Olympiad problems

2013 India IMO Training Camp, 1

For a prime $p$, a natural number $n$ and an integer $a$, we let $S_n(a,p)$ denote the exponent of $p$ in the prime factorisation of $a^{p^n} - 1$. For example, $S_1(4,3) = 2$ and $S_2(6,2) = 0$. Find all pairs $(n,p)$ such that $S_n(2013,p) = 100$.

2001 India IMO Training Camp, 1

Tags: algebra , polynomial , complex numbers , number theory proposed , number theory

Complex numbers $\alpha$ , $\beta$ , $\gamma$ have the property that $\alpha^k +\beta^k +\gamma^k$ is an integer for every natural number $k$. Prove that the polynomial \[(x-\alpha)(x-\beta )(x-\gamma )\] has integer coefficients.

2016 Bulgaria JBMO TST, 3

Tags: number theory proposed , number theory

Let $ M (x,y)=x^2+xy-2y $ , x,y are positive integers a) Solve in positive integers $ x^2+xy-2y=64 $ b) Prove that if M (x,y) is a perfect square, then x+y+2 is composite if x>2.

2007 Junior Balkan Team Selection Tests - Romania, 1

Tags: floor function , number theory proposed , number theory

Find all nonzero subsets $A$ of the set $\left\{2,3,4,5,\cdots\right\}$ such that $\forall n\in A$, we have that $n^{2}+4$ and $\left\lfloor{\sqrt{n}\right\rfloor}+1$ are both in $A$.

2010 Indonesia TST, 2

Tags: modular arithmetic , number theory , number theory proposed

Let $ A\equal{}\{n: 1 \le n \le 2009^{2009},n \in \mathbb{N} \}$ and let $ S\equal{}\{n: n \in A,\gcd \left(n,2009^{2009}\right)\equal{}1\}$. Let $ P$ be the product of all elements of $ S$. Prove that \[ P \equiv 1 \pmod{2009^{2009}}.\] [i]Nanang Susyanto, Jogjakarta[/i]

2001 Poland - Second Round, 1

Tags: modular arithmetic , arithmetic sequence , irrational number , number theory proposed , number theory

Find all integers $n\ge 3$ for which the following statement is true: Any arithmetic progression $a_1,\ldots ,a_n$ with $n$ terms for which $a_1+2a_2+\ldots+na_n$ is rational contains at least one rational term.

2004 Baltic Way, 6

Tags: geometry , 3D geometry , number theory proposed , number theory

A positive integer is written on each of the six faces of a cube. For each vertex of the cube we compute the product of the numbers on the three adjacent faces. The sum of these products is $1001$. What is the sum of the six numbers on the faces?

2010 Contests, 2

Tags: induction , number theory proposed , number theory

Determine if there are positive integers $a, b$ such that all terms of the sequence defined by \[ x_{1}= 2010,x_{2}= 2011\\ x_{n+2}= x_{n}+ x_{n+1}+a\sqrt{x_{n}x_{n+1}+b}\quad (n\ge 1) \] are integers.

2000 Baltic Way, 13

Tags: arithmetic sequence , number theory proposed , number theory

Let $a_1,a_2 ,\ldots, a_n$ be an arithmetic progression of integers such that $i|a_i$ for $i=1, 2,\ldots ,n-1$ and $n\nmid a_n$. Prove that $n$ is a prime power.

1994 Romania TST for IMO, 1:

Tags: quadratics , modular arithmetic , number theory proposed , number theory

Find the smallest nomial of this sequence that $a_1=1993^{1994^{1995}}$ and \[ a_{n+1}=\begin{cases}\frac{a_n}{2}&\text{if $n$ is even}\\a_n+7 &\text{if $n$ is odd.} \end{cases} \]

2015 Brazil National Olympiad, 3

Tags: number theory proposed , number theory , prime factorization

Given a natural $n>1$ and its prime fatorization $n=p_1^{\alpha 1}p_2^{\alpha_2} \cdots p_k^{\alpha_k}$, its [i]false derived[/i] is defined by $$f(n)=\alpha_1p_1^{\alpha_1-1}\alpha_2p_2^{\alpha_2-1}...\alpha_kp_k^{\alpha_k-1}.$$ Prove that there exist infinitely many naturals $n$ such that $f(n)=f(n-1)+1$.

