index-completion_cnt stringlengths 3 6 | conversations listlengths 2 2 | answer stringlengths 1 23 |
|---|---|---|
2-0 | [
{
"from": "human",
"value": "Let $a, b, c$ be distinct numbers such that the equations $x^2 + ax + 1 = 0$ and $x^2 + bx + c = 0$ have a common real root, and the equations $x^2 + x + a = 0$ and $x^2 + cx + b = 0$ also have a common real root. Compute the sum $a + b + c$.\nPlease reason step by step, and put... | -3 |
3-0 | [
{
"from": "human",
"value": "There are $7$ boxes arranged in a row and numbered $1$ through $7$. You have a stack of $2015$ cards, which you place one by one in the boxes. The first card is placed in box $1$, the second in box $2$, and so forth up to the seventh card which is placed in box $7$. You then sta... | 3 |
4-0 | [
{
"from": "human",
"value": "Let $S$ be the set of triples $(a,b,c)$ of non-negative integers such that $a+b+c$ is even. The value of the sum\n\\[ \\sum_{(a,b,c) \\in S} \\frac{1}{2^a 3^b 5^c} \\]\ncan be expressed as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.\n... | 37 |
5-0 | [
{
"from": "human",
"value": "For which $n$ is $n^4 + 6n^3 + 11n^2 + 3n + 31$ a perfect square?\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are to determine **for which integers $ n $** the expression\n\n$$\nn^4 + 6n^3 + 11n^2 + 3n + ... | 10 |
6-0 | [
{
"from": "human",
"value": "Four positive integers $a, b, c, d$ satisfy the condition $a < b < c < d$. Determine the smallest possible value of $d$ such that the arithmetic mean of the numbers $a, b, c$ is half of the arithmetic mean of the numbers $a, b, c, d$.\nPlease reason step by step, and put your fi... | 10 |
7-0 | [
{
"from": "human",
"value": "$n$ coins are arranged in a circle. If two neighboring coins both show heads or both show tails, you can flip both coins. How many distinct arrangements of the coins are there such that no arrangement can be transformed into another by performing these operations?\nPlease reason... | 2 |
8-0 | [
{
"from": "human",
"value": "Determine the smallest prime $p$ such that $2018!$ is divisible by $p^3$, but not divisible by $p^4$.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are asked to find the **smallest prime** $ p $ such that:\... | 509 |
9-0 | [
{
"from": "human",
"value": "Suppose you have a sphere tangent to the $xy$-plane with its center having a positive $z$-coordinate. If it is projected from a point $P=(0,b,a)$ to the $xy$-plane, it gives the conic section $y=x^2$. If we write $a=\\frac{p}{q}$ where $p$ and $q$ are integers, find $p+q$.\nPlea... | 3 |
10-0 | [
{
"from": "human",
"value": "There are $10^{2015}$ planets in an Intergalactic empire. Every two planets are connected by a two-way space line served by one of $2015$ travel companies. The Emperor would like to close $k$ of these companies such that it is still possible to reach any planet from any other pl... | 1007 |
12-0 | [
{
"from": "human",
"value": "Let $a$, $b$, $c$ be the three roots of the equation $x^3-(k+1)x^2+kx+12=0$, where $k$ is a real number. If $(a-2)^3+(b-2)^3+(c-2)^3=-18$, find the value of $k$.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "W... | 5 |
14-0 | [
{
"from": "human",
"value": "Burrito Bear has a white unit square. She inscribes a circle inside the square and paints it black. She then inscribes a square inside the black circle and paints it white. Burrito repeats this process indefinitely. The total black area can be expressed as $\\frac{a\\pi+b}{c}$. ... | 1 |
15-0 | [
{
"from": "human",
"value": "Danial went to a fruit stall that sells apples, mangoes, and papayas. Each apple costs $3$ RM, each mango costs $4$ RM, and each papaya costs $5$ RM. He bought at least one of each fruit and paid exactly $50$ RM. What is the maximum number of fruits that he could have bought?\nP... | 15 |
17-0 | [
{
"from": "human",
