data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Among the numbers from 1 to 1000, how many are divisible by 4 and do not contain the digit 4 in their representation?"
}
] | math | {
"style": "rule",
"ground_truth": "162"
} | {
"index": 4600,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the parts processed by a worker, on average, $4 \\%$ are non-standard. Find the probability that among the 30 parts taken for testing, two parts will be non-standard. What is the most probable number of non-standard parts in the considered sample of 30 parts, and what is ... | math | {
"style": "rule",
"ground_truth": "0.305"
} | {
"index": 4601,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the positive integers less than $10^{4}$, how many positive integers $n$ are there such that $2^{n} - n^{2}$ is divisible by 7?"
}
] | math | {
"style": "rule",
"ground_truth": "2857"
} | {
"index": 4602,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the positive integers less than 1000, there are how many numbers that are perfect squares but not perfect cubes?"
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 4603,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the positive integers that can be expressed as the sum of 2005 consecutive integers, which occupies the 2005th position when arranged in order?\r\n\r\n*Roland Hablutzel, Venezuela*\r\n<details><summary>Remark</summary>The original question was: Among the positive integers... | math | {
"style": "rule",
"ground_truth": "2005 * 2004 * 2005"
} | {
"index": 4604,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the rye seeds, there are $0.4\\%$ weed seeds. What is the probability of detecting 5 weed seeds when randomly selecting 5000 seeds?"
}
] | math | {
"style": "rule",
"ground_truth": "0.000055"
} | {
"index": 4605,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the scalene triangles with natural number side lengths, a perimeter not exceeding 30, and the sum of the longest and shortest sides exactly equal to twice the third side, there are ____ distinct triangles."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 4606,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the six-digit numbers formed by the digits 0, 1, 2, 3, 4, 5 without repetition, calculate the number of the numbers that are divisible by 2."
}
] | math | {
"style": "rule",
"ground_truth": "312"
} | {
"index": 4607,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the standard products of a certain factory, on average 15% are of the second grade. What is the probability that the percentage of second-grade products among 1000 standard products of this factory differs from 15% by less than 2% in absolute value?"
}
] | math | {
"style": "rule",
"ground_truth": "0.9232"
} | {
"index": 4608,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the subsets of the set $\\{1,2, \\cdots, 100\\}$, calculate the maximum number of elements a subset can have if it does not contain any pair of numbers where one number is exactly three times the other."
}
] | math | {
"style": "rule",
"ground_truth": "67"
} | {
"index": 4609,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the three-digit numbers composed of the digits $0$ to $9$, the number of numbers where the digits are arranged in strictly increasing or strictly decreasing order, calculate the total."
}
] | math | {
"style": "rule",
"ground_truth": "204"
} | {
"index": 4610,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the three-digit numbers formed by the digits 0, 1, 2, 3, 4, 5 without repetition, there are a total of numbers whose digits sum up to 9 (answer in digits)."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 4611,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the three-digit numbers formed by the digits 1, 2, 3, 4, 5, 6 with repetition allowed, how many three-digit numbers have exactly two different even digits (for example: 124, 224, 464, …)? (Answer with a number)."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 4612,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the triangles with natural number side lengths, a perimeter not exceeding 100, and the difference between the longest and shortest sides not greater than 2, there are a total of different triangles that are not congruent to each other."
}
] | math | {
"style": "rule",
"ground_truth": "190"
} | {
"index": 4613,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among the two-digit numbers less than 20, the largest prime number is ____, and the largest composite number is ____."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 4614,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Among three-digit numbers, if the digit in the tens place is smaller than both the digit in the units place and the digit in the hundreds place, then this three-digit number is called a \"concave number,\" such as 402, 745, etc. How many three-digit concave numbers with no repe... | math | {
"style": "rule",
"ground_truth": "240"
} | {
"index": 4615,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Amy and Belinda each roll a sheet of 6-inch by 8-inch paper to form a cylindrical tube. Amy tapes the two 8-inch sides together without overlap. Belinda tapes the two 6-inch sides together without overlap. What is $\\pi$ times the positive difference of the volumes of the two t... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 4616,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Amy and Ben need to eat 1000 total carrots and 1000 total muffins. The muffins can not be eaten until all the carrots are eaten. Furthermore, Amy can not eat a muffin within 5 minutes of eating a carrot and neither can Ben. If Amy eats 40 carrots per minute and 70 muffins per m... | math | {
