data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "An equilateral triangle is originally painted black. Each time the triangle is changed, the middle fourth of each black triangle turns white. After five changes, what fractional part of the original area of the black triangle remains black?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{243}{1024}"
} | {
"index": 4800,
"solution": "1. **Understanding the Problem**: We start with an equilateral triangle that is entirely black. Each time a change occurs, the middle fourth of each black triangle turns white. We need to determine the fraction of the original triangle that remains black after five changes.\n\n2. **Ana... |
DeepScaleR | [
{
"role": "user",
"content": "An equilateral triangle is placed on top of a square with each side of the square equal to one side of the triangle, forming a pentagon. What percent of the area of the pentagon is the area of the equilateral triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt{3} - 3}{13} \\times 100\\%"
} | {
"index": 4801,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An equilateral triangle is subdivided into 4 smaller equilateral triangles. Using red and yellow to paint the vertices of the triangles, each vertex must be colored and only one color can be used per vertex. If two colorings are considered the same when they can be made identic... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 4802,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An equilateral triangle lies in the Cartesian plane such that the $x$-coordinates of its vertices are pairwise distinct and all satisfy the equation $x^{3}-9 x^{2}+10 x+5=0$. Compute the side length of the triangle."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{17}"
} | {
"index": 4803,
"solution": "Let three points be $A, B$, and $C$ with $x$-coordinates $a, b$, and $c$, respectively. Let the circumcircle of $\\triangle A B C$ meet the line $y=b$ at point $P$. Then, we have $\\angle B P C=60^{\\circ} \\Longrightarrow P C=$ $\\frac{2}{\\sqrt{3}}(c-b)$. Similarly, $A P=\\frac{2}{\\... |
DeepScaleR | [
{
"role": "user",
"content": "An equilateral triangle of side length $ n$ is divided into unit triangles. Let $ f(n)$ be the number of paths from the triangle in the top row to the middle triangle in the bottom row, such that adjacent triangles in a path share a common edge and the path never travels up... | math | {
"style": "rule",
"ground_truth": "2005!"
} | {
"index": 4804,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An equilateral triangle of side length $10$ is completely filled in by non-overlapping equilateral triangles of side length $1$. How many small triangles are required?"
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 4805,
"solution": "To solve this problem, we need to determine how many small equilateral triangles of side length $1$ can fit into a larger equilateral triangle of side length $10$.\n\n#### Method 1: Summing the Number of Triangles in Each Row\n1. **Visualize the Triangle**: Imagine the large triangle d... |
DeepScaleR | [
{
"role": "user",
"content": "An equilateral triangle of side length $12$ is completely filled by non-overlapping equilateral triangles of side length $2$."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 4806,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An equilateral triangle shares a common side with a square as shown. What is the number of degrees in $m\\angle CDB$? [asy] pair A,E,C,D,B; D = dir(60); C = dir(0); E = (0,-1); B = C+E; draw(B--D--C--B--E--A--C--D--A); label(\"D\",D,N); label(\"C\",C,dir(0)); label(\"B\",B,dir(... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 4807,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An equilateral triangle with a side length of 1 is cut along a line parallel to one of its sides, resulting in a trapezoid. Let $S = \\frac{\\text{(perimeter of the trapezoid)}^2}{\\text{area of the trapezoid}}$. Find the minimum value of $S$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{32\\sqrt{3}}{3}"
} | {
"index": 4808,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An equilateral triangle with side length $12$ is completely filled with non-overlapping equilateral triangles of side length $2$. Calculate the number of small triangles required."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 4809,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An equilateral triangle with side length 10 cm is inscribed in a circle. The triangle's side is the diameter of the circle. In two opposite sectors of the circle, there are small shaded regions. If the sum of the areas of the two small shaded regions is in the form \\(a\\pi - b... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 4810,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An equivalent of the expression\n$\\left(\\frac{x^2+1}{x}\\right)\\left(\\frac{y^2+1}{y}\\right)+\\left(\\frac{x^2-1}{y}\\right)\\left(\\frac{y^2-1}{x}\\right)$, $xy \\not= 0$,\nis:"
}
] | math | {
"style": "rule",
"ground_truth": "2xy+\\frac{2}{xy}"
} | {
"index": 4811,
"solution": "1. **Start by rewriting the expression**:\n \\[\n \\left(\\frac{x^2+1}{x}\\right)\\left(\\frac{y^2+1}{y}\\right)+\\left(\\frac{x^2-1}{y}\\right)\\left(\\frac{y^2-1}{x}\\right)\n \\]\n\n2. **Simplify each term separately**:\n - For the first term:\n \\[\n \\left(\\frac{x... |
DeepScaleR | [
{
"role": "user",
"content": "An error of $.05''$ is made in the measurement of a line $25''$ long, and an error of $.3''$ is made in the measurement of a line $150''$ long. In comparison with the relative error of the first measurement, find the relative error of the second measurement."
