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

2013 Bulgaria National Olympiad, 1
Tags: modular arithmetic , diophantine equation , number theory
Find all prime numbers $p,q$, for which $p^{q+1}+q^{p+1}$ is a perfect square. [i]Proposed by P. Boyvalenkov[/i]

2007 Germany Team Selection Test, 3
Tags: divisibility , exponential , number theory , modular arithmetic
For all positive integers $n$, show that there exists a positive integer $m$ such that $n$ divides $2^{m} + m$. [i]Proposed by Juhan Aru, Estonia[/i]

1991 IMO Shortlist, 15
Tags: periodic sequence , digit , decimal representation , factorial , modular arithmetic
Let $ a_n$ be the last nonzero digit in the decimal representation of the number $ n!.$ Does the sequence $ a_1, a_2, \ldots, a_n, \ldots$ become periodic after a finite number of terms?

PEN D Problems, 1
Tags: modular arithmetic , floor function , congruence
If $p$ is an odd prime, prove that \[{k \choose p}\equiv \left\lfloor \frac{k}{p}\right\rfloor \pmod{p}.\]

1997 Brazil Team Selection Test, Problem 3
Tags: linear algebra , number theory , matrix , modular arithmetic
Find all positive integers $x>1, y$ and primes $p,q$ such that $p^{x}=2^{y}+q^{x}$

2010 International Zhautykov Olympiad, 2
Tags: number theory , combinatorics , modular arithmetic , greatest common divisor
In every vertex of a regular $n$ -gon exactly one chip is placed. At each $step$ one can exchange any two neighbouring chips. Find the least number of steps necessary to reach the arrangement where every chip is moved by $[\frac{n}{2}]$ positions clockwise from its initial position.

1971 IMO Shortlist, 15
Tags: combinatorics , modular arithmetic
Natural numbers from $1$ to $99$ (not necessarily distinct) are written on $99$ cards. It is given that the sum of the numbers on any subset of cards (including the set of all cards) is not divisible by $100$. Show that all the cards contain the same number.

2010 USA Team Selection Test, 1
Tags: algebra , calculus , induction , polynomial , greatest common divisor , number theory , modular arithmetic
Let $P$ be a polynomial with integer coefficients such that $P(0)=0$ and \[\gcd(P(0), P(1), P(2), \ldots ) = 1.\] Show there are infinitely many $n$ such that \[\gcd(P(n)- P(0), P(n+1)-P(1), P(n+2)-P(2), \ldots) = n.\]

2000 AMC 12/AHSME, 18
Tags: modular arithmetic
In year $ N$, the $ 300^\text{th}$ day of the year is a Tuesday. In year $ N \plus{} 1$, the $ 200^{\text{th}}$ day of the year is also a Tuesday. On what day of the week did the $ 100^\text{th}$ day of year $ N \minus{} 1$ occur? $ \textbf{(A)}\ \text{Thursday} \qquad \textbf{(B)}\ \text{Friday} \qquad \textbf{(C)}\ \text{Saturday} \qquad \textbf{(D)}\ \text{Sunday} \qquad \textbf{(E)}\ \text{Monday}$

1979 IMO Longlists, 23
Tags: modular arithmetic , number theory
Consider the set $E$ consisting of pairs of integers $(a, b)$, with $a \geq 1$ and $b \geq 1$, that satisfy in the decimal system the following properties: [b](i)[/b] $b$ is written with three digits, as $\overline{\alpha_2\alpha_1\alpha_0}$, $\alpha_2 \neq 0$; [b](ii)[/b] $a$ is written as $\overline{\beta_p \ldots \beta_1\beta_0}$ for some $p$; [b](iii)[/b] $(a + b)^2$ is written as $\overline{\beta_p\ldots \beta_1 \beta_0 \alpha_2 \alpha_1 \alpha_0}.$ Find the elements of $E$.

2002 Turkey MO (2nd round), 1
Tags: quadratic , prime number , modular arithmetic , number theory
Find all prime numbers $p$ for which the number of ordered pairs of integers $(x, y)$ with $0\leq x, y < p$ satisfying the condition \[y^2 \equiv  x^3 - x \pmod p\] is exactly $p.$

2002 Manhattan Mathematical Olympiad, 1
Tags: modular arithmetic
Prove that if an integer $n$ is of the form $4m+3$, where $m$ is another integer, then $n$ is not a sum of two perfect squares (a perfect square is an integer which is the square of some integer).

2000 Junior Balkan Team Selection Tests - Romania, 2
Tags: modular arithmetic , algebra , number theory
Let be a natural power of two. Find the number of numbers equivalent with $ 1 $ modulo $ 3 $ that divide it. [i]Dan Brânzei[/i]

2014 Indonesia MO, 4
Tags: number theory , modular arithmetic
A positive integer is called [i]beautiful[/i] if it can be represented in the form $\dfrac{x^2+y^2}{x+y}$ for two distinct positive integers $x,y$. A positive integer that is not beautiful is [i]ugly[/i]. a) Prove that $2014$ is a product of a beautiful number and an ugly number. b) Prove that the product of two ugly numbers is also ugly.

