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: 2008

2010 Contests, 1
Tags: modular arithmetic , number theory
Denote by $S(n)$ the sum of the digits of the positive integer $n$. Find all the solutions of the equation $n(S(n)-1)=2010.$

2000 Taiwan National Olympiad, 1
Tags: euler , quadratic , prime number , number theory , modular arithmetic , function
Suppose that for some $m,n\in\mathbb{N}$ we have $\varphi (5^m-1)=5^n-1$, where $\varphi$ denotes the Euler function. Show that $(m,n)>1$.

2006 Hungary-Israel Binational, 1
Tags: induction , modular arithmetic , number theory
If natural numbers $ x$, $ y$, $ p$, $ n$, $ k$ with $ n > 1$ odd and $ p$ an odd prime satisfy $ x^n \plus{} y^n \equal{} p^k$, prove that $ n$ is a power of $ p$.

2013 Iran MO (3rd Round), 2
Tags: combinatorics , modular arithmetic
How many rooks can be placed in an $n\times n$ chessboard such that each rook is threatened by at most $2k$ rooks? (15 points) [i]Proposed by Mostafa Einollah zadeh[/i]

2010 Putnam, A4
Tags: putnam number theory , logarithm , greatest common divisor , modular arithmetic , number theory
Prove that for each positive integer $n,$ the number $10^{10^{10^n}}+10^{10^n}+10^n-1$ is not prime.

2012 Online Math Open Problems, 25
Tags: symmetry , integration , probability , analytic geometry , modular arithmetic , calculus
Suppose 2012 reals are selected independently and at random from the unit interval $[0,1]$, and then written in nondecreasing order as $x_1\le x_2\le\cdots\le x_{2012}$. If the probability that $x_{i+1} - x_i \le \frac{1}{2011}$ for $i=1,2,\ldots,2011$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m,n$, find the remainder when $m+n$ is divided by 1000. [i]Victor Wang.[/i]

2011 Iran MO (3rd Round), 3
Tags: modular arithmetic , number theory
Let $k$ be a natural number such that $k\ge 7$. How many $(x,y)$ such that $0\le x,y<2^k$ satisfy the equation $73^{73^x}\equiv 9^{9^y} \pmod {2^k}$? [i]Proposed by Mahyar Sefidgaran[/i]

2003 France Team Selection Test, 2
Tags: number theory , analytic geometry , modular arithmetic
A lattice point in the coordinate plane with origin $O$ is called invisible if the segment $OA$ contains a lattice point other than $O,A$. Let $L$ be a positive integer. Show that there exists a square with side length $L$ and sides parallel to the coordinate axes, such that all points in the square are invisible.

2012 Vietnam Team Selection Test, 3
Tags: geometry , arithmetic sequence , combinatorial geometry , symmetry , geometric transformation , modular arithmetic , floor function
Let $p\ge 17$ be a prime. Prove that $t=3$ is the largest positive integer which satisfies the following condition: For any integers $a,b,c,d$ such that $abc$ is not divisible by $p$ and $(a+b+c)$ is divisible by $p$, there exists integers $x,y,z$ belonging to the set $\{0,1,2,\ldots , \left\lfloor \frac{p}{t} \right\rfloor - 1\}$ such that $ax+by+cz+d$ is divisible by $p$.

2015 AMC 10, 23
Tags: modular arithmetic , floor function , factorial
Let $n$ be a positive integer greater than 4 such that the decimal representation of $n!$ ends in $k$ zeros and the decimal representation of $(2n)!$ ends in $3k$ zeros. Let $s$ denote the sum of the four least possible values of $n$. What is the sum of the digits of $s$? $ \textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad\textbf{(E) }11 $

2006 Greece National Olympiad, 2
Tags: number theory , modular arithmetic
Let $n$ be a positive integer. Prove that the equation \[x+y+\frac{1}{x}+\frac{1}{y}=3n\] does not have solutions in positive rational numbers.

PEN C Problems, 3
Tags: modular arithmetic , quadratic , quadratic residue , floor function
Let $p$ be an odd prime number. Show that the smallest positive quadratic nonresidue of $p$ is smaller than $\sqrt{p}+1$.

2009 AIME Problems, 13
Tags: function , induction , pigeonhole principle , modular arithmetic , inequalities
The terms of the sequence $ (a_i)$ defined by $ a_{n \plus{} 2} \equal{} \frac {a_n \plus{} 2009} {1 \plus{} a_{n \plus{} 1}}$ for $ n \ge 1$ are positive integers. Find the minimum possible value of $ a_1 \plus{} a_2$.

