This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 85335

TNO 2024 Junior, 5
Tags: number theory
The nine digits from 1 to 9 are to be placed around a circle so that the average of any three consecutive digits is a multiple of 3. Is this possible? Justify your answer.

2002 Olympic Revenge, 5
Tags: combinatorics
In a "Hanger Party", the guests are initially dressed. In certain moments, the host chooses a guest, and the chosen guest and all his friends will wear its respective clothes if they are naked, and undress it if they are dressed. It is possible that, in some moment, the guests are naked, independent of their mutual friendships? (Suppose friendship is reciprocal.)

1967 All Soviet Union Mathematical Olympiad, 088
Tags: number theory
Prove that there exists a number divisible by $5^{1000}$ not containing a single zero in its decimal notation.

1989 Irish Math Olympiad, 4
Tags: number theory
Note that $12^2=144$ ends in two $4$s and $38^2=1444$ end in three $4$s. Determine the length of the longest string of equal nonzero digits in which the square of an integer can end.

2023 AMC 10, 6
Tags:
An integer is assigned to each vertex of a cube. The value of an edge is defined to be the sum of the values of the two vertices it touches, and the value of a face is defined to be the sum of the values of the four edges surrounding it. The value of the cube is defined as the sum of the values of its six faces. Suppose the sum of the integers assigned to the vertices is $21$. What is the value of the cube? $\textbf{(A)}~42\qquad\textbf{(B)}~63\qquad\textbf{(C)}~84\qquad\textbf{(D)}~126\qquad\textbf{(E)}~252$

2022 Moldova Team Selection Test, 3
Tags: combinatorics , number theory
Let $n$ be a positive integer. On a board there are written all integers from $1$ to $n$. Alina does $n$ moves consecutively: for every integer $m$ $(1 \leq m \leq n)$ the move $m$ consists in changing the sign of every number divisible by $m$. At the end Alina sums the numbers. Find this sum.

2018 Online Math Open Problems, 5
Tags:
A mouse has a wheel of cheese which is cut into $2018$ slices. The mouse also has a $2019$-sided die, with faces labeled $0,1,2,\ldots, 2018$, and with each face equally likely to come up. Every second, the mouse rolls the dice. If the dice lands on $k$, and the mouse has at least $k$ slices of cheese remaining, then the mouse eats $k$ slices of cheese; otherwise, the mouse does nothing. What is the expected number of seconds until all the cheese is gone? [i]Proposed by Brandon Wang[/i]

1995 Poland - First Round, 7
Tags: inequalities
Nonnegative numbers $a, b, c, p, q, r$ satisfy the conditions: $a + b + c = p + q + r = 1; ~~~~~~ p, q, r \leq \frac{1}{2}$. Prove that $8abc \leq pa + qb + rc$ and determine when equality holds.

2003 Turkey Junior National Olympiad, 3
Tags:
How many subsets of $\{1,2,3,4,5,6,7,8,9,10,11\}$ contain no two consequtive numbers?

1985 Traian Lălescu, 1.1
Tags: analytic geometry , symmetry , geometry
We are given two concurrent lines $ d_1 $ and $ d_2. $ Find, analytically, the acute angle formed by them such that for any point $ A $ the equation $ A=A_4 $ holds, where $ A_1 $ is the symmetric of $ A $ with respect to $ d_1, $ $ A_2 $ is the symmetric of $ A_1 $ with respect to $ d_2, $ $ A_3 $ is the symmetric of $ A_2 $ with respect to $ d_1, $ and $ A_4 $ is the symmetric of $ A_3 $ with respect to $ d_2. $

2022 Stanford Mathematics Tournament, 4
Tags:
Frank mistakenly believes that the number $1011$ is prime and for some integer $x$ writes down $(x+1)^{1011}\equiv x^{1011}+1\pmod{1011}$. However, it turns out that for Frank's choice of $x$, this statement is actually true. If $x$ is positive and less than $1011$, what is the sum of the possible values of $x$?

1982 Austrian-Polish Competition, 1
Tags: gcd , number theory , greatest common divisor
Find all pairs $(n, m)$ of positive integers such that $gcd ((n + 1)^m - n, (n + 1)^{m+3} - n) > 1$.

2007 Tournament Of Towns, 2
Tags: geometric transformation , combinatorics , geometry
A convex figure $F$ is such that any equilateral triangle with side $1$ has a parallel translation that takes all its vertices to the boundary of $F$. Is $F$ necessarily a circle?

2016 Saudi Arabia GMO TST, 4
Tags: game , combinatorics
There are totally $16$ teams participating in a football tournament, each team playing with every other exactly $1$ time. In each match, the winner gains $3$ points, the loser gains $0$ point and each teams gain $1$ point for the tie match. Suppose that at the end of the tournament, each team gains the same number of points. Prove that there are at least $4$ teams that have the same number of winning matches, the same number of losing matches and the same number of tie matches.

