Olympiad problems

2013 All-Russian Olympiad, 3

Find all positive integers $k$ such that for the first $k$ prime numbers $2, 3, \ldots, p_k$ there exist positive integers $a$ and $n>1$, such that $2\cdot 3\cdot\ldots\cdot p_k - 1=a^n$. [i]V. Senderov[/i]

2000 Bundeswettbewerb Mathematik, 2

Tags: number theory proposed , number theory

Prove that for every integer $n \geq 2$ there exist $n$ different positive integers such that for any two of these integers $a$ and $b$ their sum $a+b$ is divisible by their difference $a - b.$

2006 Baltic Way, 19

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

Does there exist a sequence $a_1,a_2,a_3,\ldots $ of positive integers such that the sum of every $n$ consecutive elements is divisible by $n^2$ for every positive integer $n$?

1992 Brazil National Olympiad, 5

Tags: number theory proposed , number theory

Let $d(n)=\sum_{0<d|n}{1}$. Show that, for any natural $n>1$, \[ \sum_{2 \leq i \leq n}{\frac{1}{i}} \leq \sum{\frac{d(i)}{n}} \leq \sum_{1 \leq i \leq n}{\frac{1}{i}} \]

2009 Canadian Mathematical Olympiad Qualification Repechage, 5

Tags: quadratics , algebra , quadratic formula , number theory proposed , number theory

Determine all positive integers $n$ for which $n(n + 9)$ is a perfect square.

2024 Austrian MO National Competition, 6

Tags: number theory , number theory proposed , modular arithmetic , modulo , Sum of Squares

For each prime number $p$, determine the number of residue classes modulo $p$ which can be represented as $a^2+b^2$ modulo $p$, where $a$ and $b$ are arbitrary integers. [i](Daniel Holmes)[/i]

2002 Iran MO (3rd Round), 23

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

Find all polynomials $p$ with real coefficients that if for a real $a$,$p(a)$ is integer then $a$ is integer.

2014 Contests, 2

Tags: number theory , number theory proposed

Sergei chooses two different natural numbers $a$ and $b$. He writes four numbers in a notebook: $a$, $a+2$, $b$ and $b+2$. He then writes all six pairwise products of the numbers of notebook on the blackboard. Let $S$ be the number of perfect squares on the blackboard. Find the maximum value of $S$. [i]S. Berlov[/i]

1996 All-Russian Olympiad, 3

Tags: number theory proposed , number theory

Let $x, y, p, n$, and $k$ be positive integers such that $x^n + y^n = p^k$. Prove that if $n > 1$ is odd, and $p$ is an odd prime, then $n$ is a power of $p$. [i]A. Kovaldji, V. Senderov[/i]

1998 Hungary-Israel Binational, 3

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

Let $ a, b, c, m, n$ be positive integers. Consider the trinomial $ f (x) = ax^{2}+bx+c$. Show that there exist $ n$ consecutive natural numbers $ a_{1}, a_{2}, . . . , a_{n}$ such that each of the numbers $ f (a_{1}), f (a_{2}), . . . , f (a_{n})$ has at least $ m$ different prime factors.

2003 Baltic Way, 18

Tags: number theory proposed , number theory

Every integer is to be coloured blue, green, red, or yellow. Can this be done in such a way that if $a, b, c, d$ are not all $0$ and have the same colour, then $3a-2b \neq 2c-3d$? [size=85][color=#0000FF][Mod edit: Question fixed][/color][/size]

2015 German National Olympiad, 2

Tags: number theory , number theory proposed , equation , Germany

A positive integer $n$ is called [i]smooth[/i] if there exist integers $a_1,a_2,\dotsc,a_n$ satisfying \[a_1+a_2+\dotsc+a_n=a_1 \cdot a_2 \cdot \dotsc \cdot a_n=n.\] Find all smooth numbers.