2010 ISI B.Math Entrance Exam, 1
Tags: calendar , floor function , modular arithmetic
Prove that in each year , the $13^{th}$ day of some month occurs on a Friday .

2001 IMO, 4
Tags: global , factorial , modular arithmetic , combinatorics
Let $n$ be an odd integer greater than 1 and let $c_1, c_2, \ldots, c_n$ be integers. For each permutation $a = (a_1, a_2, \ldots, a_n)$ of $\{1,2,\ldots,n\}$, define $S(a) = \sum_{i=1}^n c_i a_i$. Prove that there exist permutations $a \neq b$ of $\{1,2,\ldots,n\}$ such that $n!$ is a divisor of $S(a)-S(b)$.

2011 Poland - Second Round, 3
Tags: modular arithmetic , number theory
Prove that $\forall x_{1},x_{2},\ldots,x_{2011},y_{1},y_{2},\ldots,y_{2011}\in\mathbb{Z_{+}}$ product: \[(2x_{1}^{2}+3y_{1}^{2})(2x_{2}^{2}+3y_{2}^{2})\ldots(2x_{2011}^{2}+3y_{2011}^{2})\] is not a perfect square.

2011 Peru IMO TST, 3
Tags: divisibility , number theory , modular arithmetic
Let $a, b$ be integers, and let $P(x) = ax^3+bx.$ For any positive integer $n$ we say that the pair $(a,b)$ is $n$-good if $n | P(m)-P(k)$ implies $n | m - k$ for all integers $m, k.$ We say that $(a,b)$ is $very \ good$ if $(a,b)$ is $n$-good for infinitely many positive integers $n.$ [list][*][b](a)[/b] Find a pair $(a,b)$ which is 51-good, but not very good. [*][b](b)[/b] Show that all 2010-good pairs are very good.[/list] [i]Proposed by Okan Tekman, Turkey[/i]

1980 IMO Shortlist, 9
Tags: modular arithmetic , pascal triangle , representation , number theory
Let $p$ be a prime number. Prove that there is no number divisible by $p$ in the $n-th$ row of Pascal's triangle if and only if $n$ can be represented in the form $n = p^sq - 1$, where $s$ and $q$ are integers with $s \geq 0, 0 < q < p$.

PEN O Problems, 16
Tags: modular arithmetic
Is it possible to find $100$ positive integers not exceeding $25000$ such that all pairwise sums of them are different?

2013 NIMO Problems, 1
Tags: floor function , modular arithmetic
At ARML, Santa is asked to give rubber duckies to $2013$ students, one for each student. The students are conveniently numbered $1,2,\cdots,2013$, and for any integers $1 \le m < n \le 2013$, students $m$ and $n$ are friends if and only if $0 \le n-2m \le 1$. Santa has only four different colors of duckies, but because he wants each student to feel special, he decides to give duckies of different colors to any two students who are either friends or who share a common friend. Let $N$ denote the number of ways in which he can select a color for each student. Find the remainder when $N$ is divided by $1000$. [i]Proposed by Lewis Chen[/i]

2008 Germany Team Selection Test, 3
Tags: divisibility , function , modular arithmetic , number theory
Find all surjective functions $ f: \mathbb{N} \to \mathbb{N}$ such that for every $ m,n \in \mathbb{N}$ and every prime $ p,$ the number $ f(m + n)$ is divisible by $ p$ if and only if $ f(m) + f(n)$ is divisible by $ p$. [i]Author: Mohsen Jamaali and Nima Ahmadi Pour Anari, Iran[/i]

1972 IMO Longlists, 42
Tags: number theory , modular arithmetic
The decimal number $13^{101}$ is given. It is instead written as a ternary number. What are the two last digits of this ternary number?

2014 AIME Problems, 8
Tags: modular arithmetic , sumac , quadratic , greatest common divisor , number theory
The positive integers $N$ and $N^2$ both end in the same sequence of four digits $abcd$ when written in base 10, where digit $a$ is not zero. Find the three-digit number $abc$.

2017 Serbia National Math Olympiad, 1
Tags: modular arithmetic
Let $a$ be a positive integer.Suppose that $\forall n$ ,$\exists d$, $d\not =1$, $d\equiv 1\pmod n$ ,$d\mid n^2a-1$.Prove that $a$ is a perfect square.

1984 IMO, 2
Tags: algebra , divisibility , modular arithmetic , number theory , polynomial
Find one pair of positive integers $a,b$ such that $ab(a+b)$ is not divisible by $7$, but $(a+b)^7-a^7-b^7$ is divisible by $7^7$.