"style": "rule",
"ground_truth": "23.5"
} | {
"index": 4617,
"solution": "Amy and Ben will continuously eat carrots, then stop (not necessarily at the same time), and continuously eat muffins until no food is left. Suppose that Amy and Ben finish eating the carrots in $T_{1}$ minutes and the muffins $T_{2}$ minutes later; we wish to find the minimum value of... |
DeepScaleR | [
{
"role": "user",
"content": "Amy and Bob choose numbers from $0,1,2,\\cdots,81$ in turn and Amy choose the number first. Every time the one who choose number chooses one number from the remaining numbers. When all $82$ numbers are chosen, let $A$ be the sum of all the numbers Amy chooses, and let $B... | math | {
"style": "rule",
"ground_truth": "41"
} | {
"index": 4618,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Amy rolls six fair 8-sided dice, each numbered from 1 to 8. What is the probability that exactly three of the dice show a prime number and at least one die shows an 8?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2899900}{16777216}"
} | {
"index": 4619,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Amy works for 36 hours per week for 10 weeks during the summer, making $\\$3000$. If she works for 30 weeks during the school year at the same rate of pay and needs to make another $\\$3000$, how many hours per week must she work?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 4620,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Amy writes down four integers \\(a > b > c > d\\) whose sum is 50. The pairwise positive differences of these numbers are \\(2, 3, 4, 5, 7,\\) and \\(10\\). What is the sum of the possible values for \\(a\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"index": 4621,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Amy, Beth, and Claire each have some sweets. Amy gives one third of her sweets to Beth. Beth gives one third of all the sweets she now has to Claire. Then Claire gives one third of all the sweets she now has to Amy. All the girls end up having the same number of sweets.\n\nClai... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 4622,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those two girls but disliked by the third. In how many different ways is ... | math | {
"style": "rule",
"ground_truth": "132"
} | {
"index": 4623,
"solution": "\nWe are given that Amy, Beth, and Jo listen to four different songs and discuss which ones they like, with the conditions that no song is liked by all three and for each pair of girls, there is at least one song liked by those two but disliked by the third. We need to find the number ... |
DeepScaleR | [
{
"role": "user",
"content": "An \"$n$-pointed star\" is formed as follows: the sides of a convex polygon are numbered consecutively $1,2,\\cdots ,k,\\cdots,n,\\text{ }n\\ge 5$; for all $n$ values of $k$, sides $k$ and $k+2$ are non-parallel, sides $n+1$ and $n+2$ being respectively identical with sides $1$... | math | {
"style": "rule",
"ground_truth": "180(n-4)"
} | {
"index": 4624,
"solution": "To solve this problem, we need to determine the sum of the interior angles at the vertices of the $n$-pointed star formed by extending every second side of an $n$-sided convex polygon.\n\n1. **Understanding the Star Formation**: \n - Each vertex of the star is formed by the intersect... |
DeepScaleR | [
{
"role": "user",
"content": "An $10 \\times 25$ rectangle is divided into two congruent polygons, and these polygons are rearranged to form a rectangle again. Determine the length of the smaller side of the resulting rectangle."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 4625,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An $11 \\times 11 \\times 11$ wooden cube is formed by gluing together $11^3$ unit cubes. What is the greatest number of unit cubes that can be seen from a single point?"
}
] | math | {
"style": "rule",
"ground_truth": "331"
} | {
"index": 4626,
"solution": "To find the greatest number of unit cubes that can be seen from a single point, we consider viewing the cube from one of its corners. From this vantage point, three faces of the cube are visible: the top face, the front face, and the side face.\n\n1. **Counting the visible unit cubes o... |
DeepScaleR | [
{
"role": "user",
"content": "An $8$ by $2\\sqrt{2}$ rectangle has the same center as a circle of radius $2$. The area of the region common to both the rectangle and the circle is"
}
] | math | {
"style": "rule",
"ground_truth": "2\\pi+4"
} | {
"index": 4627,
"solution": "1. **Visualize and Analyze the Geometry**: Consider a rectangle with dimensions $8$ by $2\\sqrt{2}$ and a circle with radius $2$, both sharing the same center. We need to find the area of the region common to both the rectangle and the circle.\n\n2. **Identify Key Points and Triangles*... |
DeepScaleR | [
{
"role": "user",
"content": "An $8$-cm-by-$8$-cm square is partitioned as shown. Points $A$ and $B$ are the midpoints of two opposite sides of the square. What is the area of the shaded region?\n\n[asy]\ndraw((0,0)--(10,0));\ndraw((10,0)--(10,10));\ndraw((10,10)--(0,10));\ndraw((0,0)--(0,10));\ndraw((0,0)-... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 4628,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An $8\\times8$ array consists of the numbers $1,2,...,64$. Consecutive numbers are adjacent along a row or a column. What is the minimum value of the sum of the numbers along the diagonal?"
