data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "A $33$-gon $P_1$ is drawn in the Cartesian plane. The sum of the $x$-coordinates of the $33$ vertices equals $99$. The midpoints of the sides of $P_1$ form a second $33$-gon, $P_2$. Finally, the midpoints of the sides of $P_2$ form a third $33$-gon, $P_3$. Find the sum of t... | math | {
"style": "rule",
"ground_truth": "99"
} | {
"index": 600,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A $37$-gon $Q_1$ is drawn in the Cartesian plane, and the sum of the $x$-coordinates of its $37$ vertices equals $185$. The midpoints of the sides of $Q_1$ form a second $37$-gon, $Q_2$. Then, the midpoints of the sides of $Q_2$ form a third $37$-gon, $Q_3$. Find the sum of the... | math | {
"style": "rule",
"ground_truth": "185"
} | {
"index": 601,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A $4 \\times 4$ square is divided into $16$ unit squares. Each unit square is painted either white or black, each with a probability of $\\frac{1}{2}$, independently. The square is then rotated $180^\\circ$ about its center. After rotation, any white square that occupies a posi... | math | {
"style": "rule",
"ground_truth": "\\frac{6561}{65536}"
} | {
"index": 602,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A $4 \\times 4$ window is made out of 16 square windowpanes. How many ways are there to stain each of the windowpanes, red, pink, or magenta, such that each windowpane is the same color as exactly two of its neighbors?"
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 603,
"solution": "For the purpose of explaining this solution, let's label the squares as 11121314 21222324 31323334 41424344. Note that since the corner squares $11,14,41,44$ each only have two neighbors, each corner square is the same color as both of its neighbors. This corner square constraint heavil... |
DeepScaleR | [
{
"role": "user",
"content": "A $4$-foot by $8$-foot rectangular piece of plywood will be cut into $4$ congruent rectangles with no wood left over and no wood lost due to the cuts. What is the positive difference, in feet, between the greatest possible perimeter of a single piece and the least possible peri... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 604,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A $44$-gon $Q_1$ is constructed in the Cartesian plane, and the sum of the squares of the $x$-coordinates of the vertices equals $176$. The midpoints of the sides of $Q_1$ form another $44$-gon, $Q_2$. Finally, the midpoints of the sides of $Q_2$ form a third $44$-gon, $Q_3$. F... | math | {
"style": "rule",
"ground_truth": "44"
} | {
"index": 605,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A $4\\times 4$ block of calendar dates is shown. First, the order of the numbers in the second and the fourth rows are reversed. Then, the numbers on each diagonal are added. What will be the positive difference between the two diagonal sums?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 606,
"solution": "1. **Identify the original diagonals**: \n - The main diagonal (from top left to bottom right) consists of the numbers: $1, 9, 17, 25$.\n - The secondary diagonal (from top right to bottom left) consists of the numbers: $4, 10, 16, 22$.\n\n2. **Calculate the original diagonal sums**... |
DeepScaleR | [
{
"role": "user",
"content": "A $4\\times 4\\times h$ rectangular box contains a sphere of radius $2$ and eight smaller spheres of radius $1$. The smaller spheres are each tangent to three sides of the box, and the larger sphere is tangent to each of the smaller spheres. What is $h$?"
}
] | math | {
"style": "rule",
"ground_truth": "$2+2\\sqrt 7$"
} | {
"index": 607,
"solution": "1. **Identify the Geometry and Positioning of Spheres:**\n - The large sphere has a radius of $2$ and is centered at point $A$.\n - Each of the eight smaller spheres has a radius of $1$ and is tangent to three sides of the box. Let $C$ be the center of any one of these smaller spher... |
DeepScaleR | [
{
"role": "user",
"content": "A $5 \\times 5$ grid of squares is filled with integers. Call a rectangle corner-odd if its sides are grid lines and the sum of the integers in its four corners is an odd number. What is the maximum possible number of corner-odd rectangles within the grid?"