}
] | math | {
"style": "rule",
"ground_truth": "0.2\\%"
} | {
"index": 4812,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An escalator has \\( n \\) visible steps and descends at a constant speed. Two boys, \\( A \\) and \\( Z \\), walk down the moving escalator at a steady pace. Boy \\( A \\) walks twice as many steps per minute as boy \\( Z \\). \\( A \\) reaches the bottom after walking 27 step... | math | {
"style": "rule",
"ground_truth": "54"
} | {
"index": 4813,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An ethnographer determined that in a primitive tribe he studied, the distribution of lifespan among tribe members can be described as follows: 25% live only up to 40 years, 50% die at 50 years, and 25% live to 60 years. He then randomly selected two individuals to study in more... | math | {
"style": "rule",
"ground_truth": "53.75"
} | {
"index": 4814,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An even perfect square in the decimal system is of the form: $\\overline{a b 1 a b}$. What is this perfect square?"
}
] | math | {
"style": "rule",
"ground_truth": "76176"
} | {
"index": 4815,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An event occurs many years ago. It occurs periodically in $x$ consecutive years, then there is a break of $y$ consecutive years. We know that the event occured in $1964$ , $1986$ , $1996$ , $2008$ and it didn't occur in $1976$ , $1993$ , $2006$ , $2013$ . What is t... | math | {
"style": "rule",
"ground_truth": "2018"
} | {
"index": 4816,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An exchange point conducts two types of operations:\n\n1) Give 2 euros - receive 3 dollars and a candy as a gift;\n2) Give 5 dollars - receive 3 euros and a candy as a gift.\n\nWhen the wealthy Buratino came to the exchange point, he had only dollars. When he left, he had fewer... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 4817,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An experimenter is conducting an experiment that involves implementing five procedures in sequence. Procedure A must only occur either as the first or the last step, and procedures C and D must be implemented consecutively. The number of possible arrangements for the sequence o... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 4818,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An experimenter selects 4 out of 8 different chemical substances to place in 4 distinct bottles. If substances A and B should not be placed in bottle 1, the number of different ways of arranging them is ____."
}
] | math | {
"style": "rule",
"ground_truth": "1260"
} | {
"index": 4819,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An geometric sequence $\\{a_n\\}$ has 20 terms, where the product of the first four terms is $\\frac{1}{128}$, and the product of the last four terms is 512. The product of all terms in this geometric sequence is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 4820,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An ice ballerina rotates at a constant angular velocity at one particular point. That is, she does not translationally move. Her arms are fully extended as she rotates. Her moment of inertia is $I$ . Now, she pulls her arms in and her moment of inertia is now $\\frac{7}{10}I$... | math | {
"style": "rule",
"ground_truth": "$\\dfrac{10}{7}$"
} | {
"index": 4821,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An ice cream cone has radius 1 inch and height 4 inches, What is the number of inches in the radius of a sphere of ice cream which has the same volume as the cone?"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 4822,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An ice cream shop offers 6 kinds of ice cream. What is the greatest number of two scoop sundaes that can be made such that each sundae contains two types of ice cream and no two sundaes are the same combination?"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 4823,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An ice cream shop offers 8 different flavors of ice cream. What is the greatest number of sundaes that can be made if each sundae can consist of 1, 2, or 3 scoops, with each scoop possibly being a different type of ice cream and no two sundaes having the same combination of fla... | math | {
"style": "rule",
"ground_truth": "92"
} | {
"index": 4824,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An icosahedron is a regular polyhedron with twenty faces, all of which are equilateral triangles. If an icosahedron is rotated by $\\theta$ degrees around an axis that passes through two opposite vertices so that it occupies exactly the same region of space as before, what is t... | math | {
"style": "rule",
"ground_truth": "72^{\\circ}"
} | {
"index": 4825,
"solution": "Because this polyhedron is regular, all vertices must look the same. Let's consider just one vertex. Each triangle has a vertex angle of $60^{\\circ}$, so we must have fewer than 6 triangles; if we had 6 , there would be $360^{\\circ}$ at each vertex and you wouldn't be able to \"fold\... |
DeepScaleR | [
{
"role": "user",
"content": "An icosidodecahedron is a convex polyhedron with 20 triangular faces and 12 pentagonal faces. How many vertices does it have?"
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 4826,
"solution": "Since every edge is shared by exactly two faces, there are $(20 \\cdot 3+12 \\cdot 5) / 2=60$ edges. Using Euler's formula $v-e+f=2$, we see that there are 30 vertices."