}
] | math | {
"style": "rule",
"ground_truth": "88"
} | {
"index": 4629,
"solution": "\nWe have an \\(8 \\times 8\\) array filled with the numbers from 1 to 64, where consecutive numbers are adjacent either along a row or along a column. Our task is to find the minimum possible value of the sum of the numbers along a diagonal of this array.\n\n### Analysis\n\nLet's deno... |
DeepScaleR | [
{
"role": "user",
"content": "An $E$-shape is a geometric figure in the two-dimensional plane consisting of three rays pointing in the same direction, along with a line segment such that the endpoints of the rays all lie on the segment, the segment is perpendicular to all three rays, both endpoints of the s... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 4630,
"solution": "Define a $C$-shape to be an $E$-shape without the middle ray. Then, an $E$-shape consists of a ray and a $C$-shape. Two $C$-shapes can intersect at most 6 times, a $C$-shape and a ray can intersect at most 2 times, and two rays can intersect at most 1 time. Thus, the number of intersec... |
DeepScaleR | [
{
"role": "user",
"content": "An $\\textit{abundant number}$ is a positive integer, the sum of whose distinct proper factors is greater than the number. (The $\\textit{proper factors}$ of a number are all of its factors except the number itself.) How many numbers less than 25 are abundant numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 4631,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An $a \\times b \\times c$ rectangular box is built from $a \\cdot b \\cdot c$ unit cubes. Each unit cube is colored red, green, or yellow. Each of the $a$ layers of size $1 \\times b \\times c$ parallel to the $(b \\times c)$ faces of the box contains exactly $9$ red cubes, ex... | math | {
"style": "rule",
"ground_truth": "180"
} | {
"index": 4632,
"solution": "The total number of green cubes is given by $12a=20b\\Longrightarrow a=\\frac{5}{3}b$.\n Let $r$ be the number of red cubes on each one of the $b$ layers then the total number of red cubes is $9a=br$. Substitute $a=\\frac{5}{3}b$ gives $r=15$.\n Repeating the procedure on the number of... |
DeepScaleR | [
{
"role": "user",
"content": "An $m\\times n\\times p$ rectangular box has half the volume of an $(m + 2)\\times(n + 2)\\times(p + 2)$ rectangular box, where $m, n,$ and $p$ are integers, and $m\\le n\\le p.$ What is the largest possible value of $p$?"
}
] | math | {
"style": "rule",
"ground_truth": "130"
} | {
"index": 4633,
"solution": "Similarly as above, we solve for $p,$ but we express the denominator differently:\n\\[p=\\dfrac{2(m+2)(n+2)}{(m+2)(n+2)-4(m+n+2)} \\implies \\dfrac{1}{p}=\\dfrac{1}{2}-\\dfrac{2(m+n+2)}{(m+2)(n+2)}.\\] Hence, it suffices to maximize $\\dfrac{m+n+2}{(m+2)(n+2)},$ under the conditions th... |
DeepScaleR | [
{
"role": "user",
"content": "An $n \\times m$ maze is an $n \\times m$ grid in which each cell is one of two things: a wall, or a blank. A maze is solvable if there exists a sequence of adjacent blank cells from the top left cell to the bottom right cell going through no walls. (In particular, the top left... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 4634,
"solution": "We must have both top-left and bottom-right cells blank, and we cannot have both top-right and bottom-left cells with walls. As long as those conditions are satisfied, the maze is solvable, so the answer is 3."
} |
DeepScaleR | [
{
"role": "user",
"content": "An $n \\times n$ complex matrix $A$ is called $t$-normal if $A A^{t}=A^{t} A$ where $A^{t}$ is the transpose of $A$. For each $n$, determine the maximum dimension of a linear space of complex $n \\times n$ matrices consisting of t-normal matrices."
}
] | math | {
"style": "rule",
"ground_truth": "\\[\n\\frac{n(n+1)}{2}\n\\]"
} | {
"index": 4635,
"solution": "Answer: The maximum dimension of such a space is $\\frac{n(n+1)}{2}$. The number $\\frac{n(n+1)}{2}$ can be achieved, for example the symmetric matrices are obviously t-normal and they form a linear space with dimension $\\frac{n(n+1)}{2}$. We shall show that this is the maximal possib... |
DeepScaleR | [
{
"role": "user",
"content": "An $n$-digit positive integer is cute if its $n$ digits are an arrangement of the set $\\{1,2,...,n\\}$ and its first $k$ digits form an integer that is divisible by $k$, for $k = 1,2,...,n$. For example, $321$ is a cute $3$-digit integer because $1$ divides $3$, $2$ divides $3... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 4636,
"solution": "1. **Define the problem and constraints**: We need to find a 6-digit integer $abcdef$ such that:\n - The digits are a permutation of $\\{1, 2, 3, 4, 5, 6\\}$.\n - For each $k = 1, 2, \\ldots, 6$, the number formed by the first $k$ digits is divisible by $k$.\n\n2. **Divisibility by... |
DeepScaleR | [
{
"role": "user",
"content": "An $n$-string is a string of digits formed by writing the numbers $1,2, \\ldots, n$ in some order (in base ten). For example, one possible 10-string is $$35728910461$$ What is the smallest $n>1$ such that there exists a palindromic $n$-string?"