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 608,
"solution": "Consider any two rows and the five numbers obtained by adding the two numbers which share a given column. Suppose $a$ of these are odd and $b$ of these are even. The number of corner-odd rectangles with their sides contained in these two rows is $a b$. Since $a+b=5$, we have $a b \\leq ... |
DeepScaleR | [
{
"role": "user",
"content": "A $5 \\times 5$ square grid has the number -3 written in the upper-left square and the number 3 written in the lower-right square. In how many ways can the remaining squares be filled in with integers so that any two adjacent numbers differ by 1, where two squares are adjacent ... | math | {
"style": "rule",
"ground_truth": "250"
} | {
"index": 609,
"solution": "250 If the square in row $i$, column $j$ contains the number $k$, let its 'index' be $i+j-k$. The constraint on adjacent squares now says that if a square has index $r$, the squares to its right and below it each have index $r$ or $r+2$. The upper-left square has index 5, and the lower-... |
DeepScaleR | [
{
"role": "user",
"content": "A $5 \\times 5$ table is called regular if each of its cells contains one of four pairwise distinct real numbers, such that each of them occurs exactly once in every $2 \\times 2$ subtable.The sum of all numbers of a regular table is called the total sum of the table. With any ... | math | {
"style": "rule",
"ground_truth": "\\boxed{60}"
} | {
"index": 610,
"solution": " Solution 1 (Official solution)\nWe will prove that the maximum number of total sums is $60$ .\nThe proof is based on the following claim:\nIn a regular table either each row contains exactly two of the numbers or each column contains exactly two of the numbers.\nProof of the Claim:\nLe... |
DeepScaleR | [
{
"role": "user",
"content": "A $5 \\times 8$ rectangle can be rolled to form two different cylinders with different maximum volumes. What is the ratio of the larger volume to the smaller volume? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{5}"
} | {
"index": 611,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A $50$-gon \\(Q_1\\) is drawn in the Cartesian plane where the sum of the \\(x\\)-coordinates of the \\(50\\) vertices equals \\(150\\). A constant scaling factor \\(k = 1.5\\) applies only to the \\(x\\)-coordinates of \\(Q_1\\). The midpoints of the sides of \\(Q_1\\) form a ... | math | {
"style": "rule",
"ground_truth": "225"
} | {
"index": 612,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A $6 \\times 9$ rectangle can be rolled to form two different cylinders. Calculate the ratio of the larger volume to the smaller volume. Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 613,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A $6$-inch and $18$-inch diameter poles are placed together and bound together with wire.\nThe length of the shortest wire that will go around them is:"
}
] | math | {
"style": "rule",
"ground_truth": "12\\sqrt{3}+14\\pi"
} | {
"index": 614,
"solution": "1. **Understanding the Setup**: We have two poles with diameters of $6$ inches and $18$ inches, respectively. This means the radii are $3$ inches and $9$ inches. When bound together, the wire will wrap around the outer edges of both poles.\n\n2. **Visualizing the Geometry**: The poles a... |
DeepScaleR | [
{
"role": "user",
"content": "A $7 \\times 7$ board is either empty or contains an invisible $2 \\times 2$ ship placed \"by the cells.\" You are allowed to place detectors in some cells of the board and then activate them all at once. An activated detector signals if its cell is occupied by the ship. What i... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 615,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A $7 \\times 7$ board is either empty or contains an invisible $2 \\times 2$ ship placed \"on the cells.\" Detectors can be placed on certain cells of the board and then activated simultaneously. An activated detector signals if its cell is occupied by the ship. What is the min... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 616,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A $7\\times 1$ board is completely covered by $m\\times 1$ tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the board. Each tile is either red, blue, or green. Let $N$ be the number of tilings of the $7\\times 1$ boa... | math | {
"style": "rule",
"ground_truth": "106"
} | {
"index": 617,
"solution": "This solution is basically solution 1 with more things done at once. The game plan:\n$\\sum_{i=0}^{7} ($the amount of ways to divide the board into $i$ pieces$) \\cdot ($the amount of ways to color the respective divisions)\n The amount of ways to divide the board is just stars and bars... |
DeepScaleR | [
{
"role": "user",
"content": "A $8 \\times 8 \\times 8$ cube has three of its faces painted red and the other three faces painted blue (ensuring that any three faces sharing a common vertex are not painted the same color), and then it is cut into 512 $1 \\times 1 \\times 1$ smaller cubes. Among these 512 sm... | math | {
"style": "rule",
"ground_truth": "56"
} | {
"index": 618,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A $9 \\times 9 \\times 9$ cube is composed of twenty-seven $3 \\times 3 \\times 3$ cubes. The big cube is ‘tunneled’ as follows: First, the six $3 \\times 3 \\times 3$ cubes which make up the center of each face as well as the center $3 \\times 3 \\times 3$ cube are removed. Se... | math | {
"style": "rule",
"ground_truth": "1056"
} | {
"index": 619,
"solution": "\n#### Step-by-step Analysis:\n\n1. **Understanding the Structure**:\n - The original structure is a $9 \\times 9 \\times 9$ cube, which is composed of twenty-seven $3 \\times 3 \\times 3$ cubes.\n - The center cube of each face and the very center cube of the large cube are removed... |
DeepScaleR | [
{
"role": "user",
"content": "A $9\\times 1$ board is completely covered by $m\\times 1$ tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the board. Each tile is either red, blue, or green. Let $N$ be the number of tilings of the $9\\times 1$ boa... | math | {
"style": "rule",
"ground_truth": "838"
} | {
"index": 620,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A $9\\times 7$ rectangle is tiled with tiles of the two types: L-shaped tiles composed by three unit squares (can be rotated repeatedly with $90^\\circ$) and square tiles composed by four unit squares.\nLet $n\\ge 0$ be the number of the $2 \\times 2 $ tiles which can be used i... | math | {
"style": "rule",
"ground_truth": "0 \\text{ and } 3"
} | {
"index": 621,
"solution": "\nConsider a \\(9 \\times 7\\) rectangle that needs to be tiled using two types of tiles: L-shaped tiles, which cover three unit squares, and \\(2 \\times 2\\) square tiles, which cover four unit squares. We need to find all possible values of \\(n\\), the number of \\(2 \\times 2\\) sq... |
DeepScaleR | [
{
"role": "user",
"content": "A $\\emph{convex}$ polygon is a polygon in which every interior angle is less than 180 degrees. A $\\emph{diagonal}$ of a convex polygon is a line segment that connects two non-adjacent vertices. How many diagonals does a convex polygon with 20 sides have?"
