Olympiad problems

2010 Contests, 2

Tags: quadratics , real analysis , complex numbers , algebra , polynomial , sum of roots , number theory proposed

Find all non-negative integers $m,n,p,q$ such that \[ p^mq^n = (p+q)^2 +1 . \]

2010 Indonesia TST, 4

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

Prove that for all integers $ m$ and $ n$, the inequality \[ \dfrac{\phi(\gcd(2^m \plus{} 1,2^n \plus{} 1))}{\gcd(\phi(2^m \plus{} 1),\phi(2^n \plus{} 1))} \ge \dfrac{2\gcd(m,n)}{2^{\gcd(m,n)}}\] holds. [i]Nanang Susyanto, Jogjakarta [/i]

2008 Poland - Second Round, 3

Tags: number theory , relatively prime , number theory proposed

We have a positive integer $ n$ such that $ n \neq 3k$. Prove that there exists a positive integer $ m$ such that $ \forall_{k\in N \ k\geq m} \ k$ can be represented as a sum of digits of some multiplication of $ n$.

2008 Iran MO (3rd Round), 3

Tags: geometry , circumcircle , number theory proposed , number theory

Let $ P$ be a regular polygon. A regular sub-polygon of $ P$ is a subset of vertices of $ P$ with at least two vertices such that divides the circumcircle to equal arcs. Prove that there is a subset of vertices of $ P$ such that its intersection with each regular sub-polygon has even number of vertices.

2008 Irish Math Olympiad, 3

Tags: number theory proposed , number theory

Find $ a_3,a_4,...,a{}_2{}_0{}_0{}_8$, such that $ a_i =\pm1$ for $ i=3,...,2008$ and $ \sum\limits_{i=3}^{2008} a_i2^i = 2008$ and show that the numbers $ a_3,a_4,...,a_{2008}$ are uniquely determined by these conditions.

2014 Iran MO (3rd Round), 3

Tags: number theory proposed , number theory

Let $n$ be a positive integer. Prove that there exists a natural number $m$ with exactly $n$ prime factors, such that for every positive integer $d$ the numbers in $\{1,2,3,\ldots,m\}$ of order $d$ modulo $m$ are multiples of $\phi (d)$. (15 points)

1999 Baltic Way, 20

Tags: number theory , prime numbers , number theory proposed

Let $a,b,c$ and $d$ be prime numbers such that $a>3b>6c>12d$ and $a^2-b^2+c^2-d^2=1749$. Determine all possible values of $a^2+b^2+c^2+d^2$ .

1997 Baltic Way, 6

Tags: modular arithmetic , number theory proposed , number theory

Find all triples $(a,b,c)$ of non-negative integers satisfying $a\ge b\ge c$ and \[1\cdot a^3+9\cdot b^2+9\cdot c+7=1997 \]

2017, SRMC, 4

Tags: number theory , srmc , number theory proposed

Let $p$ be a prime number such that $p\equiv 1\pmod 9$. Show that there exist an integer $n$ such that $n^3-3n+1$ is divisible by $p$.

1988 Iran MO (2nd round), 1

Tags: number theory proposed , number theory

Let $\{a_n \}_{n=1}^{\infty}$ be a sequence such that $a_1=\frac 12$ and \[a_n=\biggl( \frac{2n-3}{2n} \biggr) a_{n-1} \qquad \forall n \geq 2.\] Prove that for every positive integer $n,$ we have $\sum_{k=1}^n a_k <1.$

2005 Poland - Second Round, 1

Tags: number theory proposed , number theory

Find all positive integers $n$ for which $n^n+1$ and $(2n)^{2n}+1$ are prime numbers.

2010 Baltic Way, 17

Tags: modular arithmetic , number theory proposed , number theory

Find all positive integers $n$ such that the decimal representation of $n^2$ consists of odd digits only.

2011 ELMO Shortlist, 1

Tags: modular arithmetic , number theory proposed , number theory

Prove that $n^3-n-3$ is not a perfect square for any integer $n$. [i]Calvin Deng.[/i]

2007 France Team Selection Test, 1

Tags: number theory proposed , number theory

For a positive integer $a$, $a'$ is the integer obtained by the following method: the decimal writing of $a'$ is the inverse of the decimal writing of $a$ (the decimal writing of $a'$ can begin by zeros, but not the one of $a$); for instance if $a=2370$, $a'=0732$, that is $732$. Let $a_{1}$ be a positive integer, and $(a_{n})_{n \geq 1}$ the sequence defined by $a_{1}$ and the following formula for $n \geq 1$: \[a_{n+1}=a_{n}+a'_{n}. \] Can $a_{7}$ be prime?