"value": "Denote by $S(n)$ the sum of the digits of the positive integer $n$. Find all the solutions of the equation $n(S(n)-1)=2010.$\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are given the equation:\n\n$$\nn(S(... | 402 |
19-0 | [
{
"from": "human",
"value": "Find all integers $k$ such that all roots of the following polynomial are also integers:\n\n$$f(x) = x^3 - (k-3)x^2 - 11x + (4k-8).$$\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are given the polynomial:\... | 5 |
20-0 | [
{
"from": "human",
"value": "A class has $25$ students. The teacher wants to stock $N$ candies and hold an Olympiad, distributing all $N$ candies based on student performance. The candies should be distributed such that students who solve the same number of tasks receive the same number of candies, and thos... | 300 |
21-0 | [
{
"from": "human",
"value": "If $a, b$ are real numbers such that $a^3 + 12a^2 + 49a + 69 = 0$ and $b^3 - 9b^2 + 28b - 31 = 0$, find $a + b$.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are given two cubic equations:\n\n1. $ a^3 + 12... | 1 |
22-0 | [
{
"from": "human",
"value": "Let $S$ be the set of triples $(a,b,c)$ of non-negative integers such that $a+b+c$ is even. Determine the value of the sum:\n\\[\n\\sum_{(a,b,c)\\in S}\\frac{1}{2^a3^b5^c}\n\\]\nThis sum can be expressed as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers... | 37 |
23-0 | [
{
"from": "human",
"value": "Some positive integers are initially written on a board, where each 2 of them are different. Each time we can do the following moves: \n1. If there are 2 numbers (written on the board) in the form $n, n+1$, we can erase them and write down $n-2$. \n2. If there are 2 numbers (w... | -3 |
24-0 | [
{
"from": "human",
"value": "Find all positive integers $n$ for which the largest prime divisor of $n^2 + 3$ is equal to the least prime divisor of $n^4 + 6$.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are given the equation:\n\n$$\... | 3 |
26-0 | [
{
"from": "human",
"value": "The integers from $1$ to $n$ are written, one on each of $n$ cards. The first player removes one card. Then the second player removes two cards with consecutive integers. After that, the first player removes three cards with consecutive integers. Finally, the second player remov... | 14 |
27-0 | [
{
"from": "human",
"value": "Joe has $1729$ randomly oriented and randomly arranged unit cubes, which are initially unpainted. He makes two cubes of side lengths $9$ and $10$ or of side lengths $1$ and $12$ (randomly chosen). These cubes are dipped into white paint. Then two more cubes of side lengths $1$ a... | 7 |
29-0 | [
{
"from": "human",
"value": "Consider two distinct positive integers $a$ and $b$ that have integer arithmetic, geometric, and harmonic means. Find the minimal value of $|a-b|$.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are given tw... | 30 |
32-0 | [
{
"from": "human",
"value": "Find the sum of all possible values of $a + b$ where $a$ and $b$ are nonnegative integers such that $4^a + 2^b + 5$ is a perfect square.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are given the expressio... | 9 |
33-0 | [
{
"from": "human",
"value": "Let $M$ be the set of those positive integers which are not divisible by $3$. The sum of $2n$ consecutive elements of $M$ is $300$. Determine $n$.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are given:\n\... | 10 |
34-0 | [
{
"from": "human",
"value": "Petya and Vasya play the following game. Petya conceives a polynomial $P(x)$ with integer coefficients. On each move, Vasya pays him a ruble and calls an integer $a$ of his choice, which has not yet been called by him. Petya has to reply with the number of distinct integer solut... | 4 |
35-0 | [
{
"from": "human",