2012 Canadian Mathematical Olympiad Qualification Repechage, 2
Tags: number theory , modular arithmetic
Given a positive integer $m$, let $d(m)$ be the number of positive divisors of $m$. Determine all positive integers $n$ such that $d(n) +d(n+ 1) = 5$.

2006 China Team Selection Test, 1
Tags: induction , polynomial , pigeonhole principle , algebra , modular arithmetic
Let $k$ be an odd number that is greater than or equal to $3$. Prove that there exists a $k^{th}$-degree integer-valued polynomial with non-integer-coefficients that has the following properties: (1) $f(0)=0$ and $f(1)=1$; and. (2) There exist infinitely many positive integers $n$ so that if the following equation: \[ n= f(x_1)+\cdots+f(x_s), \] has integer solutions $x_1, x_2, \dots, x_s$, then $s \geq 2^k-1$.

2009 AIME Problems, 8
Tags: real analysis , derivative , geometric series , modular arithmetic , calculus
Let $ S \equal{} \{2^0,2^1,2^2,\ldots,2^{10}\}$. Consider all possible positive differences of pairs of elements of $ S$. Let $ N$ be the sum of all of these differences. Find the remainder when $ N$ is divided by $ 1000$.

2005 MOP Homework, 1
Tags: combinatorics , modular arithmetic
Let $X$ be a set with $n$ elements and $0 \le k \le n$. Let $a_{n,k}$ be the maximum number of permutations of the set $X$ such that every two of them have at least $k$ common components (where a common component of $f$ and g is an $x \in X$ such that $f(x) = g(x)$). Let $b_{n,k}$ be the maximum number of permutations of the set $X$ such that every two of them have at most $k$ common components. (a) Show that $a_{n,k} \cdot b_{n,k-1} \le n!$. (b) Let $p$ be prime, and find the exact value of $a_{p,2}$.

1993 Baltic Way, 2
Tags: number theory , modular arithmetic
Do there exist positive integers $a>b>1$ such that for each positive integer $k$ there exists a positive integer $n$ for which $an+b$ is a $k$-th power of a positive integer?

2002 Kazakhstan National Olympiad, 3
Tags: modular arithmetic , extremal combinatorics , maximization , combinatorics
Let $A = (a_1, a_2, \ldots, a_{2001})$ be a sequence of positive integers. Let $m$ be the number of 3-element subsequences $(a_i,a_j,a_k)$ with $1 \leq i < j < k \leq 2001$, such that $a_j = a_i + 1$ and $a_k = a_j + 1$. Considering all such sequences $A$, find the greatest value of $m$.

2004 Tournament Of Towns, 3
Tags: algebra , modular arithmetic
Bucket $A$ contains 3 litres of syrup. Bucket $B$ contains $n$ litres of water. Bucket $C$ is empty. We can perform any combination of the following operations: - Pour away the entire amount in bucket $X$, - Pour the entire amount in bucket $X$ into bucket $Y$, - Pour from bucket $X$ into bucket $Y$ until buckets $Y$ and $Z$ contain the same amount. [b](a)[/b] How can we obtain 10 litres of 30% syrup if $n = 20$? [b](b)[/b] Determine all possible values of $n$ for which the task in (a) is possible.

2001 USA Team Selection Test, 4
Tags: induction , pigeonhole principle , combinatorics , modular arithmetic
There are 51 senators in a senate. The senate needs to be divided into $n$ committees so that each senator is on one committee. Each senator hates exactly three other senators. (If senator A hates senator B, then senator B does [i]not[/i] necessarily hate senator A.) Find the smallest $n$ such that it is always possible to arrange the committees so that no senator hates another senator on his or her committee.

2012 NIMO Problems, 2
Tags: relatively prime , modular arithmetic , number theory
Compute the number of positive integers $n < 2012$ that share exactly two positive factors with 2012. [i]Proposed by Aaron Lin[/i]

1998 Baltic Way, 4
Tags: modular arithmetic , polynomial , algebra , number theory
Let $P$ be a polynomial with integer coefficients. Suppose that for $n=1,2,3,\ldots ,1998$ the number $P(n)$ is a three-digit positive integer. Prove that the polynomial $P$ has no integer roots.

2002 Hong kong National Olympiad, 4
Tags: quadratic , number theory , modular arithmetic
Let $p$ be a prime number such that $p\equiv 1\pmod{4}$. Determine $\sum_{k=1}^{\frac{p-1}{2}}\left \lbrace \frac{k^2}{p} \right \rbrace$, where $\{x\}=x-[x]$.