\angle D F H$$ This means t... |
DeepScaleR | [
{
"role": "user",
"content": "Consider polynomials $P(x)$ of degree at most $3$, each of whose coefficients is an element of $\\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\\}$. How many such polynomials satisfy $P(-1) = -9$?\n$\\textbf{(A) } 110 \\qquad \\textbf{(B) } 143 \\qquad \\textbf{(C) } 165 \\qquad \\textbf{(D) }... | math | {
"style": "rule",
"ground_truth": "220"
} | {
"index": 6971,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider positive integers $a \\leq b \\leq c \\leq d \\leq e$. There are $N$ lists $a, b, c, d, e$ with a mean of 2023 and a median of 2023, in which the integer 2023 appears more than once, and in which no other integer appears more than once. What is the sum of the digits of... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 6972,
"solution": "Since the median of the list $a, b, c, d, e$ is 2023 and $a \\leq b \\leq c \\leq d \\leq e$, then $c=2023$. Since 2023 appears more than once in the list, then it appears 5,4,3, or 2 times. Case 1: 2023 appears 5 times Here, the list is 2023, 2023, 2023, 2023, 2023. There is 1 such li... |
DeepScaleR | [
{
"role": "user",
"content": "Consider positive integers $n$ where $D(n)$ denotes the number of pairs of different adjacent digits in the binary (base two) representation of $n$. Determine the number of positive integers less than or equal to $50$ for which $D(n) = 3$."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 6973,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider real numbers $A$ , $B$ , \\dots, $Z$ such that \\[\n EVIL = \\frac{5}{31}, \\;\n LOVE = \\frac{6}{29}, \\text{ and }\n IMO = \\frac{7}{3}.\n\\] If $OMO = \\tfrac mn$ for relatively prime positive integers $m$ and $n$ , find the value of $m+n$ .\n\n*Propos... | math | {
"style": "rule",
"ground_truth": "579"
} | {
"index": 6974,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider sequences \\(a\\) of the form \\(a=\\left(a_{1}, a_{2}, \\ldots, a_{20}\\right)\\) such that each term \\(a_{i}\\) is either 0 or 1. For each such sequence \\(a\\), we can produce a sequence \\(b=\\left(b_{1}, b_{2}, \\ldots, b_{20}\\right)\\), where \\(b_{i}= \\begin{... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"index": 6975,
"solution": "Let the two sequences be \\(b\\) and \\(\\hat{b}\\). Then, observe that given \\(a\\), if \\(b_{1}=\\hat{b}_{1}\\) and \\(b_{2}=\\hat{b}_{2}\\), then \\(b=\\hat{b}\\) (since \\(a\\) will uniquely determine the remaining elements in \\(b\\) and \\(\\hat{b}\\)). Thus, \\(b\\) and \\(\\ha... |
DeepScaleR | [
{
"role": "user",
"content": "Consider sequences of positive real numbers of the form $x, 2000, y, \\dots$ in which every term after the first is 1 less than the product of its two immediate neighbors. For how many different values of $x$ does the term 2001 appear somewhere in the sequence?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 6976,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider sequences that consist entirely of $A$'s and $B$'s and that have the property that every run of consecutive $A$'s has even length, and every run of consecutive $B$'s has odd length. Examples of such sequences are $AA$, $B$, and $AABAA$, while $BBAB$ is not such a seque... | math | {
"style": "rule",
"ground_truth": "172"
} | {
"index": 6977,
"solution": "Let $a_n$ and $b_n$ denote, respectively, the number of sequences of length $n$ ending in $A$ and $B$. If a sequence ends in an $A$, then it must have been formed by appending two $A$s to the end of a string of length $n-2$. If a sequence ends in a $B,$ it must have either been formed ... |
DeepScaleR | [
{
"role": "user",
"content": "Consider that Henry's little brother now has 10 identical stickers and 5 identical sheets of paper. How many ways can he distribute all the stickers on the sheets of paper, if only the number of stickers on each sheet matters and no sheet can remain empty?"