}
] | math | {
"style": "rule",
"ground_truth": "170"
} | {
"index": 622,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A $\\frac 1p$ -array is a structured, infinite, collection of numbers. For example, a $\\frac 13$ -array is constructed as follows:\n\\begin{align*} 1 \\qquad \\frac 13\\,\\ \\qquad \\frac 19\\,\\ \\qquad \\frac 1{27} \\qquad &\\cdots\\\\ \\frac 16 \\qquad \\frac 1{18}\\,\\ \\q... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 623,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A $\\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \\ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\\pm 1$-sequence, there exists an integer $k$ and indices $1 \\le t_1 < \\ldots < t_k \\le 2022$ so that $t_{i+1} ... | math | {
"style": "rule",
"ground_truth": "506"
} | {
"index": 624,
"solution": "\nTo solve the given problem, we first need to understand the requirements for a \\(\\pm 1\\)-sequence. We are looking for the largest integer \\( C \\) such that, for any sequence of numbers \\( a_1, a_2, \\ldots, a_{2022} \\) where each \\( a_i \\) is either \\( +1 \\) or \\( -1 \\), ... |
DeepScaleR | [
{
"role": "user",
"content": "A $\\textit{composite number}$ is a number that has two or more prime factors. The number 87 can be expressed as the sum of two composite numbers in many ways. What is the minimum positive difference between two such numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 625,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A $\\textit{palindrome}$ is a number which reads the same forward as backward, for example 313 or 1001. Ignoring the colon, how many different palindromes are possible on a 12-hour digital clock displaying only the hours and minutes? (Notice a zero may not be inserted before a ... | math | {
"style": "rule",
"ground_truth": "57"
} | {
"index": 626,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A $\\textit{palindrome}$ is a number which reads the same forward as backward. For example, 343 and 1221 are palindromes. What is the least natural number that can be added to 40,305 to create a palindrome?"
}
] | math | {
"style": "rule",
"ground_truth": "99"
} | {
"index": 627,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A $\\textit{palindrome}$ is a positive integer which reads the same forward and backward, like $12321$ or $4884$.\n\nHow many $4$-digit palindromes are there?"
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 628,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A $\\text{palindrome}$, such as $83438$, is a number that remains the same when its digits are reversed. The numbers $x$ and $x+32$ are three-digit and four-digit palindromes, respectively. What is the sum of the digits of $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 629,
"solution": "\n1. **Identify the range of $x$ and $x+32$:**\n - Since $x$ is a three-digit palindrome, the maximum value of $x$ is $999$.\n - Consequently, the maximum value of $x+32$ is $999 + 32 = 1031$.\n - The minimum value of $x+32$ is $1000$ (since it is a four-digit palindrome).\n\n2. *... |
DeepScaleR | [
{
"role": "user",
"content": "A $k \\times k$ array contains each of the numbers $1, 2, \\dots, m$ exactly once, with the remaining entries all zero. Suppose that all the row sums and column sums are equal. What is the smallest possible value of $m$ if $k = 3^n$ ($n \\in \\mathbb{N}^+$)?"
}
] | math | {
"style": "rule",
"ground_truth": "3^{n+1} - 1"
} | {
"index": 630,
"solution": "\nConsider a \\( k \\times k \\) array, where \\( k = 3^n \\) for a positive integer \\( n \\). The array contains each of the integers \\( 1, 2, \\ldots, m \\) exactly once, and the remaining entries are all zeros. We are tasked with finding the smallest possible value of \\( m \\) suc... |
DeepScaleR | [
{
"role": "user",
"content": "A $n$-gon $S-A_{1} A_{2} \\cdots A_{n}$ has its vertices colored such that each vertex is colored with one color, and the endpoints of each edge are colored differently. Given $n+1$ colors available, find the number of different ways to color the vertices. (For $n=4$, this was ... | math | {
"style": "rule",
"ground_truth": "420"
} | {
"index": 631,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A ''super ball'' is dropped from a window 16 meters above the ground. On each bounce it rises $\\frac34$ the distance of the preceding high point. The ball is caught when it reached the high point after hitting the ground for the third time. To the nearest meter, how far has it... | math | {
"style": "rule",
"ground_truth": "65"
} | {
"index": 632,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A *palindromic table* is a $3 \\times 3$ array of letters such that the words in each row and column read the same forwards and backwards. An example of such a table is shown below. \n\\[ \\begin{array}[h]{ccc}\n O & M & O \n N & M & N \n O & M & O\n\\end{array} \\]\nHow m... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 633,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A *substring* of a number $n$ is a number formed by removing some digits from the beginning and end of $n$ (possibly a different number of digits is removed from each side). Find the sum of all prime numbers $p$ that have the property that any substring of $p$ is also p... | math | {
"style": "rule",
"ground_truth": "576"
} | {
"index": 634,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A 0-1 sequence of length $2^k$ is given. Alice can pick a member from the sequence, and reveal it (its place and its value) to Bob. Find the largest number $s$ for which Bob can always pick $s$ members of the sequence, and guess all their values correctly.\n\nAlice and Bob can ... | math | {
"style": "rule",
"ground_truth": "k+1"
} | {
"index": 635,
"solution": "\nLet a sequence of length \\(2^k\\) consisting of only 0s and 1s be given. Alice selects a member from this sequence, revealing both its position and value to Bob. Our goal is to determine the largest number \\(s\\) such that Bob, by following an optimal strategy agreed upon with Alice... |
DeepScaleR | [
{
"role": "user",
"content": "A 1$\\%$ late charge was added to Jenna's bill on the $30^{\\text{th}}$ day past its due date. The resulting total was then increased by 1$\\%$ because she did not pay the bill in the next 30 days either. Her original bill was $\\$400$. Exactly how much is the bill now?"