2010 Indonesia TST, 1

Tags: algebra , polynomial , number theory , relatively prime , number theory proposed

find all pairs of relatively prime natural numbers $ (m,n) $ in such a way that there exists non constant polynomial f satisfying \[ gcd(a+b+1, mf(a)+nf(b) > 1 \] for every natural numbers $ a $ and $ b $

2005 Baltic Way, 16

Tags: number theory , greatest common divisor , number theory proposed

Let $n$ be a positive integer, let $p$ be prime and let $q$ be a divisor of $(n + 1)^p - n^p$. Show that $p$ divides $q - 1$.

2004 Silk Road, 2

Tags: number theory proposed , number theory

find all primes $p$, for which exist natural numbers, such that $p=m^2+n^2$ and $p|(m^3+n^3-4)$.

2008 Iran MO (3rd Round), 1

Tags: induction , number theory proposed , number theory

Let $ k>1$ be an integer. Prove that there exists infinitely many natural numbers such as $ n$ such that: \[ n|1^n\plus{}2^n\plus{}\dots\plus{}k^n\]

2009 Turkey MO (2nd round), 1

Tags: modular arithmetic , number theory proposed , number theory

Find all prime numbers $p$ for which $p^3-4p+9$ is a perfect square.

2001 Poland - Second Round, 1

Tags: algebra , polynomial , modular arithmetic , number theory proposed , number theory

Let $k,n>1$ be integers such that the number $p=2k-1$ is prime. Prove that, if the number $\binom{n}{2}-\binom{k}{2}$ is divisible by $p$, then it is divisible by $p^2$.

2006 Moldova Team Selection Test, 1

Tags: number theory proposed , number theory

Determine all even numbers $n$, $n \in \mathbb N$ such that \[{ \frac{1}{d_{1}}+\frac{1}{d_{2}}+ \cdots +\frac{1}{d_{k}}=\frac{1620}{1003}}, \] where $d_1, d_2, \ldots, d_k$ are all different divisors of $n$.

2001 JBMO ShortLists, 2

Tags: modular arithmetic , number theory proposed , number theory

Let $P_n \ (n=3,4,5,6,7)$ be the set of positive integers $n^k+n^l+n^m$, where $k,l,m$ are positive integers. Find $n$ such that: i) In the set $P_n$ there are infinitely many squares. ii) In the set $P_n$ there are no squares.

1990 India National Olympiad, 2

Tags: calculus , integration , quadratics , number theory proposed , number theory , Diophantine equation

Determine all non-negative integral pairs $ (x, y)$ for which \[ (xy \minus{} 7)^2 \equal{} x^2 \plus{} y^2.\]

2012 Pre - Vietnam Mathematical Olympiad, 1

Tags: modular arithmetic , number theory proposed , number theory

Let $n \geq 2$ be a positive integer. Suppose there exist non-negative integers ${n_1},{n_2},\ldots,{n_k}$ such that $2^n - 1 \mid \sum_{i = 1}^k {{2^{{n_i}}}}$. Prove that $k \ge n$.

2005 Moldova Team Selection Test, 3

Tags: calculus , integration , function , number theory proposed , number theory

\[A=3\sum_{m=1}^{n^2}(\frac12-\{\sqrt{m}\})\] where $n$ is an positive integer. Find the largest $k$ such that $n^k$ divides $[A]$.

2014 ELMO Shortlist, 10

Tags: geometry , 3D geometry , modular arithmetic , number theory proposed , number theory

Find all positive integer bases $b \ge 9$ so that the number \[ \frac{{\overbrace{11 \cdots 1}^{n-1 \ 1's}0\overbrace{77 \cdots 7}^{n-1\ 7's}8\overbrace{11 \cdots 1}^{n \ 1's}}_b}{3} \] is a perfect cube in base 10 for all sufficiently large positive integers $n$. [i]Proposed by Yang Liu[/i]

2002 Tournament Of Towns, 7

Tags: floor function , number theory proposed , number theory

Do there exist irrational numbers $a,b$ both greater than $1$, such that $\lfloor{a^m}\rfloor\neq \lfloor{b^n}\rfloor$ for all $m,n\in\mathbb{N}$ ?

2009 China Western Mathematical Olympiad, 4

Tags: modular arithmetic , induction , number theory proposed , number theory

Prove that for every given positive integer $k$, there exist infinitely many $n$, such that $2^{n}+3^{n}-1, 2^{n}+3^{n}-2,\ldots, 2^{n}+3^{n}-k$ are all composite numbers.