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

2008 China Team Selection Test, 2
Tags: modular arithmetic , relatively prime , number theory
Let $ n > 1$ be an integer, and $ n$ can divide $ 2^{\phi(n)} \plus{} 3^{\phi(n)} \plus{} \cdots \plus{} n^{\phi(n)},$ let $ p_{1},p_{2},\cdots,p_{k}$ be all distinct prime divisors of $ n$. Show that $ \frac {1}{p_{1}} \plus{} \frac {1}{p_{2}} \plus{} \cdots \plus{} \frac {1}{p_{k}} \plus{} \frac {1}{p_{1}p_{2}\cdots p_{k}}$ is an integer. ( where $ \phi(n)$ is defined as the number of positive integers $ \leq n$ that are relatively prime to $ n$.)

1998 Turkey MO (2nd round), 1
Tags: modular arithmetic , quadratic , number theory
Find all positive integers $x$ and $n$ such that ${{x}^{3}}+3367={{2}^{n}}$.

Oliforum Contest IV 2013, 1
Tags: modular arithmetic , greatest common divisor , number theory
Given a prime $p$, consider integers $0<a<b<c<d<p$ such that $a^4\equiv b^4\equiv c^4\equiv d^4\pmod{p}$. Show that \[a+b+c+d\mid a^{2013}+b^{2013}+c^{2013}+d^{2013}\]

1994 AIME Problems, 1
Tags: modular arithmetic , arithmetic sequence , number theory
The increasing sequence $3, 15, 24, 48, \ldots$ consists of those positive multiples of 3 that are one less than a perfect square. What is the remainder when the 1994th term of the sequence is divided by 1000?

2006 Czech-Polish-Slovak Match, 4
Tags: modular arithmetic , number theory
Show that for every integer $k \ge 1$ there is a positive integer $n$ such that the decimal representation of $2^n$ contains a block of exactly $k$ zeros, i.e. $2^n = \dots a00 \dots 0b \cdots$ with $k$ zeros and $a, b \ne 0$.

2025 Turkey Team Selection Test, 9
Tags: number theory , number theory with sequences , modular arithmetic , modular congruences
Let \(n\) be a positive integer. For every positive integer $1 \leq k \leq n$ the sequence ${\displaystyle {\{ a_{i}+ki\}}_{i=1}^{n }}$ is defined, where $a_1,a_2, \dots ,a_n$ are integers. Among these \(n\) sequences, for at most how many of them does all the elements of the sequence give different remainders when divided by \(n\)?

1992 IMO Longlists, 63
Tags: modular arithmetic , quadratic , number theory , divisibility
Let $a$ and $b$ be integers. Prove that $\frac{2a^2-1}{b^2+2}$ is not an integer.

2006 AIME Problems, 9
Tags: ratio , modular arithmetic , logarithm
The sequence $a_1, a_2, \ldots$ is geometric with $a_1=a$ and common ratio $r$, where $a$ and $r$ are positive integers. Given that $\log_8 a_1+\log_8 a_2+\cdots+\log_8 a_{12} = 2006,$ find the number of possible ordered pairs $(a,r)$.

1999 Baltic Way, 16
Tags: modular arithmetic , number theory
Find the smallest positive integer $k$ which is representable in the form $k=19^n-5^m$ for some positive integers $m$ and $n$.

1997 All-Russian Olympiad, 1
Tags: modular arithmetic , number theory
Find all integer solutions of the equation $(x^2 - y^2)^2 = 1 + 16y$. [i]M. Sonkin[/i]

1990 Polish MO Finals, 3
Tags: modular arithmetic , number theory
Prove that for all integers $n > 2$, \[ 3| \sum\limits_{i=0}^{[n/3]} (-1)^i C _n ^{3i} \]

2010 IMO Shortlist, 4
Tags: modular arithmetic , divisibility , number theory
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]

2010 Peru IMO TST, 9
Tags: modular arithmetic , number theory , sequence , divisibility
Find all positive integers $n$ such that there exists a sequence of positive integers $a_1$, $a_2$,$\ldots$, $a_n$ satisfying: \[a_{k+1}=\frac{a_k^2+1}{a_{k-1}+1}-1\] for every $k$ with $2\leq k\leq n-1$. [i]Proposed by North Korea[/i]