} |
DeepScaleR | [
{
"role": "user",
"content": "An ideal gas is used as the working substance of a heat engine operating cyclically. The cycle consists of three stages: isochoric pressure reduction from $3 P_{0}$ to $P_{0}$, isobaric density increase from $\\rho_{0}$ to $3 \\rho_{0}$, and a return to the initial state, repre... | math | {
"style": "rule",
"ground_truth": "1/9"
} | {
"index": 4827,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An infantry column stretched over 1 km. Sergeant Kim, riding a gyroscooter from the end of the column, reached its front and then returned to the end. During this time, the infantrymen covered 2 km 400 m. What distance did the sergeant travel during this time?"
}
] | math | {
"style": "rule",
"ground_truth": "3.6"
} | {
"index": 4828,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An infinite arithmetic progression of positive integers contains the terms 7, 11, 15, 71, 75, and 79. The first term in the progression is 7. Kim writes down all the possible values of the one-hundredth term in the progression. What is the sum of the numbers Kim writes down?"
... | math | {
"style": "rule",
"ground_truth": "714"
} | {
"index": 4829,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An infinite geometric series has a first term of $12$ and a second term of $4.$ A second infinite geometric series has the same first term of $12,$ a second term of $4+n,$ and a sum of four times that of the first series. Find the value of $n.$"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 4830,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An infinite geometric series has a first term of $15$ and a second term of $5$. A second infinite geometric series has the same first term of $15$, a second term of $5+n$, and a sum of three times that of the first series. Find the value of $n$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{20}{3}"
} | {
"index": 4831,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An infinite geometric series has a first term of $540$ and a sum of $4500$. What is its common ratio, and what is the second term of the series?"
}
] | math | {
"style": "rule",
"ground_truth": "475.2"
} | {
"index": 4832,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An infinite geometric series has a first term of \\( 416 \\) and a sum of \\( 3120 \\). What is its common ratio?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{84}{97}"
} | {
"index": 4833,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An infinite geometric series has a first term of \\( 512 \\) and its sum is \\( 8000 \\). What is its common ratio?"
}
] | math | {
"style": "rule",
"ground_truth": "0.936"
} | {
"index": 4834,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An infinite geometric series has a sum of 2020. If the first term, the third term, and the fourth term form an arithmetic sequence, find the first term."
}
] | math | {
"style": "rule",
"ground_truth": "1010(1+\\sqrt{5})"
} | {
"index": 4835,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An infinite geometric series has common ratio $-1/5$ and sum $16.$ What is the first term of the series?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{96}{5}"
} | {
"index": 4836,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An infinite geometric series has common ratio $1/8$ and sum 60. What is the first term of the series?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{105}{2}"
} | {
"index": 4837,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An infinite geometric series has common ratio $\\frac{-1}{3}$ and sum $25.$ What is the second term of the sequence?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{-100}{9}"
} | {
"index": 4838,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An infinite geometric series has first term $328$ and a sum of $2009$. What is its common ratio?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{41}{49}"
} | {
"index": 4839,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An infinite geometric series has sum 2005. A new series, obtained by squaring each term of the original series, has 10 times the sum of the original series. The common ratio of the original series is $\\frac mn$ where $m$ and $n$ are relatively prime integers. Find $m+n.$"
}
... | math | {
"style": "rule",
"ground_truth": "802"
} | {
"index": 4840,
"solution": "Let's call the first term of the original geometric series $a$ and the common ratio $r$, so $2005 = a + ar + ar^2 + \\ldots$. Using the sum formula for infinite geometric series, we have $\\;\\;\\frac a{1 -r} = 2005$. Then we form a new series, $a^2 + a^2 r^2 + a^2 r^4 + \\ldots$. We k... |
DeepScaleR | [
{
"role": "user",
"content": "An infinite sequence $ \\,x_{0},x_{1},x_{2},\\ldots \\,$ of real numbers is said to be [b]bounded[/b] if there is a constant $ \\,C\\,$ such that $ \\, \\vert x_{i} \\vert \\leq C\\,$ for every $ \\,i\\geq 0$. Given any real number $ \\,a > 1,\\,$ construct a bounded infinite s... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 4841,
"solution": "\nTo solve this problem, we need to construct a bounded sequence of real numbers \\( x_0, x_1, x_2, \\ldots \\) such that for any two distinct nonnegative integers \\( i \\) and \\( j \\), the condition\n\n\\[\n|x_i - x_j| \\cdot |i - j|^a \\geq 1\n\\]\n\nis satisfied, given \\( a > 1 ... |
DeepScaleR | [
{
"role": "user",
"content": "An infinite sequence of circles is composed such that each circle has a decreasing radius, and each circle touches the subsequent circle and the two sides of a given right angle. The ratio of the area of the first circle to the sum of the areas of all subsequent circles in the ... | math | {
"style": "rule",
"ground_truth": "$(16+12 \\sqrt{2}): 1$"
} | {
