Olympiad problems

2014 Benelux, 3

For all integers $n\ge 2$ with the following property: [list] [*] for each pair of positive divisors $k,~\ell <n$, at least one of the numbers $2k-\ell$ and $2\ell-k$ is a (not necessarily positive) divisor of $n$ as well.[/list]

1995 South africa National Olympiad, 1

Tags: quadratics , number theory unsolved , number theory

Prove that there are no integers $m$ and $n$ such that \[19m^2+95mn+2000n^2=1995.\]

2010 Vietnam Team Selection Test, 3

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

Let $S_n $ be sum of squares of the coefficient of the polynomial $(1+x)^n$. Prove that $S_{2n} +1$ is not divisible by $3.$

2009 Kurschak Competition, 2

Tags: number theory unsolved , number theory

Find all positive integer pairs $(a,b)$ for which the set of positive integers can be partitioned into sets $H_1$ and $H_2$ such that neither $a$ nor $b$ can be represented as the difference of two numbers in $H_i$ for $i=1,2$.

Oliforum Contest III 2012, 1

Tags: modular arithmetic , number theory unsolved , number theory

Prove that exist infinite integers $n$ so that $n^2$ divides $2^n+3^n$. Thanks

2010 Balkan MO Shortlist, N2

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

Solve the following equation in positive integers: $x^{3} = 2y^{2} + 1 $

2018 IFYM, Sozopol, 8

Tags: number theory , number theory unsolved

Are there infinitely many positive integers that [b]can’t[/b] be presented as a sum of no more than fifteen fourth degrees of positive integers. (For example 15 isn’t such number as it can be presented as the sum of $15.1^4$)

2006 India IMO Training Camp, 3

Tags: number theory unsolved , number theory

Let $A_1,A_2,\cdots , A_n$ be arithmetic progressions of integers, each of $k$ terms, such that any two of these arithmetic progressions have at least two common elements. Suppose $b$ of these arithmetic progressions have common difference $d_1$ and the remaining arithmetic progressions have common difference $d_2$ where $0<b<n$. Prove that \[b \le 2\left(k-\frac{d_2}{gcd(d_1,d_2)}\right)-1.\]

2010 Tournament Of Towns, 7

Tags: search , invariant , function , algorithm , number theory unsolved , number theory

A multi-digit number is written on the blackboard. Susan puts in a number of plus signs between some pairs of adjacent digits. The addition is performed and the process is repeated with the sum. Prove that regardless of what number was initially on the blackboard, Susan can always obtain a single-digit number in at most ten steps.

2000 Irish Math Olympiad, 3

Tags: number theory , number theory unsolved

Let $ f(x)\equal{}5x^{13}\plus{}13x^5\plus{}9ax$. Find the least positive integer $ a$ such that $ 65$ divides $ f(x)$ for every integer $ x$.

2010 Contests, 1

Tags: modular arithmetic , number theory unsolved , number theory

a) Replace each letter in the following sum by a digit from $0$ to $9$, in such a way that the sum is correct. $\tab$ $\tab$ $ABC$ $\tab$ $\tab$ $DEF$ [u]$+GHI$[/u] $\tab$ $\tab$ $\tab$ $J J J$ Different letters must be replaced by different digits, and equal letters must be replaced by equal digits. Numbers $ABC$, $DEF$, $GHI$ and $JJJ$ cannot begin by $0$. b) Determine how many triples of numbers $(ABC,DEF,GHI)$ can be formed under the conditions given in a).

1989 IMO Longlists, 24

Tags: geometry , quadratics , number theory unsolved , number theory

Let $ a, b, c, d$ be positive integers such that $ ab \equal{} cd$ and $ a\plus{}b \equal{} c \minus{} d.$ Prove that there exists a right-angled triangle the measure of whose sides (in some unit) are integers and whose area measure is $ ab$ square units.

2000 France Team Selection Test, 2

Tags: function , number theory unsolved , number theory

A function from the positive integers to the positive integers satisfies these properties 1. $f(ab)=f(a)f(b)$ for any two coprime positive integers $a,b$. 2. $f(p+q)=f(p)+f(q)$ for any two primes $p,q$. Prove that $f(2)=2, f(3)=3, f(1999)=1999$.

