Olympiad problems

2005 Irish Math Olympiad, 3

Let $ x$ be an integer and $ y,z,w$ be odd positive integers. Prove that $ 17$ divides $ x^{y^{z^w}}\minus{}x^{y^z}$.

2009 Greece National Olympiad, 1

Tags: number theory unsolved , number theory , IOQM

Find all positive integers $n$ such that the number \[A=\sqrt{\frac{9n-1}{n+7}}\] is rational.

2010 Postal Coaching, 5

Tags: algebra , polynomial , number theory unsolved , number theory

Let $p$ be a prime and $Q(x)$ be a polynomial with integer coeﬃcients such that $Q(0) = 0, \ Q(1) = 1$ and the remainder of $Q(n)$ is either $0$ or $1$ when divided by $p$, for every $n \in \mathbb{N}$. Prove that $Q(x)$ is of degree at least $p - 1$.

2016 Peru MO (ONEM), 4

Tags: number theory unsolved , number theory

Let $a>2$, $n>1$ integers such that $a^n-2^n$ is a perfect square. Prove that $a$ is a even number.

2013 ELMO Shortlist, 8

Tags: counting , distinguishability , modular arithmetic , algebra , polynomial , function , number theory unsolved

We define the [i]Fibonacci sequence[/i] $\{F_n\}_{n\ge0}$ by $F_0=0$, $F_1=1$, and for $n\ge2$, $F_n=F_{n-1}+F_{n-2}$; we define the [i]Stirling number of the second kind[/i] $S(n,k)$ as the number of ways to partition a set of $n\ge1$ distinguishable elements into $k\ge1$ indistinguishable nonempty subsets. For every positive integer $n$, let $t_n = \sum_{k=1}^{n} S(n,k) F_k$. Let $p\ge7$ be a prime. Prove that \[ t_{n+p^{2p}-1} \equiv t_n \pmod{p} \] for all $n\ge1$. [i]Proposed by Victor Wang[/i]

1972 IMO Longlists, 31

Tags: number theory unsolved , number theory

Find values of $n\in \mathbb{N}$ for which the fraction $\frac{3^n-2}{2^n-3}$ is reducible.

2014 India Regional Mathematical Olympiad, 6

Tags: inequalities , modular arithmetic , number theory unsolved , number theory

For any natural number, let $S(n)$ denote sum of digits of $n$. Find the number of $3$ digit numbers for which $S(S(n)) = 2$.

2011 Morocco National Olympiad, 4

Tags: modular arithmetic , number theory unsolved , number theory

Let $a, b, c, d, m, n$ be positive integers such that $a^{2}+b^{2}+c^{2}+d^{2}=1989$, $n^{2}=max\left \{ a,b,c,d \right \}$ and $a+b+c+d=m^{2}$. Find the values of $m$ and $n$.

2000 China National Olympiad, 2

Tags: number theory unsolved , number theory

Find all positive integers $n$ such that there exists integers $n_1,\ldots,n_k\ge 3$, for some integer $k$, satisfying \[n=n_1n_2\cdots n_k=2^{\frac{1}{2^k}(n_1-1)\cdots (n_k-1)}-1.\]

2006 Brazil National Olympiad, 4

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

A positive integer is [i]bold[/i] iff it has $8$ positive divisors that sum up to $3240$. For example, $2006$ is bold because its $8$ positive divisors, $1$, $2$, $17$, $34$, $59$, $118$, $1003$ and $2006$, sum up to $3240$. Find the smallest positive bold number.

2005 Irish Math Olympiad, 5

Tags: search , number theory unsolved , number theory

Suppose that $ m$ and $ n$ are odd integers such that $ m^2\minus{}n^2\plus{}1$ divides $ n^2\minus{}1$. Prove that $ m^2\minus{}n^2\plus{}1$ is a perfect square.

2005 Hungary-Israel Binational, 1

Tags: number theory unsolved , number theory

Does there exist a sequence of $2005$ consecutive positive integers that contains exactly $25$ prime numbers?