"index": 4842,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An infinite sequence of real numbers $a_{1}, a_{2}, \\ldots$ satisfies the recurrence $$a_{n+3}=a_{n+2}-2 a_{n+1}+a_{n}$$ for every positive integer $n$. Given that $a_{1}=a_{3}=1$ and $a_{98}=a_{99}$, compute $a_{1}+a_{2}+\\cdots+a_{100}$."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 4843,
"solution": "A quick telescope gives that $a_{1}+\\cdots+a_{n}=2 a_{1}+a_{3}+a_{n-1}-a_{n-2}$ for all $n \\geq 3$: $$\\begin{aligned} \\sum_{k=1}^{n} a_{k} & =a_{1}+a_{2}+a_{3}+\\sum_{k=1}^{n-3}\\left(a_{k}-2 a_{k+1}+2 a_{k+2}\\right) \\\\ & =a_{1}+a_{2}+a_{3}+\\sum_{k=1}^{n-3} a_{k}-2 \\sum_{k=2}^... |
DeepScaleR | [
{
"role": "user",
"content": "An infinite sheet of paper is divided into equal squares, some of which are colored red. In each $2\\times3$ rectangle, there are exactly two red squares. Now consider an arbitrary $9\\times11$ rectangle. How many red squares does it contain? (The sides of all considered re... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"index": 4844,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An influenza outbreak occurred in regions $A$, $B$, and $C$. The percentages of people with influenza in these three regions are $6\\%$, $5\\%$, and $4\\%$ respectively. It is assumed that the populations of these three regions are in the ratio of $3:5:2$. Now, a person is rand... | math | {
"style": "rule",
"ground_truth": "\\frac{18}{51}"
} | {
"index": 4845,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An influenza outbreak occurred in three areas, $A$, $B$, and $C$, where $6\\%$, $5\\%$, and $4\\%$ of the population in each area have the flu, respectively. Assuming the population ratios in these three areas are $6:5:4$, if a person is randomly selected from these three areas... | math | {
"style": "rule",
"ground_truth": "\\frac{77}{1500}"
} | {
"index": 4846,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An inscribed circle is drawn inside isosceles trapezoid \\(ABCD\\) with \\(AB = CD\\). Let \\(M\\) be the point where the circle touches side \\(CD\\), \\(K\\) be the intersection point of the circle with segment \\(AM\\), and \\(L\\) be the intersection point of the circle wit... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 4847,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An inscribed dodecagon. A convex dodecagon is inscribed in a circle. The lengths of some six sides of the dodecagon are $\\sqrt{2}$, and the lengths of the remaining six sides are $\\sqrt{24}$. What is the radius of the circle?"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{38}"
} | {
"index": 4848,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An insect has just told me that she has laid $154_6$ eggs. In base 10, how many eggs did she lay?"
}
] | math | {
"style": "rule",
"ground_truth": "70"
} | {
"index": 4849,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An insect lives on the surface of a regular tetrahedron with edges of length 1. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge. What is the length of the shortest such trip? (Note: Two edges of a tetrahed... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 4850,
"solution": "1. **Understanding the Problem**: We need to find the shortest path on the surface of a regular tetrahedron from the midpoint of one edge to the midpoint of an opposite edge. The tetrahedron has edges of length 1.\n\n2. **Unfolding the Tetrahedron**: To visualize the problem, we can un... |
DeepScaleR | [
{
"role": "user",
"content": "An integer $n>1$ is given . Find the smallest positive number $m$ satisfying the following conditions： for any set $\\{a,b\\}$ $\\subset \\{1,2,\\cdots,2n-1\\}$ ,there are non-negative integers $ x, y$ ( not all zero) such that $2n|ax+by$ and $x+y\\leq m.$"
}
] | math | {
"style": "rule",
"ground_truth": "n"
} | {
"index": 4851,
"solution": "\nGiven an integer \\( n > 1 \\), we aim to find the smallest positive number \\( m \\) satisfying the following conditions: for any set \\(\\{a, b\\} \\subset \\{1, 2, \\ldots, 2n-1\\}\\), there exist non-negative integers \\( x \\) and \\( y \\) (not both zero) such that \\( 2n \\mid... |
DeepScaleR | [
{
"role": "user",
"content": "An integer $N$ is selected at random in the range $1 \\leq N \\leq 2020$. What is the probability that the remainder when $N^{16}$ is divided by $5$ is $1$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{5}"
} | {
"index": 4852,
"solution": "To solve this problem, we will use modular arithmetic to determine the remainder when $N^{16}$ is divided by $5$ for each possible residue class of $N$ modulo $5$. We then count the number of cases where this remainder is $1$.\n\n1. **Case Analysis for Each Residue Class Modulo $5$:**\... |
DeepScaleR | [
{
"role": "user",
"content": "An integer $N$ is selected at random in the range $1 \\leq N \\leq 2030$. Calculate the probability that the remainder when $N^{12}$ is divided by $7$ is $1$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{6}{7}"
} | {
"index": 4853,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An integer $n$ is chosen uniformly at random from the set $\\{1,2,3, \\ldots, 2023!\\}$. Compute the probability that $$\\operatorname{gcd}\\left(n^{n}+50, n+1\\right)=1$$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{265}{357}"
} | {
"index": 4854,
"solution": "If $n$ is even, we need $\\operatorname{gcd}(n+1,51)=1$. If $n$ is odd, we need $\\operatorname{gcd}(n+1,49)=1$. Thus, the answer is $$\\frac{1}{2}\\left(\\frac{\\varphi(49)}{49}+\\frac{\\varphi(51)}{51}\\right)=\\frac{265}{357}$$"
} |
DeepScaleR | [
{
"role": "user",
"content": "An integer $n$ is decreased by 2 and then multiplied by 5. If the result is 85, what is the value of $n$?"