}
] | math | {
"style": "rule",
"ground_truth": "408.04"
} | {
"index": 636,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A 1-liter carton of milk used to cost 80 rubles. Recently, in an effort to cut costs, the manufacturer reduced the carton size to 0.9 liters and increased the price to 99 rubles. By what percentage did the manufacturer's revenue increase?"
}
] | math | {
"style": "rule",
"ground_truth": "37.5"
} | {
"index": 637,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A 10 by 10 checkerboard has alternating black and white squares. How many distinct squares, with sides on the grid lines of the checkerboard (both horizontal and vertical) and containing at least 7 black squares, can be drawn on the checkerboard?"
}
] | math | {
"style": "rule",
"ground_truth": "140"
} | {
"index": 638,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A 10 by 10 checkerboard has alternating black and white squares. How many distinct squares, with sides on the grid lines of the checkerboard (horizontal and vertical) and containing at least 6 black squares, can be drawn on the checkerboard?"
}
] | math | {
"style": "rule",
"ground_truth": "140"
} | {
"index": 639,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A 10-cm-by-10-cm square is partitioned such that points $A$ and $B$ are on two opposite sides of the square at one-third and two-thirds the length of the sides, respectively. What is the area of the new shaded region formed by connecting points $A$, $B$, and their reflections a... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 640,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A 10-digit arrangement $ 0,1,2,3,4,5,6,7,8,9$ is called *beautiful* if (i) when read left to right, $ 0,1,2,3,4$ form an increasing sequence, and $ 5,6,7,8,9$ form a decreasing sequence, and (ii) $ 0$ is not the leftmost digit. For example, $ 9807123654$ is a beautifu... | math | {
"style": "rule",
"ground_truth": "126"
} | {
"index": 641,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A 100 foot long moving walkway moves at a constant rate of 6 feet per second. Al steps onto the start of the walkway and stands. Bob steps onto the start of the walkway two seconds later and strolls forward along the walkway at a constant rate of 4 feet per second. Two seconds ... | math | {
"style": "rule",
"ground_truth": "52"
} | {
"index": 642,
"solution": "Clearly we have people moving at speeds of $6,8$ and $10$ feet/second. Notice that out of the three people, Cy is at the largest disadvantage to begin with and since all speeds are close, it is hardest for him to catch up. Furthermore, Bob is clearly the farthest along. Thus it is reaso... |
DeepScaleR | [
{
"role": "user",
"content": "A 100-digit number has the form \\(a = 1777 \\ldots 76\\) (with 98 digits of 7 in the middle). The number \\(\\frac{1}{a}\\) is represented as an infinite repeating decimal. Find its period and justify your answer."
}
] | math | {
"style": "rule",
"ground_truth": "99"
} | {
"index": 643,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A 10x10 arrangement of alternating black and white squares has a black square $R$ in the second-bottom row and a white square $S$ in the top-most row. Given that a marker is initially placed at $R$ and can move to an immediately adjoining white square on the row above either to... | math | {
"style": "rule",
"ground_truth": "70"
} | {
"index": 644,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A 12-hour digital clock has a glitch such that whenever it is supposed to display a 5, it mistakenly displays a 7. For example, when it is 5:15 PM the clock incorrectly shows 7:77 PM. What fraction of the day will the clock show the correct time?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{33}{40}"
} | {
"index": 645,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A 12-slice pizza was made with only pepperoni and mushroom toppings, and every slice has at least one topping. Only six slices have pepperoni, and exactly ten slices have mushrooms. How many slices have both pepperoni and mushrooms?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 646,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A 16-step path is to go from $(-4,-4)$ to $(4,4)$ with each step increasing either the $x$-coordinate or the $y$-coordinate by 1. How many such paths stay outside or on the boundary of the square $-2 < x < 2$, $-2 < y < 2$ at each step?"