2004 All-Russian Olympiad, 4

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

Is there a natural number $ n > 10^{1000}$ which is not divisible by 10 and which satisfies: in its decimal representation one can exchange two distinct non-zero digits such that the set of prime divisors does not change.

2000 Hungary-Israel Binational, 2

Tags: algebra , binomial theorem , number theory unsolved , number theory

Prove or disprove: For any positive integer $k$ there exists an integer $n > 1$ such that the binomial coeffcient $\binom{n}{i}$ is divisible by $k$ for any $1 \leq i \leq n-1.$

1996 South africa National Olympiad, 1

Tags: floor function , number theory unsolved , number theory

Find the highest power of $2$ that divides exactly into $1996!=1\times2\times\cdots\times1996$.

2014 South East Mathematical Olympiad, 3

Tags: modular arithmetic , number theory unsolved , number theory

Let $p$ be a primes ,$x,y,z $ be positive integers such that $x<y<z<p$ and $\{\frac{x^3}{p}\}=\{\frac{y^3}{p}\}=\{\frac{z^3}{p}\}$. Prove that $(x+y+z)|(x^5+y^5+z^5).$

2014 India IMO Training Camp, 3

Tags: function , number theory unsolved , number theory

For integers $a,b$ we define $f((a,b))=(2a,b-a)$ if $a<b$ and $f((a,b))=(a-b,2b)$ if $a\geq b$. Given a natural number $n>1$ show that there exist natural numbers $m,k$ with $m<n$ such that $f^{k}((n,m))=(m,n)$,where $f^{k}(x)=f(f(f(...f(x))))$,$f$ being composed with itself $k$ times.

2006 Tuymaada Olympiad, 1

Tags: number theory unsolved , number theory

Seven different odd primes are given. Is it possible that for any two of them, the difference of their eight powers to be divisible by all the remaining ones ? [i]Proposed by F. Petrov, K. Sukhov[/i]

1985 IMO Longlists, 64

Tags: number theory unsolved , number theory

Let $p$ be a prime. For which $k$ can the set $\{1, 2, \dots , k\}$ be partitioned into $p$ subsets with equal sums of elements ?

2007 Pre-Preparation Course Examination, 20

Tags: number theory unsolved , number theory

Let $m,n$ be two positive integers and $m \geq 2$. We know that for every positive integer $a$ such that $\gcd(a,n)=1$ we have $n|a^m-1$. Prove that $n \leq 4m(2^m-1)$.

2004 China Team Selection Test, 2

Tags: LaTeX , number theory unsolved , number theory

Let $p_1, p_2, \ldots, p_{25}$ are primes which don’t exceed 2004. Find the largest integer $T$ such that every positive integer $\leq T$ can be expressed as sums of distinct divisors of $(p_1\cdot p_2 \cdot \ldots \cdot p_{25})^{2004}.$

2005 Canada National Olympiad, 5

Tags: number theory unsolved , number theory

Let's say that an ordered triple of positive integers $(a,b,c)$ is [i]$n$-powerful[/i] if $a\le b\le c,\gcd (a,b,c)=1$ and $a^n+b^n+c^n$ is divisible by $a+b+c$. For example, $(1,2,2)$ is $5$-powerful. $a)$ Determine all ordered triples (if any) which are $n$-powerful for all $n\ge 1$. $b)$ Determine all ordered triples (if any) which are $2004$-powerful and $2005$-powerful, but not $2007$-powerful.

2013 Baltic Way, 20

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

Find all polynomials $f$ with non-negative integer coefficients such that for all primes $p$ and positive integers $n$ there exist a prime $q$ and a positive integer $m$ such that $f(p^n)=q^m$.

2002 Iran Team Selection Test, 9

Tags: function , calculus , integration , inequalities , number theory unsolved , number theory

$\pi(n)$ is the number of primes that are not bigger than $n$. For $n=2,3,4,6,8,33,\dots$ we have $\pi(n)|n$. Does exist infinitely many integers $n$ that $\pi(n)|n$?