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 4855,
"solution": "We undo each of the operations in reverse order. The final result, 85, was obtained by multiplying a number by 5. This number was $85 \\div 5=17$. The number 17 was obtained by decreasing $n$ by 2. Thus, $n=17+2=19$."
} |
DeepScaleR | [
{
"role": "user",
"content": "An integer $n$ is said to be square-free if the only perfect square that divides $n$ is $1^2$. How many positive odd integers greater than 1 and less than $100$ are square-free?"
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 4856,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An integer $x$ is chosen so that $3 x+1$ is an even integer. Which of the following must be an odd integer?"
}
] | math | {
"style": "rule",
"ground_truth": "7x+4"
} | {
"index": 4857,
"solution": "If $x$ is an integer for which $3 x+1$ is even, then $3 x$ is odd, since it is 1 less than an even integer. If $3 x$ is odd, then $x$ must be odd (since if $x$ is even, then $3 x$ would be even). If $x$ is odd, then $7 x$ is odd (odd times odd equals odd) and so $7 x+4$ is odd (odd plu... |
DeepScaleR | [
{
"role": "user",
"content": "An integer $x$ is chosen so that $3x+1$ is an even integer. Which of the following must be an odd integer? (A) $x+3$ (B) $x-3$ (C) $2x$ (D) $7x+4$ (E) $5x+3$"
}
] | math | {
"style": "rule",
"ground_truth": "7x+4"
} | {
"index": 4858,
"solution": "Solution 1: If $x=1$, then $3x+1=4$, which is an even integer. In this case, the five given choices are (A) $x+3=4$, (B) $x-3=-2$, (C) $2x=2$, (D) $7x+4=11$, (E) $5x+3=8$. Of these, the only odd integer is (D). Therefore, since $x=1$ satisfies the initial criteria, then (D) must be the... |
DeepScaleR | [
{
"role": "user",
"content": "An integer between $1000$ and $9999$, inclusive, is called balanced if the sum of its two leftmost digits equals the sum of its two rightmost digits. How many balanced integers are there?"
}
] | math | {
"style": "rule",
"ground_truth": "615"
} | {
"index": 4859,
"solution": "If the common sum of the first two and last two digits is $n$, such that $1 \\leq n \\leq 9$, there are $n$ choices for the first two digits and $n + 1$ choices for the second two digits (since zero may not be the first digit). This gives $\\sum_{n = 1}^9 n(n + 1) = 330$ balanced numbe... |
DeepScaleR | [
{
"role": "user",
"content": "An integer between $1000$ and $9999$, inclusive, is chosen at random. What is the probability that it is an odd integer whose digits are all distinct?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{56}{225}"
} | {
"index": 4860,
"solution": "To solve this problem, we need to calculate the total number of integers between $1000$ and $9999$ and then find how many of these integers meet the given conditions (odd and all digits distinct).\n\n1. **Total number of integers between $1000$ and $9999$:**\n - These integers are al... |
DeepScaleR | [
{
"role": "user",
"content": "An integer greater than 9 and less than 100 is randomly chosen. What is the probability that its digits are different?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{10}"
} | {
"index": 4861,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An integer has exactly 4 prime factors, and the sum of the squares of these factors is 476. Find this integer."
}
] | math | {
"style": "rule",
"ground_truth": "1989"
} | {
"index": 4862,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An integer is called \"octal\" if it is divisible by $8$ or if at least one of its digits is $8$ . \nHow many integers between $1$ and $100$ are octal?"