"value": "Determine the greatest power of $2$ that is a factor of $3^{15} + 3^{11} + 3^{6} + 1$.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are given the expression:\n\n$$\n3^{15} + 3^{11} + 3^6 + 1\n$$\n\nWe are ... | 64 |
36-0 | [
{
"from": "human",
"value": "On a board, the numbers from 1 to 2009 are written. A couple of them are erased and instead of them, the remainder of the sum of the erased numbers divided by 13 is written on the board. After a few repetitions of this process, only three numbers are left, of which two are 9 and... | 8 |
37-0 | [
{
"from": "human",
"value": "For positive integers $n$, let $s(n)$ be the sum of the digits of $n$. Over all four-digit positive integers $n$, which value of $n$ maximizes the ratio $\\frac{s(n)}{n}$?\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"... | 1099 |
38-0 | [
{
"from": "human",
"value": "Suppose that $n$ persons meet in a meeting, and that each of the persons is acquainted with exactly $8$ others. Any two acquainted persons have exactly $4$ common acquaintances, and any two non-acquainted persons have exactly $2$ common acquaintances. Find all possible values of... | 21 |
40-0 | [
{
"from": "human",
"value": "Find a necessary and sufficient condition on the positive integer $n$ such that the equation $x^n + (2 + x)^n + (2 - x)^n = 0$ has a rational root.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are given th... | 1 |
41-0 | [
{
"from": "human",
"value": "Let $ABCDEFGHIJ$ be a regular 10-sided polygon inscribed in a circle with center $O$ and radius $5$. The diagonals $AD$ and $BE$ intersect at point $P$, and the diagonals $AH$ and $BI$ intersect at point $Q$. Calculate the length of segment $PQ$.\nPlease reason step by step, and... | 5 |
42-0 | [
{
"from": "human",
"value": "Determine all integers $m$ for which all solutions of the equation $3x^3 - 3x^2 + m = 0$ are rational.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are given the cubic equation:\n\n$$\n3x^3 - 3x^2 + m = 0\... | 0 |
43-0 | [
{
"from": "human",
"value": "Three boxes contain 600 balls each. The first box contains 600 identical red balls, the second box contains 600 identical white balls, and the third box contains 600 identical blue balls. From these three boxes, 900 balls are chosen. In how many ways can the balls be chosen? For... | 270901 |
45-0 | [
{
"from": "human",
"value": "Let $A$, $B$, $C$, and $D$ be points on a circle such that $AB = 11$ and $CD = 19$. Point $P$ is on segment $AB$ with $AP = 6$, and $Q$ is on segment $CD$ with $CQ = 7$. The line through $P$ and $Q$ intersects the circle at $X$ and $Y$. If $PQ = 27$, find $XY$.\nPlease reason st... | 31 |
46-0 | [
{
"from": "human",
"value": "For any real numbers $x$ and $y$ that satisfy the equations:\n\\[\nx + y - xy = 155\n\\]\nand\n\\[\nx^2 + y^2 = 325\n\\]\nFind $|x^3 - y^3|$.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are given two equa... | 4375 |
47-0 | [
{
"from": "human",
"value": "There are $2023$ distinct points on a plane, which are colored either white or red. For each white point, there are exactly two red points such that the distance from the white point to each of these red points is $2023$. Find the minimum number of red points.\nPlease reason ste... | 45 |
48-0 | [
{
"from": "human",
"value": "Find all positive integers $k$ for which the equation:\n$$ \\text{lcm}(m,n) - \\text{gcd}(m,n) = k(m-n) $$\nhas no solution in positive integers $(m,n)$ with $m \\neq n$.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"v... | 2 |
49-0 | [
{
"from": "human",
"value": "Determine the largest $k$ such that for all competitive graphs with $2019$ vertices, if the difference between the in-degree and out-degree of any vertex is less than or equal to $k$, then this graph is strongly connected.\nPlease reason step by step, and put your final answer w... | 1 |
52-0 | [
{
"from": "human",