}
] | math | {
"style": "rule",
"ground_truth": "1698"
} | {
"index": 647,
"solution": "To solve this problem, we need to count the number of paths from $(-4,-4)$ to $(4,4)$ that do not enter the square defined by $-2 \\leq x \\leq 2$ and $-2 \\leq y \\leq 2$. We can consider paths that either pass through the second quadrant or the fourth quadrant.\n\n#### Paths through t... |
DeepScaleR | [
{
"role": "user",
"content": "A 2 by 2003 rectangle consists of unit squares as shown below. The middle unit square of each row is shaded. If a rectangle from the figure is chosen at random, what is the probability that the rectangle does not include a shaded square? Express your answer as a common fract... | math | {
"style": "rule",
"ground_truth": "\\dfrac{1001}{2003}"
} | {
"index": 648,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A 20-quart container is fully filled with water. Five quarts are removed and replaced with pure antifreeze liquid. Then, five quarts of the mixture are removed and replaced with pure antifreeze. This process is repeated three more times (for a total of five times). Determine th... | math | {
"style": "rule",
"ground_truth": "\\frac{243}{1024}"
} | {
"index": 649,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A 20-step path is to go from $(-5,-5)$ to $(5,5)$ with each step increasing either the $x$-coordinate or the $y$-coordinate by 1. How many such paths stay outside or on the boundary of the rectangle $-3 \\le x \\le 3$, $-1 \\le y \\le 1$ at each step?"
}
] | math | {
"style": "rule",
"ground_truth": "2126"
} | {
"index": 650,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A 24-hour digital clock shows the time in hours and minutes. How many times in one day will it display all four digits 2, 0, 1, and 9 in some order?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 651,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A 24-hour digital clock shows the time in hours and minutes. How many times in one day will it display all four digits 2, 0, 1, and 9 in some order?\n\nA) 6 \nB) 10 \nC) 12 \nD) 18 \nE) 24"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 652,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A 24-hour digital clock shows times $h: m: s$, where $h, m$, and $s$ are integers with $0 \\leq h \\leq 23$, $0 \\leq m \\leq 59$, and $0 \\leq s \\leq 59$. How many times $h: m: s$ satisfy $h+m=s$?"
}
] | math | {
"style": "rule",
"ground_truth": "1164"
} | {
"index": 653,
"solution": "We are solving $h+m=s$ in $0 \\leq s \\leq 59,0 \\leq m \\leq 59$, and $0 \\leq h \\leq 23$. If $s \\geq 24$, each $h$ corresponds to exactly 1 solution, so we get $24(59-23)=24(36)$ in this case. If $s \\leq 23$, we want the number of nonnegative integer solutions to $h+m \\leq 23$, wh... |
DeepScaleR | [
{
"role": "user",
"content": "A 3 by 2 rectangle is split into four congruent right-angled triangles. Those four triangles are rearranged to form a rhombus. What is the ratio of the perimeter of the rectangle to the perimeter of the rhombus?"
}
] | math | {
"style": "rule",
"ground_truth": "1: 1"
} | {
"index": 654,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A 30 foot ladder is placed against a vertical wall of a building. The foot of the ladder is 11 feet from the base of the building. If the top of the ladder slips 6 feet, then the foot of the ladder will slide how many feet?"
}
] | math | {
"style": "rule",
"ground_truth": "9.49"
} | {
"index": 655,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A 3x3x3 cube is made of $27$ normal dice. Each die's opposite sides sum to $7$. What is the smallest possible sum of all of the values visible on the $6$ faces of the large cube?"
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 656,
"solution": "1. **Understanding the Cube Configuration**:\n - A 3x3x3 cube consists of 27 smaller cubes.\n - The smaller cubes on the corners have three faces visible.\n - The smaller cubes on the edges have two faces visible.\n - The smaller cubes in the center of each face have one face vi... |
DeepScaleR | [
{
"role": "user",
"content": "A 4-inch by 6-inch picture is enlarged for framing by tripling its dimensions. A 2-inch-wide border is then placed around each side of the enlarged picture, as shown. Thin metal framing is sold only in increments of one foot. What is the minimum number of linear feet of... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 657,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A 40-foot by 10-foot rectangular garden is enclosed by a fence. To make the garden larger, while using the same amount of fencing, its shape is changed to a square. How many square feet larger than the old garden is the new garden?"
}
] | math | {
"style": "rule",
"ground_truth": "225"
} | {
"index": 658,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A 5 by 5 grid of unit squares is partitioned into 5 pairwise incongruent rectangles with sides lying on the gridlines. Find the maximum possible value of the product of their areas."