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 4863,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An integer is called a \"good number\" if it has 8 positive divisors and the sum of these 8 positive divisors is 3240. For example, 2006 is a good number because the sum of its divisors 1, 2, 17, 34, 59, 118, 1003, and 2006 is 3240. Find the smallest good number."
}
] | math | {
"style": "rule",
"ground_truth": "1614"
} | {
"index": 4864,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An integer is called parity-monotonic if its decimal representation $a_{1}a_{2}a_{3}\\cdots a_{k}$ satisfies $a_{i}<a_{i+1}$ if $a_{i}$ is odd, and $a_{i}>a_{i+1}$ if $a_{i}$ is even. How many four-digit parity-monotonic integers are there?"
}
] | math | {
"style": "rule",
"ground_truth": "640"
} | {
"index": 4865,
"solution": "This problem can be solved via recursion since we are \"building a string\" of numbers with some condition. We want to create a new string by adding a new digit at the front so we avoid complications($0$ can't be at the front and no digit is less than $9$). There are $4$ options to add... |
DeepScaleR | [
{
"role": "user",
"content": "An integer is called snakelike if its decimal representation $a_1a_2a_3\\cdots a_k$ satisfies $a_i<a_{i+1}$ if $i$ is odd and $a_i>a_{i+1}$ if $i$ is even. How many snakelike integers between 1000 and 9999 have four distinct digits?"
}
] | math | {
"style": "rule",
"ground_truth": "882"
} | {
"index": 4866,
"solution": "Let's create the snakelike number from digits $a < b < c < d$, and, if we already picked the digits there are 5 ways to do so, as said in the first solution. And, let's just pick the digits from 0-9. This get's a total count of $5\\cdot{10 \\choose 4}$ But, this over-counts since it co... |
DeepScaleR | [
{
"role": "user",
"content": "An integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition.\n[quote]For example, 4 can be partitioned in five distinct ways:\n4\n3 + 1\n2 + 2\n2 + 1 + 1\n1 + 1 + 1 + 1[/qu... | math | {
"style": "rule",
"ground_truth": "1, 3, 5"
} | {
"index": 4867,
"solution": "\nWe need to determine all positive integers \\( n \\) such that \n\\[\np(n) + p(n+4) = p(n+2) + p(n+3),\n\\]\nwhere \\( p(n) \\) denotes the partition function, which counts the number of ways \\( n \\) can be partitioned into positive integers.\n\nTo solve this, we consider the equiv... |
DeepScaleR | [
{
"role": "user",
"content": "An integer, whose decimal representation reads the same left to right and right to left, is called symmetrical. For example, the number 513151315 is symmetrical, while 513152315 is not. How many nine-digit symmetrical numbers exist such that adding the number 11000 to them leav... | math | {
"style": "rule",
"ground_truth": "8100"
} | {
"index": 4868,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An international scientific research cooperation project has been jointly developed and completed by two Americans, one Frenchman, and one Chinese. Now, two individuals are randomly selected to announce the results. The probability that the selected pair includes the Chinese in... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 4869,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An inverted cone with base radius $12 \\mathrm{cm}$ and height $18 \\mathrm{cm}$ is full of water. The water is poured into a tall cylinder whose horizontal base has radius of $24 \\mathrm{cm}$. What is the height in centimeters of the water in the cylinder?"
}
] | math | {
"style": "rule",
"ground_truth": "1.5"
} | {
"index": 4870,
"solution": "1. **Calculate the volume of the water in the cone**: \n The formula for the volume $V$ of a cone is given by:\n \\[\n V = \\frac{1}{3} \\pi r^2 h\n \\]\n where $r$ is the radius of the base and $h$ is the height of the cone. For the given cone, $r = 12 \\text{ cm}$ and $h =... |
DeepScaleR | [
{
"role": "user",
"content": "An inverted frustum with a bottom diameter of 12 and height of 18, filled with water, is emptied into another cylindrical container with a bottom diameter of 24. Assuming the cylindrical container is sufficiently tall, what will be the height of the water level in the cylindric... | math | {
"style": "rule",
"ground_truth": "1.5"
} | {
"index": 4871,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An investment of $\\$$10,000 is made in a government bond that will pay 6$\\%$ interest compounded annually. At the end of five years, what is the total number of dollars in this investment? Express your answer to the nearest whole number."