"value": "Suppose integers $a < b < c$ satisfy the following conditions:\n\\[\n a + b + c = 95 \\\\\n a^2 + b^2 + c^2 = 3083\n\\]\nFind the value of $c$.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are given:\n\n$$... | 37 |
53-0 | [
{
"from": "human",
"value": "Let $a$, $b$, $c$ be real numbers such that $a^2 - 2 = 3b - c$, $b^2 + 4 = 3 + a$, and $c^2 + 4 = 3a - b$. Find $a^4 + b^4 + c^4$.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are given three equations inv... | 18 |
55-0 | [
{
"from": "human",
"value": "Let $P$ be a regular $2006$-gon. A diagonal is called *good* if its endpoints divide the boundary of $P$ into two parts, each composed of an odd number of sides of $P$. The sides of $P$ are also called *good*.\n\nSuppose $P$ has been dissected into triangles by $2003$ diagonals,... | 1003 |
57-0 | [
{
"from": "human",
"value": "Find all positive integers $n$ for which all positive divisors of $n$ can be placed into the cells of a rectangular table under the following constraints:\n- Each cell contains a distinct divisor.\n- The sums of all rows are equal.\n- The sums of all columns are equal.\nPlease r... | 1 |
58-0 | [
{
"from": "human",
"value": "One day, students in school organized an exchange among themselves with the following rates:\n- $11$ strawberries exchange for $14$ raspberries.\n- $22$ cherries exchange for $21$ raspberries.\n- $10$ cherries exchange for $3$ bananas.\n- $5$ pears exchange for $2$ bananas.\n\nT... | 7 |
59-0 | [
{
"from": "human",
"value": "In triangle $ABC$, let $D$ be the foot of the altitude from $A$. Suppose that $AD = 4$, $BD = 3$, $CD = 2$, and $AB$ is extended past $B$ to a point $E$ such that $BE = 5$. Determine the value of $CE^2$.\nPlease reason step by step, and put your final answer within \\boxed{}."
... | 80 |
60-0 | [
{
"from": "human",
"value": "Find all integers $n$ such that $\\frac{5^n - 1}{3}$ is a prime or a perfect square of an integer.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are asked to find all integers $ n $ such that:\n\n$$\n\\frac... | 0 |
61-0 | [
{
"from": "human",
"value": "Find the sum of all positive integers $x$ such that $|x^2-x-6|$ has exactly 4 positive integer divisors.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are given the expression:\n\n$$\n|x^2 - x - 6|\n$$\n\na... | 10 |
63-0 | [
{
"from": "human",
"value": "The triangle $ABC$ has sides $AB = 137$, $AC = 241$, and $BC = 200$. There is a point $D$ on $BC$ such that both incircles of triangles $ABD$ and $ACD$ touch $AD$ at the same point $E$. Determine the length of $CD$.\nPlease reason step by step, and put your final answer within \... | 152 |
65-0 | [
{
"from": "human",
"value": "Find all odd natural numbers $n$ such that $d(n)$ is the largest divisor of the number $n$ different from $n$. Here, $d(n)$ represents the number of divisors of the number $n$, including $1$ and $n$.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
... | 9 |
66-0 | [
{
"from": "human",
"value": "Determine all natural numbers $n$ for which the number $A = n^4 + 4n^3 + 5n^2 + 6n$ is a perfect square of a natural number.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are given the expression:\n\n$$\nA ... | 1 |
67-0 | [
{
"from": "human",
"value": "Let's consider a set of distinct positive integers with a sum equal to 2023. Among these integers, there are a total of $d$ even numbers and $m$ odd numbers. Determine the maximum possible value of $2d + 4m$.\nPlease reason step by step, and put your final answer within \\boxed{... | 200 |
69-0 | [
{
"from": "human",
"value": "Let $f(x)$ be a quadratic polynomial such that $f(f(1)) = f(-f(-1)) = 0$ and $f(1) \\neq -f(-1)$. Suppose furthermore that the quadratic $2f(x)$ has coefficients that are nonzero integers. Find $f(0)$.\nPlease reason step by step, and put your final answer within \\boxed{}."