2010 Canadian Mathematical Olympiad Qualification Repechage, 4

Tags: number theory , prime numbers , number theory proposed

Determine the smallest positive integer $m$ with the property that $m^3-3m^2+2m$ is divisible by both $79$ and $83$.

1992 Baltic Way, 3

Tags: geometry , 3D geometry , arithmetic sequence , algebra , factorization , sum of cubes , number theory proposed

Find an infinite non-constant arithmetic progression of natural numbers such that each term is neither a sum of two squares, nor a sum of two cubes (of natural numbers).

1993 IberoAmerican, 1

Tags: number theory proposed , number theory

A number is called [i]capicua[/i] if when it is written in decimal notation, it can be read equal from left to right as from right to left; for example: $8, 23432, 6446$. Let $x_1<x_2<\cdots<x_i<x_{i+1},\cdots$ be the sequence of all capicua numbers. For each $i$ define $y_i=x_{i+1}-x_i$. How many distinct primes contains the set $\{y_1,y_2, \ldots\}$?

2014 ELMO Shortlist, 7

Tags: number theory , number theory proposed , Hi

Find all triples $(a,b,c)$ of positive integers such that if $n$ is not divisible by any prime less than $2014$, then $n+c$ divides $a^n+b^n+n$. [i]Proposed by Evan Chen[/i]

2002 Iran MO (3rd Round), 14

Tags: number theory proposed , number theory

A subset $S$ of $\mathbb N$ is [i]eventually linear[/i] iff there are $k,N\in\mathbb N$ that for $n>N,n\in S\Longleftrightarrow k|n$. Let $S$ be a subset of $\mathbb N$ that is closed under addition. Prove that $S$ is eventually linear.

2012 JBMO TST - Macedonia, 4

Tags: number theory proposed , number theory

Find all primes $p$ and $q$ such that $(p+q)^p = (q-p)^{(2q-1)}$

2013 India IMO Training Camp, 1

Tags: number theory proposed , number theory

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 $S_n(2013,p) = 100$.

2004 ITAMO, 1

Tags: number theory proposed , number theory

Observing the temperatures recorded in Cesenatico during the December and January, Stefano noticed an interesting coincidence: in each day of this period, the low temperature is equal to the sum of the low temperatures the preceeding day and the succeeding day. Given that the low temperatures in December $3$ and January $31$ were $5^\circ \text C$ and $2^\circ \text C$ respectively, find the low temperature in December $25$.

1994 Baltic Way, 10

Tags: absolute value , number theory proposed , number theory

How many positive integers satisfy the following three conditions: a) All digits of the number are from the set $\{1,2,3,4,5\}$; b) The absolute value of the difference between any two consecutive digits is $1$; c) The integer has $1994$ digits?

1973 Miklós Schweitzer, 3

Tags: logarithms , floor function , number theory , prime numbers , number theory proposed

Find a constant $ c > 1$ with the property that, for arbitrary positive integers $ n$ and $ k$ such that $ n>c^k$, the number of distinct prime factors of $ \binom{n}{k}$ is at least $ k$. [i]P. Erdos[/i]

1999 Baltic Way, 18

Tags: modular arithmetic , number theory proposed , number theory

Let $m$ be a positive integer such that $m=2\pmod{4}$. Show that there exists at most one factorization $m=ab$ where $a$ and $b$ are positive integers satisfying \[0<a-b<\sqrt{5+4\sqrt{4m+1}}\]

1993 All-Russian Olympiad, 1

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

For integers $x$, $y$, and $z$, we have $(x-y)(y-z)(z-x)=x+y+z$. Prove that $27|x+y+z$.

2010 Kazakhstan National Olympiad, 6

Tags: modular arithmetic , number theory proposed , number theory

Let numbers $1,2,3,...,2010$ stand in a row at random. Consider row, obtain by next rule: For any number we sum it and it's number in a row (For example for row $( 2,7,4)$ we consider a row $(2+1;7+2;4+3)=(3;9;7)$ ); Proved, that in resulting row we can found two equals numbers, or two numbers, which is differ by $2010$