2002 Indonesia MO, 5
Tags: modular arithmetic , algebra
Nine of the numbers $4, 5, 6, 7, 8, 12, 13, 16, 18, 19$ are going to be inputted to the empty cells in the following table: $\begin{array} {|c|c|c|} \cline{1-3} 10 & & \\ \cline{1-3} & & 9 \\ \cline{1-3} & 3 & \\ \cline{1-3} 11 & & 17 \\ \cline{1-3} & 20 & \\ \cline{1-3} \end{array}$ such that each row sums to the same number, and each column sums to the same number. Determine all possible arrangements.

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.

2010 BMO TST, 1
Tags: modular arithmetic , number theory
[b]a) [/b]Is the number $ 1111\cdots11$ (with $ 2010$ ones) a prime number? [b]b)[/b] Prove that every prime factor of $ 1111\cdots11$ (with $ 2011$ ones) is of the form $ 4022j\plus{}1$ where $ j$ is a natural number.

2012 Regional Olympiad of Mexico Center Zone, 2
Tags: modular arithmetic , number theory
Let $m, n$ integers such that: $(n-1)^3+n^3+(n+1)^3=m^3$ Prove that 4 divides $n$

2009 Ukraine Team Selection Test, 7
Tags: number theory , modular arithmetic , sequence , divisibility
Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be distinct positive integers, $ n\ge 3$. Prove that there exist distinct indices $ i$ and $ j$ such that $ a_i \plus{} a_j$ does not divide any of the numbers $ 3a_1$, $ 3a_2$, $ \ldots$, $ 3a_n$. [i]Proposed by Mohsen Jamaali, Iran[/i]

2005 QEDMO 1st, 1 (Z4)
Tags: geometry , 3d geometry , induction , modular arithmetic , number theory
Prove that every integer can be written as sum of $5$ third powers of integers.

2012 IMO Shortlist, N8
Tags: gauss , modular arithmetic , number theory , divisibility , prime
Prove that for every prime $p>100$ and every integer $r$, there exist two integers $a$ and $b$ such that $p$ divides $a^2+b^5-r$.

2011 Morocco TST, 1
Tags: binomial coefficient , number theory , summation , divisibility , modular arithmetic
Prove that for any n natural, the number \[ \sum \limits_{k=0}^{n} \binom{2n+1}{2k+1} 2^{3k} \] cannot be divided by $5$.

2005 Baltic Way, 9
Tags: geometry , rectangle , modular arithmetic , combinatorics
A rectangle is divided into $200\times 3$ unit squares. Prove that the number of ways of splitting this rectangle into rectangles of size $1\times 2$ is divisible by $3$.

2007 ITest, 11
Tags: modular arithmetic
Consider the "tower of power" $2^{2^{2^{.^{.^{.^2}}}}}$, where there are $2007$ twos including the base. What is the last (units) digit of this number? $\textbf{(A) }0\hspace{14em}\textbf{(B) }1\hspace{14em}\textbf{(C) }2$ $\textbf{(D) }3\hspace{14em}\textbf{(E) }4\hspace{14em}\textbf{(F) }5$ $\textbf{(G) }6\hspace{14em}\textbf{(H) }7\hspace{14em}\textbf{(I) }8$ $\textbf{(J) }9\hspace{14.3em}\textbf{(K) }2007$

PEN H Problems, 50
Tags: diophantine equation , modular arithmetic
Show that the equation $y^{2}=x^{3}+2a^{3}-3b^2$ has no solution in integers if $ab \neq 0$, $a \not\equiv 1 \; \pmod{3}$, $3$ does not divide $b$, $a$ is odd if $b$ is even, and $p=t^2 +27u^2$ has a solution in integers $t,u$ if $p \vert a$ and $p \equiv 1 \; \pmod{3}$.

2006 USA Team Selection Test, 4
Tags: induction , modular arithmetic , number theory
Let $n$ be a positive integer. Find, with proof, the least positive integer $d_{n}$ which cannot be expressed in the form \[\sum_{i=1}^{n}(-1)^{a_{i}}2^{b_{i}},\] where $a_{i}$ and $b_{i}$ are nonnegative integers for each $i.$