}... | 3 |
70-0 | [
{
"from": "human",
"value": "Find all natural numbers $a > 1$ with the property that every prime divisor of $a^6 - 1$ also divides at least one of the numbers $a^3 - 1$, $a^2 - 1$.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are aske... | 2 |
71-0 | [
{
"from": "human",
"value": "Given that nonzero real numbers $x$ and $y$ satisfy the equations $x + \\frac{1}{y} = 3$ and $y + \\frac{1}{x} = 4$, find the value of $xy + \\frac{1}{xy}$.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are... | 10 |
72-0 | [
{
"from": "human",
"value": "Find the smallest positive integer $k$ such that $k + 25973$ is a palindrome, a number that reads the same forward and backward.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are asked to find the **smalles... | 89 |
73-0 | [
{
"from": "human",
"value": "Find all values of the positive integer $m$ such that there exist polynomials $P(x)$, $Q(x)$, and $R(x,y)$ with real coefficients satisfying the condition: For every pair of real numbers $a$ and $b$ satisfying $a^m - b^2 = 0$, we always have $P(R(a,b)) = a$ and $Q(R(a,b)) = b$.\... | 1 |
77-0 | [
{
"from": "human",
"value": "Five congruent circles have centers at the vertices of a regular pentagon so that each of the circles is tangent to its two neighbors. A sixth circle, congruent to the other five, is placed tangent to two of the five. If this sixth circle is allowed to roll without slipping arou... | 720 |
78-0 | [
{
"from": "human",
"value": "The thousands digit of a five-digit number which is divisible by $37$ and $173$ is $3$. What is the hundreds digit of this number?\n\n\\[\n\\textbf{a)}\\ 0 \\quad \\textbf{b)}\\ 2 \\quad \\textbf{c)}\\ 4 \\quad \\textbf{d)}\\ 6 \\quad \\textbf{e)}\\ 8\n\\]\nPlease reason step by... | 2 |
80-0 | [
{
"from": "human",
"value": "Let $ABC$ be a triangle in the $xy$ plane, where $B$ is at the origin $(0,0)$. Extend $BC$ to $D$ such that $BC: CD = 1:1$, extend $CA$ to $E$ such that $CA: AE = 1:2$, and extend $AB$ to $F$ such that $AB: BF = 1:3$. Let $G(32,24)$ be the centroid of the triangle $ABC$ and $K$ ... | 40 |
81-0 | [
{
"from": "human",
"value": "Real nonzero numbers $x, y, z$ satisfy $x + y + z = 0$. Furthermore, it is given that:\n\\[\nA = \\frac{x}{y} + \\frac{y}{z} + \\frac{z}{x} = \\frac{x}{z} + \\frac{z}{y} + \\frac{y}{x} + 1\n\\]\nDetermine the value of $A$.\nPlease reason step by step, and put your final answer w... | -1 |
82-0 | [
{
"from": "human",
"value": "Find all prime numbers $p$ such that $4p^2 + 1$ and $6p^2 + 1$ are also prime numbers.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are given that $ p $ is a **prime number**, and we want to find all such ... | 5 |
83-0 | [
{
"from": "human",
"value": "Find the sum of all positive integers $x$ such that $3 \\times 2^x = n^2 - 1$ for some positive integer $n$.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are asked to find the sum of all **positive integer... | 7 |
85-0 | [
{
"from": "human",
"value": "Given a pentagon $ABCDE$ with the following properties:\n\n1. Angles $\\angle A$ and $\\angle C$ are right angles.\n2. The side lengths are $|AB| = 4$, $|BC| = 5$, $|CD| = 10$, and $|DE| = 6$.\n3. Point $C'$ is the reflection of point $C$ across line $BD$ and lies on line segmen... | 90 |
88-0 | [
{
"from": "human",