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 4637,
"solution": "The following is such a string for $n=19$ : $$ 9|18| 7|16| 5|14| 3|12| 1|10| 11|2| 13|4| 15|6| 17|8| 19 $$ where the vertical bars indicate breaks between the numbers. On the other hand, to see that $n=19$ is the minimum, notice that only one digit can occur an odd number of times in a... |
DeepScaleR | [
{
"role": "user",
"content": "An 18 inch by 24 inch painting is mounted in a wooden frame where the width of the wood at the top and bottom of the frame is twice the width of the wood at the sides. If the area of the frame is equal to the area of the painting, find the ratio of the shorter side to the longe... | math | {
"style": "rule",
"ground_truth": "2:3"
} | {
"index": 4638,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An 18 inch by 24 inch painting is mounted in a wooden frame where the width of the wood at the top and bottom of the frame is twice the width of the wood at the sides. If the area of the frame is equal to the area of the painting, what is the ratio of the shorter side to the lo... | math | {
"style": "rule",
"ground_truth": "2:3"
} | {
"index": 4639,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An 18-slice pizza was made with only pepperoni and mushroom toppings, and every slice has at least one topping. Exactly ten slices have pepperoni, and exactly ten slices have mushrooms. How many slices have both pepperoni and mushrooms?"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 4640,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An 8 by 8 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 5 black squares, can be drawn on the checkerboard?\n\n[asy]\ndraw((0,0)--(8,0)--(8,8)--(... | math | {
"style": "rule",
"ground_truth": "73"
} | {
"index": 4641,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An 8 by 8 grid of numbers obeys the following pattern: 1) The first row and first column consist of all 1s. 2) The entry in the $i$th row and $j$th column equals the sum of the numbers in the $(i-1)$ by $(j-1)$ sub-grid with row less than $i$ and column less than $j$. What is t... | math | {
"style": "rule",
"ground_truth": "2508"
} | {
"index": 4642,
"solution": "Let $x_{i, j}$ be the number in the $i$th row and the $j$th column. Then if $i, j \\geq 2, x_{i+1, j+1}-x_{i+1, j}-x_{i, j+1}+x_{i, j}$ only counts the term $x_{i, j}$ since every other term is added and subtracted the same number of times. Thus $x_{i+1, j+1}=x_{i+1, j}+x_{i, j+1}$ whe... |
DeepScaleR | [
{
"role": "user",
"content": "An 8-by-8 square is divided into 64 unit squares in the usual way. Each unit square is colored black or white. The number of black unit squares is even. We can take two adjacent unit squares (forming a 1-by-2 or 2-by-1 rectangle), and flip their colors: black becomes white a... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 4643,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An 8-foot by 10-foot floor is tiled with square tiles of size 1 foot by 1 foot. Each tile has a pattern consisting of four white quarter circles of radius 1/2 foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floo... | math | {
"style": "rule",
"ground_truth": "80 - 20\\pi"
} | {
"index": 4644,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An 8.5-meter ladder is leaning against a vertical wall. How many meters is its base from the wall if the ladder reaches 7.5 meters up the wall?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 4645,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An ATM password at Fred's Bank consists of four digits from $0$ to $9$. No password may begin with the sequence \"123\", and if a password begins with \"123\", the fourth digit cannot be $4$ or $5$. Calculate the number of valid passwords that are possible."
}
] | math | {
"style": "rule",
"ground_truth": "9992"
} | {
"index": 4646,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An ATM password at Fred's Bank is composed of four digits from $0$ to $9$, with repeated digits allowable. If no password may begin with the sequence $9,1,1,$ then how many passwords are possible?"
}
] | math | {
"style": "rule",
"ground_truth": "9990"
} | {
"index": 4647,
"solution": "\nTo find the total number of possible ATM passwords, we need to consider the constraints given in the problem. The password is a four-digit number, where each digit can range from $0$ to $9$. However, the password cannot begin with the sequence $9,1,1$.\n\n**Step 1: Calculate the tota... |
DeepScaleR | [
{
"role": "user",
"content": "An American traveling in Italy wishes to exchange American money (dollars) for Italian money (lire). If 3000 lire = 1.60, how much lire will the traveler receive in exchange for 1.00?"
}
] | math | {
"style": "rule",
"ground_truth": "1875"
} | {
"index": 4648,
"solution": "1. **Identify the given exchange rate**: The problem states that 3000 lire is equivalent to 1.60 dollars. We can express this as a ratio:\n \\[\n \\frac{3000 \\text{ lire}}{1.60 \\text{ dollars}}\n \\]\n\n2. **Set up a proportion to find the amount of lire for 1 dollar**: We need... |
DeepScaleR | [
{
"role": "user",
"content": "An Englishman owns a plot of land in Russia. He knows that, in the units familiar to him, the size of his plot is three acres. The cost of the land is 250,000 rubles per hectare. It is known that 1 acre = 4840 square yards, 1 yard = 0.9144 meters, and 1 hectare = 10,000 square ... | math | {
"style": "rule",
"ground_truth": "303514"
} | {
"index": 4649,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An HMMT party has $m$ MIT students and $h$ Harvard students for some positive integers $m$ and $h$, For every pair of people at the party, they are either friends or enemies. If every MIT student has 16 MIT friends and 8 Harvard friends, and every Harvard student has 7 MIT enem... | math | {
"style": "rule",
"ground_truth": "342"
} | {
"index": 4650,
"solution": "We count the number of MIT-Harvard friendships. Each of the $m$ MIT students has 8 Harvard friends, for a total of $8 m$ friendships. Each of the $h$ Harvard students has $m-7$ MIT friends, for a total of $h(m-7)$ friendships. So, $8 m=h(m-7) \\Longrightarrow m h-8 m-7 h=0 \\Longrighta... |
DeepScaleR | [
{
"role": "user",