}
] | math | {
"style": "rule",
"ground_truth": "2304"
} | {
"index": 659,
"solution": "The greatest possible value for the product is $3 \\cdot 4 \\cdot 4 \\cdot 6 \\cdot 8=2304$, achieved when the rectangles are $3 \\times 1,1 \\times 4,2 \\times 2,2 \\times 3,4 \\times 2$. To see that this is possible, orient these rectangles so that the first number is the horizontal d... |
DeepScaleR | [
{
"role": "user",
"content": "A 5-digit natural number \\(abcde\\) is called a \"\\(\\pi_1\\)\" number if and only if it satisfies \\(a < b < c\\) and \\(c > d > e\\). Determine the total number of \"\\(\\pi_1\\)\" numbers among all 5-digit numbers."
}
] | math | {
"style": "rule",
"ground_truth": "2142"
} | {
"index": 660,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A 5-dimensional ant starts at one vertex of a 5-dimensional hypercube of side length 1. A move is when the ant travels from one vertex to another vertex at a distance of $\\sqrt{2}$ away. How many ways can the ant make 5 moves and end up on the same vertex it started at?"
}
] | math | {
"style": "rule",
"ground_truth": "6240"
} | {
"index": 661,
"solution": "We let the cube lie in $\\mathbb{R}^{5}$ with each corner with coordinates 1 or 0. Assume the ant starts at $(0,0,0,0,0)$. Every move the ant adds or subtracts 1 to two of the places. Note that this means the ant can only land on a vertex with the sum of its coordinates an even number. ... |
DeepScaleR | [
{
"role": "user",
"content": "A 50-card deck consists of 4 cards labeled \" $i$ \" for $i=1,2, \\ldots, 12$ and 2 cards labeled \" 13 \". If Bob randomly chooses 2 cards from the deck without replacement, what is the probability that his 2 cards have the same label?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{73}{1225}"
} | {
"index": 662,
"solution": "All pairs of distinct cards (where we distinguish cards even with the same label) are equally likely. There are $\\binom{2}{2}+12\\binom{4}{2}=73$ pairs of cards with the same label and $\\binom{50}{2}=100 \\cdot \\frac{49}{4}=1225$ pairs of cards overall, so the desired probability is ... |
DeepScaleR | [
{
"role": "user",
"content": "A 5x5x5 cube is formed by assembling 125 unit cubes. Nine unit squares are painted on each of the six faces of the cube according to the pattern shown. How many of the 125 unit cubes have no paint on them? [asy]\nfill((2,0)--(3,0)--(3,5)--(2,5)--cycle,gray(.7));\nfill((0,2)--(0... | math | {
"style": "rule",
"ground_truth": "83"
} | {
"index": 663,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A 6 cm by 8 cm rectangle is inscribed in a circle. What is the number of centimeters in the circumference of the circle? Express your answer in terms of $\\pi$.\n\n[asy]import graph;\ndefaultpen(linewidth(0.7));\ndraw(Circle((0,0),20));\ndraw((-16,-12)--(16,-12)--(16,12)--(-16,... | math | {
"style": "rule",
"ground_truth": "10\\pi"
} | {
"index": 664,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A 6 m by 8 m rectangular field has a fence around it. There is a post at each of the four corners of the field. Starting at each corner, there is a post every 2 m along each side of the fence. How many posts are there?"
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 665,
"solution": "A rectangle that is 6 m by 8 m has perimeter $2 \\times(6 \\mathrm{~m}+8 \\mathrm{~m})=28 \\mathrm{~m}$. If posts are put in every 2 m around the perimeter starting at a corner, then we would guess that it will take $\\frac{28 \\mathrm{~m}}{2 \\mathrm{~m}}=14$ posts."
} |
DeepScaleR | [
{
"role": "user",
"content": "A 60-degree angle contains five circles, where each subsequent circle (starting from the second) touches the previous one. By how many times is the sum of the areas of all five circles greater than the area of the smallest circle?"
}
] | math | {
"style": "rule",
"ground_truth": "7381"
} | {
"index": 666,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A 6x6x6 cube is formed by assembling 216 unit cubes. Ten unit squares are painted on each of the six faces of the cube, leaving some rows and columns unpainted. Specifically, two non-adjacent columns and two non-adjacent rows on each face are left unpainted. How many of the 216... | math | {
"style": "rule",
"ground_truth": "168"
} | {
"index": 667,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A 6x6x6 cube is formed by assembling 216 unit cubes. Two 1x6 stripes are painted on each of the six faces of the cube parallel to the edges, with one stripe along the top edge and one along the bottom edge of each face. How many of the 216 unit cubes have no paint on them?"
}... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 668,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A 7' × 11' table sits in the corner of a square room. The table is to be rotated so that the side formerly 7' now lies along what was previously the end side of the longer dimension. Determine the smallest integer value of the side S of the room needed to accommodate this move.... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 669,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A 9 by 9 checkerboard has alternating black and white squares. How many distinct squares, with sides on the grid lines of the checkerboard (horizontal and vertical) and containing at least 6 black squares, can be drawn on the checkerboard?"