2011 Saudi Arabia Pre-TST, 4.2
Tags: pentagon , cyclic , geometry
Pentagon $ABCDE$ is inscribed in a circle. Distances from point $E$ to lines $AB$ , $BC$ and $CD$ are equal to $a, b$ and $c$, respectively. Find the distance from point $E$ to line $AD$.

2015 Princeton University Math Competition, B3
Tags: algebra
Andrew and Blair are bored in class and decide to play a game. They pick a pair $(a, b)$ with $1 \le a, b \le 100$. Andrew says the next number in the geometric series that begins with $a,b$ and Blair says the next number in the arithmetic series that begins with $a,b$. For how many pairs $(a, b)$ is Andrew's number minus Blair's number a positive perfect square?

VMEO IV 2015, 10.3
Tags: number theory , divide , divisible
Given a positive integer $k$. Find the condition of positive integer $m$ over $k$ such that there exists only one positive integer $n$ satisfying $$n^m | 5^{n^k} + 1,$$

2012 Hanoi Open Mathematics Competitions, 7
Tags: number theory , perfect square
Prove that the number $a =\overline{{1...1}{5...5}6}$ is a perfect square (where $1$s are $2012$ in total and $5$s are $2011$ in total)

1988 AIME Problems, 4
Tags:
Suppose that $|x_i| < 1$ for $i = 1, 2, \dots, n$. Suppose further that \[ |x_1| + |x_2| + \dots + |x_n| = 19 + |x_1 + x_2 + \dots + x_n|. \] What is the smallest possible value of $n$?

1999 Brazil Team Selection Test, Problem 2
Tags: geometric inequality , inequalities , geometry , ratio
In a triangle $ABC$, the bisector of the angle at $A$ of a triangle $ABC$ intersects the segment $BC$ and the circumcircle of $ABC$ at points $A_1$ and $A_2$, respectively. Points $B_1,B_2,C_1,C_2$ are analogously defined. Prove that $$\frac{A_1A_2}{BA_2+CA_2}+\frac{B_1B_2}{CB_2+AB_2}+\frac{C_1C_2}{AC_2+BC_2}\ge\frac34.$$

2017 AMC 10, 14
Tags: percent
Every week Roger pays for a movie ticket and a soda out of his allowance. Last week, Roger's allowance was $A$ dollars. The cost of his movie ticket was $20\%$ of the difference between $A$ and the cost of his soda, while the cost of his soda was $5\%$ of the difference between $A$ and the cost of his movie ticket. To the nearest whole percent, what fraction of $A$ did Roger pay for his movie ticket and soda? $ \textbf{(A) }9\%\qquad \textbf{(B) } 19\%\qquad \textbf{(C) } 22\%\qquad \textbf{(D) } 23\%\qquad \textbf{(E) }25\%$

2019 Dutch IMO TST, 3
Tags: functional inequality , find all functions , algebra
Find all functions $f : Z \to Z$ satisfying the following two conditions: (i) for all integers $x$ we have $f(f(x)) = x$, (ii) for all integers $x$ and y such that $x + y$ is odd, we have $f(x) + f(y) \ge x + y$.

2019 USAMO, 3
Tags: sob
Let $K$ be the set of all positive integers that do not contain the digit $7$ in their base-$10$ representation. Find all polynomials $f$ with nonnegative integer coefficients such that $f(n)\in K$ whenever $n\in K$. [i]Proposed by Titu Andreescu, Cosmin Pohoata, and Vlad Matei[/i]

2024 China Team Selection Test, 6
Tags: combinatorics , combinatorial geometry
Let $m,n>2$ be integers. A regular ${n}$-sided polygon region $\mathcal T$ on a plane contains a regular ${m}$-sided polygon region with a side length of ${}{}{}1$. Prove that any regular ${m}$-sided polygon region $\mathcal S$ on the plane with side length $\cos{\pi}/[m,n]$ can be translated inside $\mathcal T.$ In other words, there exists a vector $\vec\alpha,$ such that for each point in $\mathcal S,$ after translating the vector $\vec\alpha$ at that point, it fall into $\mathcal T.$ Note: The polygonal area includes both the interior and boundaries. [i]Created by Bin Wang[/i]

Mathley 2014-15, 4
Tags: circumcenter , equal segments , orthocenter , isosceles , trigonometry
Points $E, F$ are in the plane of triangle $ABC$ so that triangles $ABE$ and $ACF$ are the opposite directed, and the two triangles are isosceles in that $BE = AE, AF = CF$. Let $H, K$ be the orthocenter of triangle $ABE, ACF$ respectively. Points $M, N$ are the intersections of $BE$ and $CF, CK$ and $CH$. Prove that $MN$ passes through the center of the circumcircle of triangle $ABC$. Nguyen Minh Ha, High School for Education, Hanoi Pedagogical University