1951 Miklós Schweitzer, 12

Tags: function , number theory proposed , number theory

By number-theoretical functions, we will understand integer-valued functions defined on the set of all integers. Are there number-theoretical functions $ f_0(x),f_1(x),f_2(x),\dots$ such that every number theoretical function $ F(x)$ can be uniquely represented in the form $ F(x)\equal{}\sum_{k\equal{}0}^{\infty}a_kf_k(x)$, $ a_0,a_1,a_2,\dots$ being integers?

1988 Iran MO (2nd round), 3

Tags: function , number theory proposed , number theory

Let $f : \mathbb N \to \mathbb N$ be a function satisfying \[f(f(m)+f(n))=m+n, \quad \forall m,n \in \mathbb N.\] Prove that $f(x)=x$ for all $x \in \mathbb N$.

1998 USAMO, 1

Tags: modular arithmetic , number theory proposed , number theory

Suppose that the set $\{1,2,\cdots, 1998\}$ has been partitioned into disjoint pairs $\{a_i,b_i\}$ ($1\leq i\leq 999$) so that for all $i$, $|a_i-b_i|$ equals $1$ or $6$. Prove that the sum \[ |a_1-b_1|+|a_2-b_2|+\cdots +|a_{999}-b_{999}| \] ends in the digit $9$.

2006 Vietnam Team Selection Test, 3

Tags: induction , number theory proposed , number theory

The real sequence $\{a_n|n=0,1,2,3,...\}$ defined $a_0=1$ and \[ a_{n+1}=\frac{1}{2}\left (a_{n}+\frac{1}{3 \cdot a_{n}} \right ). \] Denote \[ A_n=\frac{3}{3 \cdot a_n^2-1}. \] Prove that $A_n$ is a perfect square and it has at least $n$ distinct prime divisors.

2007 International Zhautykov Olympiad, 3

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

Show that there are an infinity of positive integers $n$ such that $2^{n}+3^{n}$ is divisible by $n^{2}$.

2011 All-Russian Olympiad Regional Round, 10.5

Tags: number theory proposed , number theory

Find all $a$ such that for any positive integer $n$, the number $an(n+2)(n+3)(n+4)$ is an integer. (Author: O. Podlipski) [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=57&t=427802](similar to Problem 5 of grade 9)[/url] Same problem for grades 10 and 11

2019 Turkey EGMO TST, 4

Tags: number theory , number theory proposed

Let $\sigma (n)$ shows the number of positive divisors of $n$. Let $s(n)$ be the number of positive divisors of $n+1$ such that for every divisor $a$, $a-1$ is also a divisor of $n$. Find the maximum value of $2s(n)- \sigma (n) $.

1995 All-Russian Olympiad, 3

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

Does there exist a sequence of natural numbers in which every natural number occurs exactly once, such that for each $k = 1, 2, 3, \dots$ the sum of the first $k$ terms of the sequence is divisible by $k$? [i]A. Shapovalov[/i]

2011 Turkey Team Selection Test, 3

Tags: floor function , limit , number theory , greatest common divisor , induction , prime factorization , number theory proposed

Let $t(n)$ be the sum of the digits in the binary representation of a positive integer $n,$ and let $k \geq 2$ be an integer. [b]a.[/b] Show that there exists a sequence $(a_i)_{i=1}^{\infty}$ of integers such that $a_m \geq 3$ is an odd integer and $t(a_1a_2 \cdots a_m)=k$ for all $m \geq 1.$ [b]b.[/b] Show that there is an integer $N$ such that $t(3 \cdot 5 \cdots (2m+1))>k$ for all integers $m \geq N.$

2014 Iran Team Selection Test, 3

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

prove for all $k> 1$ equation $(x+1)(x+2)...(x+k)=y^{2}$ has finite solutions.

2014 China National Olympiad, 2

Tags: number theory proposed , number theory

For the integer $n>1$, define $D(n)=\{ a-b\mid ab=n, a>b>0, a,b\in\mathbb{N} \}$. Prove that for any integer $k>1$, there exists pairwise distinct positive integers $n_1,n_2,\ldots,n_k$ such that $n_1,\ldots,n_k>1$ and $|D(n_1)\cap D(n_2)\cap\cdots\cap D(n_k)|\geq 2$.