"value": "Triangle $ABC$ is right-angled at $A$. The circle with center $A$ and radius $AB$ intersects $BC$ and $AC$ internally at points $D$ and $E$ respectively. Given that $BD = 20$ and $DC = 16$, determine $AC^2$.\nPlease reason step by step, and put your final answer within \\boxed{}... | 936 |
89-0 | [
{
"from": "human",
"value": "Petya gave Vasya a number puzzle. Petya chose a digit $X$ and said, \"I am thinking of a three-digit number that is divisible by 11. The hundreds digit is $X$ and the tens digit is 3. Find the units digit.\" Vasya was excited because he knew how to solve this problem, but then r... | 4 |
90-0 | [
{
"from": "human",
"value": "A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five.\n\nAda has been told to write down five haikus plus two more every hour. This means she needs to write down five in the first hour, seven in the second hour, nine in the third hour, and s... | 4 |
91-0 | [
{
"from": "human",
"value": "The quadrilateral $ABCD$ satisfies the following conditions: \\( \\angle ABC = \\angle BCD = 150^{\\circ} \\). Additionally, \\( AB = 18 \\) and \\( BC = 24 \\). Equilateral triangles \\( \\triangle APB \\), \\( \\triangle BQC \\), and \\( \\triangle CRD \\) are constructed outs... | 10 |
92-0 | [
{
"from": "human",
"value": "We are given a coin of diameter $\\frac{1}{2}$ and a checkerboard of $1 \\times 1$ squares with an area of $2010 \\times 2010$. We toss the coin such that it lands completely on the checkerboard. If the probability that the coin doesn't touch any of the lattice lines is $\\frac{... | 6029 |
93-0 | [
{
"from": "human",
"value": "Graph $G_1$ of a quadratic trinomial $y = px^2 + qx + r$ with real coefficients intersects the graph $G_2$ of a quadratic trinomial $y = x^2$ at points $A$ and $B$. The intersection of tangents to $G_2$ at points $A$ and $B$ is point $C$. If $C$ lies on $G_1$, find all possible ... | 2 |
95-0 | [
{
"from": "human",
"value": "The radius $r$ of a circle with center at the origin is an odd integer. There is a point $(p^m, q^n)$ on the circle, with $p, q$ being prime numbers and $m, n$ being positive integers. Determine $r$.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
... | 5 |
98-0 | [
{
"from": "human",
"value": "Solve for $x$:\n\\[\nv - w + x - y + z = 79 \\\\\nv + w + x + y + z = -1 \\\\\nv + 2w + 4x + 8y + 16z = -2 \\\\\nv + 3w + 9x + 27y + 81z = -1 \\\\\nv + 5w + 25x + 125y + 625z = 79.\n\\]\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from"... | 24 |
99-0 | [
{
"from": "human",
"value": "You have a 4-digit whole number that is a perfect square. Another number is constructed by making the following changes to the digits of the original number:\n\n1. Add 1 to the unit's digit.\n2. Subtract 1 from the ten's digit.\n3. Add 1 to the hundred's digit.\n4. Subtract 1 fr... | 3025 |
102-0 | [
{
"from": "human",
"value": "Given a regular octagon $ABCDEFGH$ with side length $3$. By drawing the four diagonals $AF$, $BE$, $CH$, and $DG$, the octagon is divided into a square, four triangles, and four rectangles. Find the sum of the areas of the square and the four triangles.\nPlease reason step by st... | 18 |
103-0 | [
{
"from": "human",
"value": "The prime numbers $a$, $b$, and $c$ satisfy the equation $a + b^2 = 4c^2$. Determine the sum of all possible values of $a + b + c$.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are given the equation:\n\n$... | 31 |