2011 Kosovo Team Selection Test, 3

Tags: modular arithmetic , number theory unsolved , number theory

Let $n$ be a natural number, for which we define $S(n)=\{1+g+g^2+...+g^{n-1}|g\in{\mathbb{N}},g\geq2\}$ $a)$ Prove that: $S(3)\cap S(4)=\varnothing$ $b)$ Determine: $S(3)\cap S(5)$

1994 Turkey MO (2nd round), 5

Tags: quadratics , calculus , integration , number theory unsolved , number theory

Find the set of all ordered pairs $(s,t)$ of positive integers such that \[t^{2}+1=s(s+1).\]

2005 Federal Competition For Advanced Students, Part 2, 1

Tags: number theory , least common multiple , number theory unsolved

Find all triples $(a,b,c)$ of natural numbers, such that $LCM(a,b,c)=a+b+c$

2014 Romania Team Selection Test, 4

Tags: induction , modular arithmetic , number theory , relatively prime , prime factorization , number theory unsolved

Let $k$ be a nonzero natural number and $m$ an odd natural number . Prove that there exist a natural number $n$ such that the number $m^n+n^m$ has at least $k$ distinct prime factors.

2005 Czech-Polish-Slovak Match, 6

Tags: modular arithmetic , number theory , Diophantine equation , prime factorization , number theory unsolved

Determine all pairs of integers $(x, y)$ satisfying the equation \[y(x + y) = x^3- 7x^2 + 11x - 3.\]

1981 Vietnam National Olympiad, 2

Tags: algebra , polynomial , calculus , integration , number theory unsolved , number theory

Consider the polynomials \[f(p) = p^{12} - p^{11} + 3p^{10} + 11p^3 - p^2 + 23p + 30;\] \[g(p) = p^3 + 2p + m.\] Find all integral values of $m$ for which $f$ is divisible by $g$.

2010 Dutch BxMO TST, 3

Tags: number theory unsolved , number theory

Let $N$ be the number of ordered 5-tuples $(a_{1}, a_{2}, a_{3}, a_{4}, a_{5})$ of positive integers satisfying $\frac{1}{a_{1}}+\frac{1}{a_{2}}+\frac{1}{a_{3}}+\frac{1}{a_{4}}+\frac{1}{a_{5}}=1$ Is $N$ even or odd? Oh and [b]HINTS ONLY[/b], please do not give full solutions. Thanks.

2009 Albania Team Selection Test, 4

Tags: number theory , relatively prime , number theory unsolved , auyesl

Find all the natural numbers $m,n$ such that $1+5 \cdot 2^m=n^2$.

2012 Indonesia TST, 4

Tags: function , number theory unsolved , number theory

Let $\mathbb{N}$ be the set of positive integers. For every $n \in \mathbb{N}$, define $d(n)$ as the number of positive divisors of $n$. Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that: a) $d(f(x)) = x$ for all $x \in \mathbb{N}$ b) $f(xy)$ divides $(x-1)y^{xy-1}f(x)$ for all $x,y \in \mathbb{N}$

2012 Indonesia TST, 4

Tags: number theory unsolved , number theory

The Fibonacci sequence $\{F_n\}$ is defined by $F_1 = F_2 = 1$ and $F_{n+2} = F_{n+1} + F_n$ for all positive integers $n$. Determine all triplets of positive integers $(k,m,n)$ such that $F_n = F_m^k$.

2000 All-Russian Olympiad, 2

Tags: ceiling function , logarithms , modular arithmetic , number theory unsolved , number theory

Tanya chose a natural number $X\le100$, and Sasha is trying to guess this number. He can select two natural numbers $M$ and $N$ less than $100$ and ask about $\gcd(X+M,N)$. Show that Sasha can determine Tanya's number with at most seven questions.

1992 Cono Sur Olympiad, 1

Tags: number theory unsolved , number theory

Find a positive integrer number $n$ such that, if yor put a number $2$ on the left and a number $1$ on the right, the new number is equal to $33n$.

2012 Olympic Revenge, 4

Tags: number theory unsolved , number theory

Say that two sets of positive integers $S, T$ are $\emph{k-equivalent}$ if the sum of the $i$th powers of elements of $S$ equals the sum of the $i$th powers of elements of $T$, for each $i= 1, 2, \ldots, k$. Given $k$, prove that there are infinitely many numbers $N$ such that $\{1,2,\ldots,N^{k+1}\}$ can be divided into $N$ subsets, all of which are $k$-equivalent to each other.