}
] | math | {
"style": "rule",
"ground_truth": "13382"
} | {
"index": 4872,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An investor has an open brokerage account with an investment company. In 2021, the investor received the following income from securities:\n\n- Dividends from shares of the company PAO “Winning” amounted to 50,000 rubles.\n- Coupon income from government bonds OFZ amounted to 4... | math | {
"style": "rule",
"ground_truth": "11050"
} | {
"index": 4873,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An investor placed $\\$15,000$ in a three-month term deposit that offered a simple annual interest rate of $8\\%$. After three months, the total value of the deposit was reinvested in another three-month term deposit. After these three months, the investment amounted to $\\$15,... | math | {
"style": "rule",
"ground_truth": "11.36"
} | {
"index": 4874,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An investor placed \\$12,000 in a one-year fixed deposit that yielded a simple annual interest rate of 8%. After one year, the total amount was reinvested in another one-year deposit. At the end of the second year, the total amount was \\$13,500. If the annual interest rate of ... | math | {
"style": "rule",
"ground_truth": "4.17\\%"
} | {
"index": 4875,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An investor put $\\$12,\\!000$ into a three-month savings certificate that paid a simple annual interest rate of $8\\%$. After three months, she reinvested the resulting amount in another three-month certificate. The investment value after another three months was $\\$12,\\!435... | math | {
"style": "rule",
"ground_truth": "6.38\\%"
} | {
"index": 4876,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An island has $10$ cities, where some of the possible pairs of cities are connected by roads. A *tour route* is a route starting from a city, passing exactly eight out of the other nine cities exactly once each, and returning to the starting city. (In other words, it is a loo... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 4877,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An isosceles right triangle $A B C$ has area 1. Points $D, E, F$ are chosen on $B C, C A, A B$ respectively such that $D E F$ is also an isosceles right triangle. Find the smallest possible area of $D E F$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"index": 4878,
"solution": "Without loss of generality, suppose that $A B$ is the hypotenuse. If $F$ is the right angle, then $F$ must be the midpoint of $A B$. To prove this, let $X$ and $Y$ be the feet from $F$ to $B C$ and $A C$. Since $\\angle X F Y=\\angle D F E=90^{\\circ}$, we have $\\angle X F D=\\angle Y... |
DeepScaleR | [
{
"role": "user",
"content": "An isosceles right triangle has a leg length of 36 units. Starting from the right angle vertex, an infinite series of equilateral triangles is drawn consecutively on one of the legs. Each equilateral triangle is inscribed such that their third vertices always lie on the hypoten... | math | {
"style": "rule",
"ground_truth": "324"
} | {
"index": 4879,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An isosceles right triangle is removed from each corner of a square piece of paper to form a rectangle. If $AB = 15$ units in the new configuration, what is the combined area of the four removed triangles?"
}
] | math | {
"style": "rule",
"ground_truth": "112.5"
} | {
"index": 4880,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An isosceles right triangle is removed from each corner of a square piece of paper, as shown, to create a rectangle. If $AB = 12$ units, what is the combined area of the four removed triangles, in square units? [asy]\nunitsize(5mm);\ndefaultpen(linewidth(.7pt)+fontsize(8pt));\n... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 4881,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An isosceles right triangle with legs of length $8$ is partitioned into $16$ congruent triangles as shown. The shaded area is"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 4882,
"solution": "\n#### Step 1: Understand the problem\nWe are given an isosceles right triangle with legs of length $8$. This triangle is partitioned into $16$ congruent smaller triangles, and we need to find the area of the shaded region which consists of $10$ of these smaller triangles.\n\n#### Step... |
DeepScaleR | [
{
"role": "user",
"content": "An isosceles right triangle with side lengths in the ratio 1:1:\\(\\sqrt{2}\\) is inscribed in a circle with a radius of \\(\\sqrt{2}\\). What is the area of the triangle and the circumference of the circle?"
}
] | math | {
"style": "rule",
"ground_truth": "2\\pi\\sqrt{2}"
} | {
"index": 4883,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An isosceles trapezoid $A B C D$ with bases $A B$ and $C D$ has $A B=13, C D=17$, and height 3. Let $E$ be the intersection of $A C$ and $B D$. Circles $\\Omega$ and $\\omega$ are circumscribed about triangles $A B E$ and $C D E$. Compute the sum of the radii of $\\Omega$ and $... | math | {
"style": "rule",
"ground_truth": "39"
} | {
"index": 4884,
"solution": "Let $\\Omega$ have center $O$ and radius $R$ and let $\\omega$ have center $P$ and radius $M$. Let $Q$ be the intersection of $A B$ and $O E$. Note that $O E$ is the perpendicular bisector of $A B$ because the trapezoid is isosceles. Also, we see $O E$ is the circumradius of $\\Omega$.... |
DeepScaleR | [
{
"role": "user",
"content": "An isosceles trapezoid \\(ABCD\\) is circumscribed around a circle. The lateral sides \\(AB\\) and \\(CD\\) are tangent to the circle at points \\(M\\) and \\(N\\), respectively, and \\(K\\) is the midpoint of \\(AD\\). In what ratio does the line \\(BK\\) divide the segment \\... | math | {
"style": "rule",
"ground_truth": "1:3"
} | {
"index": 4885,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An isosceles trapezoid has legs of length 30 cm each, two diagonals of length 40 cm each and the longer base is 50 cm. What is the trapezoid's area in sq cm?"