"content": "An Ultraman is fighting a group of monsters. It is known that Ultraman has one head and two legs. Initially, each monster has two heads and five legs. During the battle, some monsters split, with each splitting monster creating two new monsters, each with one head and six legs ... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 4651,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An [i]animal[/i] with $n$ [i]cells[/i] is a connected figure consisting of $n$ equal-sized cells[1].\n\nA [i]dinosaur[/i] is an animal with at least $2007$ cells. It is said to be [i]primitive[/i] it its cells cannot be partitioned into two or more dinosaurs. Find with proof th... | math | {
"style": "rule",
"ground_truth": "4n-3"
} | {
"index": 4652,
"solution": "\nA **dinosaur** is a **polyomino** having at least 2007 cells that is also **primitive**, meaning it cannot be split into smaller dinosaurs. We need to determine the maximum number of cells in a primitive dinosaur.\n\nTo tackle this problem, let's consider a primitive dinosaur with \\... |
DeepScaleR | [
{
"role": "user",
"content": "An \\( n \\)-digit number \\( x \\) has the following property: if the last digit of \\( x \\) is moved to the front, the result is \\( 2x \\). Find the smallest possible value of \\( n \\)."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 4653,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An abstract animal lives in groups of two and three.\n\nIn a forest, there is one group of two and one group of three. Each day, a new animal arrives in the forest and randomly chooses one of the inhabitants. If the chosen animal belongs to a group of three, that group splits i... | math | {
"style": "rule",
"ground_truth": "4/7"
} | {
"index": 4654,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An abundant number is a positive integer such that the sum of its proper divisors is greater than the number itself. Find the smallest abundant number that is not a multiple of 10."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 4655,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An abundant number is a positive integer such that the sum of its proper divisors is greater than the number itself. The number 12 is an abundant number since $1 + 2 + 3 + 4 + 6 > 12$. What is the smallest abundant number that is not a multiple of 6?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 4656,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An academy has $200$ students and $8$ teachers. The class sizes are as follows: $80, 40, 40, 20, 10, 5, 3, 2$. Calculate the average number of students per class as seen by a randomly picked teacher, represented by $t$, and the average number of students per class from the pers... | math | {
"style": "rule",
"ground_truth": "-25.69"
} | {
"index": 4657,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An acronym XYZ is drawn within a 2x4 rectangular grid with grid lines spaced 1 unit apart. The letter X is formed by two diagonals crossing in a $1 \\times 1$ square. Y consists of a vertical line segment and two slanted segments each forming 45° with the vertical line, making ... | math | {
"style": "rule",
"ground_truth": "5 + 4\\sqrt{2} + \\sqrt{5}"
} | {
"index": 4658,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An acute isosceles triangle, $ABC$, is inscribed in a circle. Through $B$ and $C$, tangents to the circle are drawn, meeting at point $D$. If $\\angle ABC = \\angle ACB = 2 \\angle D$ and $\\angle BAC = k \\pi$ in radians, then find $k$.\n\n[asy]\nimport graph;\n\nunitsize(2 ... | math | {
"style": "rule",
"ground_truth": "3/7"
} | {
"index": 4659,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An acute isosceles triangle, $ABC$, is inscribed in a circle. Through $B$ and $C$, tangents to the circle are drawn, meeting at point $D$. If $\\angle ABC = \\angle ACB = 3 (\\angle D$) and $\\angle BAC = t \\pi$ in radians, then find $t$.\n\n[asy]\nimport graph;\n\nunitsize(2 ... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{11}"
} | {
"index": 4660,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An acute isosceles triangle, $ABC$, is inscribed in a circle. Through $B$ and $C$, tangents to the circle are drawn, meeting at point $D$. If $\\angle ABC = \\angle ACB = 3 \\angle D$ and $\\angle BAC = k \\pi$ in radians, then find $k$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{11}"
} | {
"index": 4661,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An acute isosceles triangle, $ABC$, is inscribed in a circle. Through $B$ and $C$, tangents to the circle are drawn, meeting at point $D$. If $\\angle ABC = \\angle ACB = 3 \\angle D$ and $\\angle BAC = m \\pi$ in radians, determine the value of $m$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{11}"
} | {
"index": 4662,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An acute isosceles triangle, \\( ABC \\), is inscribed in a circle. Through \\( B \\) and \\( C \\), tangents to the circle are drawn, meeting at point \\( D \\). If \\( \\angle ABC = \\angle ACB = 3 \\angle D \\) and \\( \\angle BAC = k \\pi \\) in radians, then find \\( k \\)... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{11}"
} | {
"index": 4663,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An acute triangle $ABC$ is inscribed in a circle of radius 1 with centre $O;$ all the angles of $ABC$ are greater than $45^\\circ.$ \n$B_{1}$ is the foot of perpendicular from $B$ to $CO,$ $B_{2}$ is the foot of perpendicular from $B_{1}$ to $AC.$ \nSimilarly, $C_{1}$ is the fo... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 4664,
"solution": "\nGiven an acute triangle \\( ABC \\) inscribed in a circle with radius 1 and center \\( O \\), where all angles of \\( \\triangle ABC \\) are greater than \\( 45^\\circ \\), we are tasked with finding the circumradius of triangle \\( A_3B_3C_3 \\). The construction is defined as follo... |
DeepScaleR | [
{
"role": "user",
"content": "An airline company is planning to introduce a network of connections between the ten different airports of Sawubonia. The airports are ranked by priority from first to last (with no ties). We call such a network *feasible* if it satisfies the following conditions:\n\n\n\n- All ... | math | {
"style": "rule",
"ground_truth": "512"
} | {
"index": 4665,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An airplane has three sections: First Class (24 seats), Business Class ($25\\%$ of the total number of seats) and Economy ($\\frac{2}{3}$ of the total number of seats). How many seats does the plane have?"