}
] | math | {
"style": "rule",
"ground_truth": "126"
} | {
"index": 6978,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider that for integers from 1 to 1500, $x_1+2=x_2+4=x_3+6=\\cdots=x_{1500}+3000=\\sum_{n=1}^{1500}x_n + 3001$. Find the value of $\\left\\lfloor|S|\\right\\rfloor$, where $S=\\sum_{n=1}^{1500}x_n$."
}
] | math | {
"style": "rule",
"ground_truth": "1500"
} | {
"index": 6979,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the $12$-sided polygon $ABCDEFGHIJKL$, as shown. Each of its sides has length $4$, and each two consecutive sides form a right angle. Suppose that $\\overline{AG}$ and $\\overline{CH}$ meet at $M$. What is the area of quadrilateral $ABCM$?"
}
] | math | {
"style": "rule",
"ground_truth": "88/5"
} | {
"index": 6980,
"solution": "\n1. **Identify the Key Points and Setup:**\n - We are given a 12-sided polygon with each side of length 4 and each angle being a right angle.\n - We need to find the area of quadrilateral $ABCM$ where lines $\\overline{AG}$ and $\\overline{CH}$ intersect at point $M$.\n\n2. **Calc... |
DeepScaleR | [
{
"role": "user",
"content": "Consider the 100th, 101st, and 102nd rows of Pascal's triangle, denoted as sequences $(p_i)$, $(q_i)$, and $(r_i)$ respectively. Calculate:\n\\[\n\\sum_{i = 0}^{100} \\frac{q_i}{r_i} - \\sum_{i = 0}^{99} \\frac{p_i}{q_i}.\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 6981,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the 800-digit integer\n$$\n234523452345 \\cdots 2345 .\n$$\n\nThe first \\( m \\) digits and the last \\( n \\) digits of the above integer are crossed out so that the sum of the remaining digits is 2345. Find the value of \\( m+n \\)."
}
] | math | {
"style": "rule",
"ground_truth": "130"
} | {
"index": 6982,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the L-shaped region formed by three unit squares joined at their sides, as shown below. Two points $A$ and $B$ are chosen independently and uniformly at random from inside the region. The probability that the midpoint of $\\overline{AB}$ also lies inside this L-shaped ... | math | {
"style": "rule",
"ground_truth": "035"
} | {
"index": 6983,
"solution": "Consider this diagram:\n First, the one of points must be in the uppermost box and the other in the rightmost box. This happens with probability 2/3*1/3=2/9.\n We need the midpoints of the $x$ coordinates to be greater than $1$ but less than $2.$\n We need the midpoints of the $y$ coor... |
DeepScaleR | [
{
"role": "user",
"content": "Consider the addition problem: \\begin{tabular}{ccccc} & C & A & S & H \\\\ + & & & M & E \\\\ \\hline O & S & I & D & E \\end{tabular} where each letter represents a base-ten digit, and $C, M, O \\neq 0$. (Distinct letters are allowed to represent the same digit) How many ways... | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 6984,
"solution": "Clearly, $C A S H$ and $M E$ cannot add up to 11000 or more, so $O=1$ and $S=0$. By examining the units digit, we find that $H=0$. Then $C A S H+M E<9900+99<10000$, so there are no solutions."
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the arithmetic sequence $1$, $4$, $7$, $10$, $13$, $\\ldots$. Find the $15^{\\text{th}}$ term in the sequence."
}
] | math | {
"style": "rule",
"ground_truth": "43"
} | {
"index": 6985,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the arithmetic sequence defined by the set $\\{2, 5, 8, 11, 14, 17, 20\\}$. Determine the total number of different integers that can be expressed as the sum of three distinct members of this set."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 6986,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the cards $A, 2, \\cdots, J, Q, K$ as the numbers $1, 2, \\cdots, 11, 12, 13$. If we take the 13 cards of spades and 13 cards of hearts together and randomly draw 2 cards, what is the probability that the two cards are of the same suit and the product of the two number... | math | {
"style": "rule",
"ground_truth": "2/65"
} | {
"index": 6987,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the case when all numbers are equal. $\\frac{5}{4} n + \\frac{5}{4} = n$. If the first number is -5, then all numbers will be equal to -5. The same applies to all cases where the first number is equal to $-5 + 1024n$, $n \\in \\mathbb{Z}$."