}
] | math | {
"style": "rule",
"ground_truth": "91"
} | {
"index": 670,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A 92-digit natural number \\( n \\) has its first 90 digits given: from the 1st to the 10th digit are ones, from the 11th to the 20th are twos, and so on, from the 81st to the 90th are nines. Find the last two digits of \\( n \\), given that \\( n \\) is divisible by 72."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 671,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A 9x9 chessboard has its squares labeled such that the label of the square in the ith row and jth column is given by $\\frac{1}{2 \\times (i + j - 1)}$. We need to select one square from each row and each column. Find the minimum sum of the labels of the nine chosen squares."
... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 672,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A Ferris wheel rotates at a constant speed, completing one revolution every 12 minutes. The lowest point of the Ferris wheel is 2 meters above the ground, and the highest point is 18 meters above the ground. Let P be a fixed point on the circumference of the Ferris wheel. Start... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 673,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A Gareth sequence is a sequence of numbers where each number after the second is the non-negative difference between the two previous numbers. For example, if a Gareth sequence begins 15, 12, then:\n\n- The third number in the sequence is \\(15 - 12 = 3\\),\n- The fourth number... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"index": 674,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A Martian traffic light consists of six identical bulbs arranged in two horizontal rows (one below the other) with three bulbs in each row. A rover driver in foggy conditions can distinguish the number and relative positions of the lit bulbs on the traffic light (for example, i... | math | {
"style": "rule",
"ground_truth": "44"
} | {
"index": 675,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A Mediterranean polynomial has only real roots and it is of the form\n\\[ P(x) = x^{10}-20x^9+135x^8+a_7x^7+a_6x^6+a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0 \\] with real coefficients $a_0\\ldots,a_7$ . Determine the largest real number that occurs as a root of some Mediterranean... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 676,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A Mersenne prime is defined to be a prime number of the form $2^n - 1$, where $n$ must itself be a prime. For example, since $2^3 - 1 = 7$, and 3 is a prime number, 7 is a Mersenne prime. What is the largest Mersenne prime less than 200?"
}
] | math | {
"style": "rule",
"ground_truth": "127"
} | {
"index": 677,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A Moskvich car was sent to transport mail from the post office to the airfield. The plane carrying the mail landed earlier than expected, and the delivered mail was sent to the post office by a passing truck. After driving for 30 minutes, the truck met the Moskvich on the road,... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 678,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A Saxon silver penny, from the reign of Ethelbert II in the eighth century, was sold in 2014 for £78000. A design on the coin depicts a circle surrounded by four equal arcs, each a quarter of a circle. The width of the design is 2 cm. What is the radius of the small circle, in ... | math | {
"style": "rule",
"ground_truth": "2 - \\sqrt{2}"
} | {
"index": 679,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A Senate committee consists of 10 Republicans and 8 Democrats. In how many ways can we form a subcommittee that has at most 5 members, including exactly 3 Republicans and at least 2 Democrats?"
}
] | math | {
"style": "rule",
"ground_truth": "10080"
} | {
"index": 680,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A Senate committee has 5 Democrats and 5 Republicans. Assuming all politicians are distinguishable, in how many ways can they sit around a circular table without restrictions? (Two seatings are considered the same if one is a rotation of the other.)"
}
] | math | {
"style": "rule",
"ground_truth": "362,\\!880"
} | {
"index": 681,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A Senate committee has 8 Republicans and 6 Democrats. In how many ways can we form a subcommittee with 3 Republicans and 2 Democrats?"
}
] | math | {
"style": "rule",
"ground_truth": "840"
} | {
"index": 682,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A Sudoku matrix is defined as a $9 \\times 9$ array with entries from \\{1,2, \\ldots, 9\\} and with the constraint that each row, each column, and each of the nine $3 \\times 3$ boxes that tile the array contains each digit from 1 to 9 exactly once. A Sudoku matrix is chosen a... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{21}"
} | {
"index": 683,
"solution": "The third row must contain the digit 1, and it cannot appear in the leftmost three squares. Therefore, the digit 1 must fall into one of the six squares shown below that are marked with $\\star$. By symmetry, each starred square has an equal probability of containing the digit 1 (To see... |
DeepScaleR | [
{
"role": "user",
"content": "A TV station broadcasts 5 advertisements in a row, among which there are 3 different commercial advertisements and 2 different World Expo promotional advertisements. The last advertisement broadcasted is a World Expo promotional advertisement, and the methods in which the 2 Wor... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 684,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A TV station is broadcasting 5 advertisements in a row, including 3 different commercial advertisements and 2 different public service advertisements. The last advertisement cannot be a commercial one, and the two public service advertisements cannot be broadcast consecutively.... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 685,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A TV station is broadcasting 5 different advertisements, among which there are 3 different commercial advertisements and 2 different Olympic promotional advertisements. It is required that the last advertisement must be an Olympic promotional advertisement, and the two Olympic ... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 686,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A Tim number is a five-digit positive integer with the following properties:\n1. It is a multiple of 15.\n2. Its hundreds digit is 3.\n3. Its tens digit is equal to the sum of its first three (leftmost) digits.\n\nHow many Tim numbers are there?"