}
] | math | {
"style": "rule",
"ground_truth": "288"
} | {
"index": 4666,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An alarm clock gains 9 minutes each day. When going to bed at 22:00, the precise current time is set on the clock. At what time should the alarm be set so that it rings exactly at 6:00? Explain your answer."
}
] | math | {
"style": "rule",
"ground_truth": "6:03"
} | {
"index": 4667,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An alloy consists of zinc and copper in the ratio of $1:2$, and another alloy contains the same metals in the ratio of $2:3$. How many parts of the two alloys can be combined to obtain a third alloy containing the same metals in the ratio of $17:27$?"
}
] | math | {
"style": "rule",
"ground_truth": "9/35"
} | {
"index": 4668,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An aluminum can in the shape of a cylinder is closed at both ends. Its surface area is $300 \\mathrm{~cm}^{2}$. If the radius of the can were doubled, its surface area would be $900 \\mathrm{~cm}^{2}$. If instead the height of the can were doubled, what would its surface area b... | math | {
"style": "rule",
"ground_truth": "450 \\mathrm{~cm}^{2}"
} | {
"index": 4669,
"solution": "Suppose that the original can has radius $r \\mathrm{~cm}$ and height $h \\mathrm{~cm}$. Since the surface area of the original can is $300 \\mathrm{~cm}^{2}$, then $2 \\pi r^{2}+2 \\pi r h=300$. When the radius of the original can is doubled, its new radius is $2 r \\mathrm{~cm}$, and... |
DeepScaleR | [
{
"role": "user",
"content": "An aluminum cube with an edge length of \\( l = 10 \\) cm is heated to a temperature of \\( t_{1} = 100^{\\circ} \\mathrm{C} \\). After this, it is placed on ice, which has a temperature of \\( t_{2} = 0^{\\circ} \\mathrm{C} \\). Determine the maximum depth to which the cube ca... | math | {
"style": "rule",
"ground_truth": "0.0818"
} | {
"index": 4670,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An aluminum part and a copper part have the same volume. The density of aluminum is $\\rho_{A} = 2700 \\, \\text{kg/m}^3$, and the density of copper is $\\rho_{M} = 8900 \\, \\text{kg/m}^3$. Find the mass of the aluminum part, given that the masses of the parts differ by $\\Del... | math | {
"style": "rule",
"ground_truth": "26.13"
} | {
"index": 4671,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An amoeba is placed in a puddle one day, and on that same day it splits into two amoebas. The next day, each new amoeba splits into two new amoebas, and so on, so that each day every living amoeba splits into two new amoebas. After one week, how many amoebas are in the puddle?... | math | {
"style": "rule",
"ground_truth": "128"
} | {
"index": 4672,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An amoeba is placed in a puddle one day, and on that same day, it splits into two amoebas with a probability of 0.8. Each subsequent day, every amoeba in the puddle has a probability of 0.8 to split into two new amoebas. After one week, assuming no amoebas die, how many amoebas... | math | {
"style": "rule",
"ground_truth": "26.8435456"
} | {
"index": 4673,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An amusement park has a collection of scale models, with a ratio of $1: 20$, of buildings and other sights from around the country. The height of the United States Capitol is $289$ feet. What is the height in feet of its duplicate to the nearest whole number?"
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 4674,
"solution": "1. **Identify the scale ratio and the actual height of the building**: The scale model is built at a ratio of $1:20$, and the actual height of the United States Capitol is $289$ feet.\n\n2. **Calculate the height of the scale model**: To find the height of the scale model, we divide th... |
DeepScaleR | [
{
"role": "user",
"content": "An amusement park has a series of miniature buildings and landscape models from various places in the United States, scaled at $1: 20$. If the height of the United States Capitol Building is 289 feet, what is the nearest integer height of its model in feet?"
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 4675,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An ancient Greek was born on January 7, 40 B.C., and died on January 7, 40 A.D. Calculate the number of years he lived."
}
] | math | {
"style": "rule",
"ground_truth": "79"
} | {
"index": 4676,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An ancient Greek was born on January 7, 40 B.C., and died on January 7, 40 A.D. How many years did he live?"