105-0 | [
{
"from": "human",
"value": "Find all prime numbers $p$ such that $(p-3)^p + p^2$ is a perfect square.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are given the expression:\n\n$$\n(p - 3)^p + p^2\n$$\n\nand we are asked to **find all... | 3 |
107-0 | [
{
"from": "human",
"value": "Several cells are marked in a $100 \\times 100$ table. Vasya wants to split the square into several rectangles such that each rectangle does not contain more than two marked cells, and there are at most $k$ rectangles containing less than two cells. What is the smallest $k$ such... | 100 |
110-0 | [
{
"from": "human",
"value": "30 volleyball teams have participated in a league. Any two teams have played a match with each other exactly once. At the end of the league, a match is called **unusual** if the winner of the match has fewer wins than the loser. A team is called **astonishing** if all its matche... | 1 |
111-0 | [
{
"from": "human",
"value": "In the arrow-shaped polygon (refer to the provided figure), the angles at vertices $A$, $C$, $D$, $E$, and $F$ are right angles. The side lengths are given as follows: $BC = FG = 5$, $CD = FE = 20$, $DE = 10$, and $AB = AG$. Determine the area of the polygon that is closest to o... | 300 |
112-0 | [
{
"from": "human",
"value": "Let $u$, $v$, and $w$ be real numbers in geometric progression such that $u > v > w$. Suppose $u^{40} = v^n = w^{60}$. Find the value of $n$.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are given that $ u... | 48 |
113-0 | [
{
"from": "human",
"value": "For a positive integer $n$, let $p(n)$ denote the smallest prime divisor of $n$. Find the maximum number of divisors $m$ can have if $p(m)^4 > m$.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are given:\n\... | 6 |
114-0 | [
{
"from": "human",
"value": "Determine all real values of $A$ for which there exist distinct complex numbers $x_1$, $x_2$ such that the following three equations hold:\n\\[\nx_1(x_1+1) = A \\\\\nx_2(x_2+1) = A \\\\\nx_1^4 + 3x_1^3 + 5x_1 = x_2^4 + 3x_2^3 + 5x_2.\n\\]\nPlease reason step by step, and put you... | -7 |
115-0 | [
{
"from": "human",
"value": "Find the smallest positive integer $n$ such that there are at least three distinct ordered pairs $(x,y)$ of positive integers satisfying the equation $x^2 - y^2 = n$.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value... | 45 |
116-0 | [
{
"from": "human",
"value": "A function $f$ is defined on integers such that:\n\n- $f(n) = n + 3$ if $n$ is odd.\n- $f(n) = \\frac{n}{2}$ if $n$ is even.\n\nIf $k$ is an odd integer, determine the values for which $f(f(f(k))) = k$.\nPlease reason step by step, and put your final answer within \\boxed{}."
... | 1 |
117-0 | [
{
"from": "human",
"value": "Let $a, b, c$ be positive integers such that $a, b, c, a+b-c, a+c-b, b+c-a, a+b+c$ are 7 distinct primes. The sum of two of $a, b, c$ is 800. If $d$ is the difference between the largest prime and the smallest prime among these 7 primes, find the maximum value of $d$.\nPlease re... | 1594 |
119-0 | [
{
"from": "human",
"value": "On a semicircle with diameter $|AB|=d$, there are points $C$ and $D$ such that $|BC|=|CD|=a$ and $|DA|=b$, where $a$, $b$, and $d$ are different positive integers. Find the minimum possible value of $d$.\nPlease reason step by step, and put your final answer within \\boxed{}."