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 687,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A U-shaped number is a special type of three-digit number where the units digit and the hundreds digit are equal and greater than the tens digit. For example, 818 is a U-shaped number. How many U-shaped numbers are there?"
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 688,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A Vandal and a Moderator are editing a Wikipedia article. The article originally is error-free. Each day, the Vandal introduces one new error into the Wikipedia article. At the end of the day, the moderator checks the article and has a $2 / 3$ chance of catching each individual... | math | {
"style": "rule",
"ground_truth": "\\frac{416}{729}"
} | {
"index": 689,
"solution": "Consider the error that was introduced on day 1. The probability that the Moderator misses this error on all three checks is $1 / 3^{3}$, so the probability that this error gets removed is $1-\\frac{1}{3^{3}}$. Similarly, the probability that the moderator misses the other two errors ar... |
DeepScaleR | [
{
"role": "user",
"content": "A Yule log is shaped like a right cylinder with height $10$ and diameter $5$ . Freya cuts it parallel to its bases into $9$ right cylindrical slices. After Freya cut it, the combined surface area of the slices of the Yule log increased by $a\\pi$ . Compute $a$ ."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 690,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A [i]permutation[/i] of the set of positive integers $[n] = \\{1, 2, . . . , n\\}$ is a sequence $(a_1 , a_2 , \\ldots, a_n ) $ such that each element of $[n]$ appears precisely one time as a term of the sequence. For example, $(3, 5, 1, 2, 4)$ is a permutation of $[5]$. Let $P... | math | {
"style": "rule",
"ground_truth": "4489"
} | {
"index": 691,
"solution": "\nTo solve this problem, we will analyze the given condition involving permutations and perfect squares to determine the smallest \\( n \\) such that \\( P(n) \\), the number of permutations of \\([n] = \\{1, 2, \\ldots, n\\}\\) where \\( ka_k \\) is a perfect square for all \\( 1 \\leq... |
DeepScaleR | [
{
"role": "user",
"content": "A [i]site[/i] is any point $(x, y)$ in the plane such that $x$ and $y$ are both positive integers less than or equal to 20.\n\nInitially, each of the 400 sites is unoccupied. Amy and Ben take turns placing stones with Amy going first. On her turn, Amy places a new red stone on ... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 692,
"solution": "\nLet us consider the problem where Amy and Ben take turns placing stones on a 20x20 grid consisting of sites \\((x, y)\\) where \\(x\\) and \\(y\\) are integers between 1 and 20 inclusive. Amy's condition for placing a red stone is that the distance between any two red stones is not eq... |
DeepScaleR | [
{
"role": "user",
"content": "A \\( 5 \\mathrm{~cm} \\) by \\( 5 \\mathrm{~cm} \\) pegboard and a \\( 10 \\mathrm{~cm} \\) by \\( 10 \\mathrm{~cm} \\) pegboard each have holes at the intersection of invisible horizontal and vertical lines that occur in \\( 1 \\mathrm{~cm} \\) intervals from each edge. Pegs ... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 693,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A \\(10 \\times 1\\) rectangular pavement is to be covered by tiles which are either green or yellow, each of width 1 and of varying integer lengths from 1 to 10. Suppose you have an unlimited supply of tiles for each color and for each of the varying lengths. How many distinct... | math | {
"style": "rule",
"ground_truth": "1022"
} | {
"index": 694,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A \\(101 \\times 101\\) grid is given, where all cells are initially colored white. You are allowed to choose several rows and paint all the cells in those rows black. Then, choose exactly the same number of columns and invert the color of all cells in those columns (i.e., chan... | math | {
"style": "rule",
"ground_truth": "5100"
} | {
"index": 695,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A \\(3 \\times 3 \\times 3\\) cube composed of 27 unit cubes rests on a horizontal plane. Determine the number of ways of selecting two distinct unit cubes (order is irrelevant) from a \\(3 \\times 3 \\times 1\\) block with the property that the line joining the centers of the ... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 696,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A \\(3\\times 5\\) rectangle and a \\(4\\times 6\\) rectangle need to be contained within a square without any overlapping at their interior points, and the square's sides are parallel to the sides of the given rectangles. Determine the smallest possible area of this square."
... | math | {
"style": "rule",
"ground_truth": "81"
} | {
"index": 697,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A \\(4 \\times 4\\) Sudoku grid is filled with digits so that each column, each row, and each of the four \\(2 \\times 2\\) sub-grids that compose the grid contains all of the digits from 1 to 4. \n\nFind the total number of possible \\(4 \\times 4\\) Sudoku grids."
}
] | math | {
"style": "rule",
"ground_truth": "288"
} | {
"index": 698,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A and B are playing a guessing game. First, A thinks of a number, denoted as $a$, then B guesses the number A was thinking of, denoting B's guess as $b$. Both $a$ and $b$ belong to the set $\\{0,1,2,…,9\\}$. The probability that $|a-b|\\leqslant 1$ is __________."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{7}{25}"
} | {
"index": 699,
"solution": ""
} |
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