}
] | math | {
"style": "rule",
"ground_truth": "79"
} | {
"index": 4677,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An ancient civilization has a tribe of 12 members organized hierarchically. The tribe has one main chief, two supporting chiefs (Senior and Junior), and each supporting chief has three inferior officers. If the tribe has 12 members in total, in how many ways can the leadership ... | math | {
"style": "rule",
"ground_truth": "2217600"
} | {
"index": 4678,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An ancient human tribe had a hierarchical system where there existed one chief with 2 supporting chiefs (supporting chief A and supporting chief B), each of whom had 2 equal, inferior officers. If the tribe at one point had 10 members, what is the number of different ways to ch... | math | {
"style": "rule",
"ground_truth": "151200"
} | {
"index": 4679,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An angle $x$ is chosen at random from the interval $0^\\circ < x < 90^\\circ.$ Let $p$ be the probability that the numbers $\\sin^2 x, \\cos^2 x,$ and $\\sin x \\cos x$ are not the lengths of the sides of a triangle. Given that $p = d/n,$ where $d$ is the number of degrees in $... | math | {
"style": "rule",
"ground_truth": "92"
} | {
"index": 4680,
"solution": "Note that the three expressions are symmetric with respect to interchanging $\\sin$ and $\\cos$, and so the probability is symmetric around $45^\\circ$. Thus, take $0 < x < 45$ so that $\\sin x < \\cos x$. Then $\\cos^2 x$ is the largest of the three given expressions and those three l... |
DeepScaleR | [
{
"role": "user",
"content": "An angle can be represented by two uppercase letters on its sides and the vertex letter, such as $\\angle A O B$ (where “ $\\angle$ \" represents an angle), or by $\\angle O$ if the vertex has only one angle. In the triangle $\\mathrm{ABC}$ shown below, $\\angle B A O=\\angle C... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 4681,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An angle is drawn on a set of equally spaced parallel lines as shown. The ratio of the area of shaded region $C$ to the area of shaded region $B$ is 11/5. Find the ratio of shaded region $D$ to the area of shaded region $A.$\n[asy] defaultpen(linewidth(0.7)+fontsize(10)); for(i... | math | {
"style": "rule",
"ground_truth": "408"
} | {
"index": 4682,
"solution": "Let the distances from the apex to the parallel lines be $x$ and $y$ and the distance between the intersections be $a,b.$ We know the area ratio means $\\frac{(x+4a)(y+4b)-(x+3a)(y+3b)}{(x+2a)(y+2b)-(x+a)(y+b)} =\\frac{5}{11}$ which simplifying yields $ab = 3ay+3bx.$ The ratio we seek ... |
DeepScaleR | [
{
"role": "user",
"content": "An ant has one sock and one shoe for each of its six legs, and on one specific leg, both the sock and shoe must be put on last. Find the number of different orders in which the ant can put on its socks and shoes."
}
] | math | {
"style": "rule",
"ground_truth": "10!"
} | {
"index": 4683,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An ant makes a sequence of moves on a cube where a move consists of walking from one vertex to an adjacent vertex along an edge of the cube. Initially the ant is at a vertex of the bottom face of the cube and chooses one of the three adjacent vertices to move to as its first mo... | math | {
"style": "rule",
"ground_truth": "49"
} | {
"index": 4684,
"solution": "Note that we don't care which exact vertex the ant is located at, just which level (either top face or bottom face). Consider the ant to be on any of the two levels and having moved at least one move. Define $p_i$ to be the probability that after $i$ moves, the ant ends up on the level... |
DeepScaleR | [
{
"role": "user",
"content": "An ant moves on the following lattice, beginning at the dot labeled $A$. Each minute he moves to one of the dots neighboring the dot he was at, choosing from among its neighbors at random. What is the probability that after 5 minutes he is at the dot labeled $B$? [asy]\ndraw((-... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 4685,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An ant situated at point \\( A \\) decides to walk 1 foot east, then \\( \\frac{1}{2} \\) foot northeast, then \\( \\frac{1}{4} \\) foot east, then \\( \\frac{1}{8} \\) foot northeast, then \\( \\frac{1}{16} \\) foot east, and so on (that is, the ant travels alternately between... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3} \\sqrt{2 \\sqrt{2}+5}"
} | {
"index": 4686,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An ant starts at one vertex of an octahedron and moves along the edges according to a similar rule: at each vertex, the ant chooses one of the four available edges with equal probability, and all choices are independent. What is the probability that after six moves, the ant end... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{128}"
} | {
"index": 4687,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An ant starts at the origin of a coordinate plane. Each minute, it either walks one unit to the right or one unit up, but it will never move in the same direction more than twice in the row. In how many different ways can it get to the point $(5,5)$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "84"
} | {
"index": 4688,
"solution": "We can change the ant's sequence of moves to a sequence $a_{1}, a_{2}, \\ldots, a_{10}$, with $a_{i}=0$ if the $i$-th step is up, and $a_{i}=1$ if the $i$-th step is right. We define a subsequence of moves $a_{i}, a_{i+1}, \\ldots, a_{j}$, ( $i \\leq j$ ) as an up run if all terms of t... |