... | 8 |
120-0 | [
{
"from": "human",
"value": "Suppose in the plane, there are 10 pairwise nonparallel lines intersecting one another. What is the maximum possible number of polygons (with finite areas) that can be formed?\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
... | 36 |
121-0 | [
{
"from": "human",
"value": "The average age of the participants in a mathematics competition (gymnasts and high school students) increases by exactly one month if three high school age students $18$ years each are included in the competition or if three gymnasts aged $12$ years each are excluded from the c... | 108 |
124-0 | [
{
"from": "human",
"value": "All six-digit natural numbers from $100000$ to $999999$ are written on the page in ascending order without spaces. What is the largest value of $k$ for which the same $k$-digit number can be found in at least two different places in this string?\nPlease reason step by step, and ... | 6 |
125-0 | [
{
"from": "human",
"value": "Let $f(n)$ be a function that fulfills the following properties:\n\n- For each natural $n$, $f(n)$ is an integer greater than or equal to $0$.\n- $f(n) = 2010$, if $n$ ends in $7$. For example, $f(137) = 2010$.\n- If $a$ is a divisor of $b$, then: $f\\left(\\frac{b}{a}\\right) =... | 0 |
126-0 | [
{
"from": "human",
"value": "Find all integers $n > 1$ such that every prime that divides $n^6 - 1$ also divides $n^5 - n^3 - n^2 + 1$.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are given the problem:\n\n> **Find all integers $ n >... | 2 |
127-0 | [
{
"from": "human",
"value": "Let $q$ be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three lines of real numbers on the blackboard:\n- In the first line, Gugu writes down every number of the form $a-b$, where $a$ and $b$ are two (not necessarily ... | 0 |
128-0 | [
{
"from": "human",
"value": "Let $n$ be a positive integer. Find as many zeros as possible for the last digits of the following expression: $1^n + 2^n + 3^n + 4^n$.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are given the expression... | 2 |
129-0 | [
{
"from": "human",
"value": "Find all positive integers $n$ such that $n = 100 \\times d(n)$, where $d(n)$ represents the number of positive divisors of $n$.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are given the equation:\n\n$$\n... | 2000 |
130-0 | [
{
"from": "human",
"value": "In $\\triangle ABC$, we have $AC = BC = 7$ and $AB = 2$. Suppose that $D$ is a point on line $AB$ such that $B$ lies between $A$ and $D$ and $CD = 8$. What is the length of the segment $BD$?\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"... | 3 |
131-0 | [
{
"from": "human",
"value": "Find the number of ordered triplets $(a, b, c)$ of positive integers such that $a < b < c$ and $abc = 2008$.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are asked to find the number of **ordered triplets*... | 4 |
132-0 | [
{
"from": "human",
"value": "Solve in the real numbers the equation $3^{x+1} = (x-1)(x-3)$.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are given the equation:\n\n$$\n3^{x+1} = (x - 1)(x - 3)\n$$\n\n---\n\n### **Step 1: Understand th... | 0 |
133-0 | [
{
"from": "human",
"value": "Let $f(x) = x^3 + x + 1$. Suppose $g$ is a cubic polynomial such that $g(0) = -1$, and the roots of $g$ are the squares of the roots of $f$. Find $g(9)$.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
"value": "We are gi... | 899 |
134-0 | [
{
"from": "human",
"value": "A large gathering of people stand in a triangular array with $2020$ rows, such that the first row has $1$ person, the second row has $2$ people, and so on. Every day, the people in each row infect all of the people adjacent to them in their own row. Additionally, the people at t... | 2019 |
135-0 | [
{
"from": "human",
"value": "Vitya cut the chessboard along the borders of the cells into pieces of the same perimeter. It turned out that not all of the received parts are equal. What is the largest possible number of parts that Vitya could get?\nPlease reason step by step, and put your final answer within... | 32 |
137-0 | [
{
"from": "human",
"value": "In rectangle $ABCD$, diagonal $AC$ is intersected by the angle bisector from $B$ at point $B'$ and by the angle bisector from $D$ at point $D'$. Similarly, diagonal $BD$ is intersected by the angle bisector from $A$ at point $A'$ and by the angle bisector from $C$ at point $C'$.... | 7 |
139-0 | [
{
"from": "human",
"value": "What is the smallest positive integer $n$ such that $2013^n$ ends in $001$? In other words, find the smallest $n$ for which the rightmost three digits of $2013^n$ are $001$.\nPlease reason step by step, and put your final answer within \\boxed{}."
},
{
"from": "gpt",
... | 100 |
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