}
] | math | {
"style": "rule",
"ground_truth": "768"
} | {
"index": 4886,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An isosceles trapezoid has sides labeled as follows: \\(AB = 25\\) units, \\(BC = 12\\) units, \\(CD = 11\\) units, and \\(DA = 12\\) units. Compute the length of the diagonal \\( AC \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{419}"
} | {
"index": 4887,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An isosceles trapezoid is circumscribed around a circle. The longer base of the trapezoid is $16$, and one of the base angles is $\\arcsin(.8)$. Find the area of the trapezoid.\n$\\textbf{(A)}\\ 72\\qquad \\textbf{(B)}\\ 75\\qquad \\textbf{(C)}\\ 80\\qquad \\textbf{(D)}\\ 90\\q... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"index": 4888,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An isosceles trapezoid is circumscribed around a circle. The longer base of the trapezoid is $20$, and one of the base angles is $\\arcsin(0.6)$. Find the area of the trapezoid given that the height of the trapezoid is $9$."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 4889,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An isosceles triangle $ABC$, with $AB=AC$, is fixed in the plane. A point $P$ is randomly placed within the triangle. The length of each equal side $AB$ and $AC$ is 6 units and the base $BC$ is 8 units. What is the probability that the area of triangle $PBC$ is more than one-th... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 4890,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An isosceles triangle $ABP$ with sides $AB = AP = 3$ inches and $BP = 4$ inches is placed inside a square $AXYZ$ with a side length of $8$ inches, such that $B$ is on side $AX$. The triangle is rotated clockwise about $B$, then $P$, and so on along the sides of the square until... | math | {
"style": "rule",
"ground_truth": "\\frac{32\\pi}{3}"
} | {
"index": 4891,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An isosceles triangle $DEF$ has $DE = DF = 5\\sqrt{3}$, and a circle with radius $3\\sqrt{3}$ is tangent to line $DE$ at $E$ and to line $DF$ at $F$. What is the area of the circle that passes through vertices $D$, $E$, and $F$?\nA) $63\\pi$\nB) $48\\pi$\nC) $72\\pi$\nD) $36\\p... | math | {
"style": "rule",
"ground_truth": "48\\pi"
} | {
"index": 4892,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An isosceles triangle and a rectangle both have perimeters of 60 inches. The rectangle's length is twice its width. What is the ratio of the length of one of the equal sides of the triangle to the width of the rectangle? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"index": 4893,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An isosceles triangle and a rectangle have the same area. The base of the triangle is equal to the width of the rectangle, and this dimension is 10 units. The length of the rectangle is twice its width. What is the height of the triangle, $h$, in terms of the dimensions of the ... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 4894,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An isosceles triangle has its vertex at $(0,5)$ and a base between points $(3,5)$ and $(13,5)$. The two equal sides are each 10 units long. If the third vertex (top vertex) is in the first quadrant, what is the y-coordinate?"
}
] | math | {
"style": "rule",
"ground_truth": "5 + 5\\sqrt{3}"
} | {
"index": 4895,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An isosceles triangle has side lengths 8 cm, 8 cm and 10 cm. The longest side of a similar triangle is 25 cm. What is the perimeter of the larger triangle, in centimeters?"
}
] | math | {
"style": "rule",
"ground_truth": "65"
} | {
"index": 4896,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An isosceles triangle has sides with lengths that are composite numbers, and the square of the sum of the lengths is a perfect square. What is the smallest possible value for the square of its perimeter?"
}
] | math | {
"style": "rule",
"ground_truth": "256"
} | {
"index": 4897,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An isosceles triangle is a triangle with two sides of equal length. How many of the five triangles on the square grid below are isosceles?\n[asy] for(int a=0; a<12; ++a) { draw((a,0)--(a,6)); } for(int b=0; b<7; ++b) { draw((0,b)--(11,b)); } draw((0,6)--(2,6)--(1,4)--c... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 4898,
"solution": "To determine how many of the triangles are isosceles, we need to calculate the lengths of the sides of each triangle and check if at least two sides are equal in length.\n\n#### Triangle 1:\nVertices: $(0,6), (2,6), (1,4)$\n- Side 1: Distance between $(0,6)$ and $(2,6)$ = $2 - 0 = 2$\n... |
DeepScaleR | [
{
"role": "user",
"content": "An isosceles triangle with a base of $\\sqrt{2}$ has medians intersecting at a right angle. Calculate the area of this triangle."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 4899,
"solution": ""
} |
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