DeepScaleR | [
{
"role": "user",
"content": "An ant starts at the origin, facing in the positive $x$-direction. Each second, it moves 1 unit forward, then turns counterclockwise by $\\sin ^{-1}\\left(\\frac{3}{5}\\right)$ degrees. What is the least upper bound on the distance between the ant and the origin? (The least upp... | math | {
"style": "rule",
"ground_truth": "\\sqrt{10}"
} | {
"index": 4689,
"solution": "We claim that the points the ant visits lie on a circle of radius $\\frac{\\sqrt{10}}{2}$. We show this by saying that the ant stays a constant distance $\\frac{\\sqrt{10}}{2}$ from the point $\\left(\\frac{1}{2}, \\frac{3}{2}\\right)$. Suppose the ant moves on a plane $P$. Consider a ... |
DeepScaleR | [
{
"role": "user",
"content": "An ant starts at the point $(0,0)$ in the Cartesian plane. In the first minute, the ant faces towards $(1,0)$ and walks one unit. Each subsequent minute, the ant chooses an angle $\\theta$ uniformly at random in the interval $\\left[-90^{\\circ}, 90^{\\circ}\\right]$, and then ... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 4690,
"solution": "Let $\\alpha_{k}$ be a random variable that represents the turn made after step $k$, choosing $\\alpha_{k}$ uniformly at random on the complex plane among the arc of the unit circle containing 1 from $-i$ to $i$. It is well known that $\\mathbb{E}\\left[\\alpha_{k}\\right]=\\frac{2}{\\... |
DeepScaleR | [
{
"role": "user",
"content": "An ant starts at the point \\((1,0)\\). Each minute, it walks from its current position to one of the four adjacent lattice points until it reaches a point \\((x, y)\\) with \\(|x|+|y| \\geq 2\\). What is the probability that the ant ends at the point \\((1,1)\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "7/24"
} | {
"index": 4691,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An ant starts from vertex \\( A \\) of rectangular prism \\( ABCD-A_1B_1C_1D_1 \\) and travels along the surface to reach vertex \\( C_1 \\) with the shortest distance being 6. What is the maximum volume of the rectangular prism?"
}
] | math | {
"style": "rule",
"ground_truth": "12\\sqrt{3}"
} | {
"index": 4692,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An ant starts out at $(0,0)$. Each second, if it is currently at the square $(x, y)$, it can move to $(x-1, y-1),(x-1, y+1),(x+1, y-1)$, or $(x+1, y+1)$. In how many ways can it end up at $(2010,2010)$ after 4020 seconds?"
}
] | math | {
"style": "rule",
"ground_truth": "$\\binom{4020}{1005}^{2}$"
} | {
"index": 4693,
"solution": "Note that each of the coordinates either increases or decreases the x and y coordinates by 1. In order to reach 2010 after 4020 steps, each of the coordinates must be increased 3015 times and decreased 1005 times. A permutation of 3015 plusses and 1005 minuses for each of $x$ and $y$ u... |
DeepScaleR | [
{
"role": "user",
"content": "An apartment and an office are sold for $15,000 each. The apartment was sold at a loss of 25% and the office at a gain of 25%. Determine the net effect of the transactions."
}
] | math | {
"style": "rule",
"ground_truth": "2000"
} | {
"index": 4694,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An apartment building consists of 20 rooms numbered $1,2, \\ldots, 20$ arranged clockwise in a circle. To move from one room to another, one can either walk to the next room clockwise (i.e. from room $i$ to room $(i+1)(\\bmod 20))$ or walk across the center to the opposite room... | math | {
"style": "rule",
"ground_truth": "257"
} | {
"index": 4695,
"solution": "One way is to walk directly from room 10 to 20 . Else, divide the rooms into 10 pairs $A_{0}=(10,20), A_{1}=(1,11), A_{2}=(2,12), \\ldots, A_{9}=(9,19)$. Notice that - each move is either between rooms in $A_{i}$ and $A_{(i+1)(\\bmod 10)}$ for some $i \\in\\{0,1, \\ldots, 9\\}$, or bet... |
DeepScaleR | [
{
"role": "user",
"content": "An arbitrary point \\( E \\) inside the square \\( ABCD \\) with side length 1 is connected by line segments to its vertices. Points \\( P, Q, F, \\) and \\( T \\) are the points of intersection of the medians of triangles \\( BCE, CDE, DAE, \\) and \\( ABE \\) respectively. Fi... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{9}"
} | {
"index": 4696,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An archipelago consists of $N \\geqslant 7$ islands. Any two islands are connected by at most one bridge. It is known that no more than 5 bridges lead from each island, and among any 7 islands, there are always at least two connected by a bridge. What is the maximum possible va... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 4697,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An archipelago consists of \\( N \\geq 7 \\) islands. Any two islands are connected by at most one bridge. It is known that no more than 5 bridges lead from each island, and among any 7 islands, there are always two that are connected by a bridge. What is the largest possible v... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 4698,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An archipelago consists of \\( N \\geq 7 \\) islands. Any two islands are connected by no more than one bridge. It is known that no more than 5 bridges lead from each island, and among any 7 islands, there are necessarily two that are connected by a bridge. What is the maximum ... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 4699,
"solution": ""
} |
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