}
] | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"index": 6988,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the cube whose vertices are the eight points $(x, y, z)$ for which each of $x, y$, and $z$ is either 0 or 1 . How many ways are there to color its vertices black or white such that, for any vertex, if all of its neighbors are the same color then it is also that color? ... | math | {
"style": "rule",
"ground_truth": "118"
} | {
"index": 6989,
"solution": "Divide the 8 vertices of the cube into two sets $A$ and $B$ such that each set contains 4 vertices, any two of which are diagonally adjacent across a face of the cube. We do casework based on the number of vertices of each color in set $A$. - Case 1: 4 black. Then all the vertices in $... |
DeepScaleR | [
{
"role": "user",
"content": "Consider the cubes whose vertices lie on the surface of a given cube. Which one is the smallest among them?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{\\sqrt{2}}"
} | {
"index": 6990,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the curve $y=x^{n+1}$ (where $n$ is a positive integer) and its tangent at the point (1,1). Let the x-coordinate of the intersection point between this tangent and the x-axis be $x_n$.\n(Ⅰ) Let $a_n = \\log{x_n}$. Find the value of $a_1 + a_2 + \\ldots + a_9$.\n(Ⅱ) Def... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 6991,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the dark square in an array of unit squares, part of which is shown. The first ring of squares around this center square contains 8 unit squares. The second ring contains 16 unit squares. If we continue this process, then what is the number of unit squares in the $100^... | math | {
"style": "rule",
"ground_truth": "800"
} | {
"index": 6992,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the decimal function denoted by $\\{ x \\} = x - \\lfloor x \\rfloor$ which represents the decimal part of a number $x$. Find the sum of the five smallest positive solutions to the equation $\\{x\\} = \\frac{1}{\\lfloor x \\rfloor}$. Express your answer as a mixed numb... | math | {
"style": "rule",
"ground_truth": "21\\frac{9}{20}"
} | {
"index": 6993,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the eighth-sphere $\\left\\{(x, y, z) \\mid x, y, z \\geq 0, x^{2}+y^{2}+z^{2}=1\\right\\}$. What is the area of its projection onto the plane $x+y+z=1$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi \\sqrt{3}}{4}"
} | {
"index": 6994,
"solution": "Consider the three flat faces of the eighth-ball. Each of these is a quarter-circle of radius 1, so each has area $\\frac{\\pi}{4}$. Furthermore, the projections of these faces cover the desired area without overlap. To find the projection factor one can find the cosine of the angle $\... |
DeepScaleR | [
{
"role": "user",
"content": "Consider the ellipse $C\\_1$: $\\frac{x^{2}}{2}+y^{2}=1$ and the ellipse $C\\_2$: $\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1$ ($a > b > 0$). One focus of ellipse $C\\_2$ has the coordinates $(\\sqrt{5},0)$. The line $l$ with slope $1$ intersects ellipse $C\\_2$ at points $A$ ... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"index": 6995,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the ellipse $\\frac{x^2}{16} + \\frac{y^2}{12} = 1$ whose left and right intersection points are $F_1$ and $F_2$, respectively. Let point $P$ be on the ellipse and satisfy $\\vec{PF_1} \\cdot \\vec{PF_2} = 9$. Find the value of $|\\vec{PF_1}| \\cdot |\\vec{PF_2}|$."
... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 6996,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the ellipse $\\frac{x^{2}}{6} + \\frac{y^{2}}{2} = 1$ and the hyperbola $\\frac{x^{2}}{3} - y^{2} = 1$ with common foci $F_{1}$ and $F_{2}$. Let $P$ be one of the intersection points of the two curves. Find the value of $\\cos \\angle F_{1}PF_{2}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 6997,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the ellipse (C) with the equation \\\\(\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1 (a > b > 0)\\\\) and an eccentricity of \\\\(\\frac{\\sqrt{3}}{3}\\\\). A line passing through the right focus (F2(c,0)) perpendicular to the x-axis intersects the ellipse at points A a... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{2}"
} | {
"index": 6998,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider the ellipse \\[9(x-1)^2 + y^2 = 36.\\]Let $A$ be one of the endpoints of its major axis, and let $B$ be one of the endpoints of its minor axis. Find the distance $AB.$"
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{10}"
} | {
"index": 6999,
"solution": ""
} |
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