Olympiad problems

2004 Uzbekistan National Olympiad, 2

Lenth of a right angle triangle sides are posive integer. Prove that double area of the triangle divides 12.

2006 South africa National Olympiad, 3

Tags: number theory unsolved , number theory

Determine all positive integers whose squares end in $196$.

2005 MOP Homework, 6

Tags: number theory unsolved , number theory

A positive integer $n$ is good if $n$ can be written as the sum of $2004$ positive integers $a_1$, $a_2$, ..., $a_{2004}$ such that $1 \le a_1 < a_2<...<a_{2004}$ and $a_i$ divides $a_{i+1}$ for $i=1$, $2$, ..., $2003$. Show that there are only finitely many positive integers that are not good.

2010 Contests, 2

Tags: number theory unsolved , number theory

Let $k$ be an integer $\geq 3$. Sequence $\{a_n\}$ satisfies that $a_k = 2k$ and for all $n > k$, we have \[a_n = \begin{cases} a_{n-1}+1 & \text{if } (a_{n-1},n) = 1 \\ 2n & \text{if } (a_{n-1},n) > 1 \end{cases} \] Prove that there are infinitely many primes in the sequence $\{a_n - a_{n-1}\}$.

1989 APMO, 2

Tags: LaTeX , modular arithmetic , number theory , greatest common divisor , number theory unsolved

Prove that the equation \[ 6(6a^2 + 3b^2 + c^2) = 5n^2 \] has no solutions in integers except $a = b = c = n = 0$.

2002 Indonesia MO, 6

Tags: number theory unsolved , number theory

Find all primes $p$ such that $4p^2+1$ and $6p^2+1$ are both primes.

2012 Portugal MO, 1

Tags: floor function , number theory unsolved , number theory

Find the number of positive integers $n$ such that $1\leq n\leq 1000$ and $n$ is divisible by $\lfloor \sqrt[3]{n} \rfloor$.

2004 India IMO Training Camp, 2

Tags: trigonometry , number theory unsolved , number theory

Show that the only solutions of te equation \[ p^{k} + 1 = q^{m} \], in positive integers $k,q,m > 1$ and prime $p$ are (i) $(p,k,q,m) = (2,3,3,2)$ (ii) $k=1 , q=2,$and $p$ is a prime of the form $2^{m} -1$, $m > 1 \in \mathbb{N}$

2004 Austrian-Polish Competition, 4

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

Determine all $n \in \mathbb{N}$ for which $n^{10} + n^5 + 1$ is prime.

2005 MOP Homework, 7

Tags: number theory , prime numbers , number theory unsolved

Let $A$ be a finite subset of prime numbers and $a> 1$ be a positive integer. Show that the number of positive integers $m$ for which all prime divisors of $a^m-1$ are in $A$ is finite.

2000 Cono Sur Olympiad, 3

Tags: number theory unsolved , number theory

Is there a positive integer divisible by the product of its digits such that this product is greater than $10^{2000}$?

1991 Federal Competition For Advanced Students, P2, 3

Tags: modular arithmetic , number theory unsolved , number theory

$ (a)$ Prove that $ 91$ divides $ n^{37}\minus{}n$ for all integers $ n$. $ (b)$ Find the largest $ k$ that divides $ n^{37}\minus{}n$ for all integers $ n$.

2008 Mexico National Olympiad, 2

Tags: geometry , 3D geometry , greatest common divisor , number theory unsolved , number theory

We place $8$ distinct integers in the vertices of a cube and then write the greatest common divisor of each pair of adjacent vertices on the edge connecting them. Let $E$ be the sum of the numbers on the edges and $V$ the sum of the numbers on the vertices. a) Prove that $\frac23E\le V$. b) Can $E=V$?

1984 IMO Longlists, 36

Tags: inequalities , number theory unsolved , number theory

The set $\{1, 2, \cdots, 49\}$ is divided into three subsets. Prove that at least one of these subsets contains three different numbers $a, b, c$ such that $a + b = c$.

2010 Contests, 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.

2015 Kazakhstan National Olympiad, 4

Tags: number theory unsolved , number theory , Kazakhstan

$P_k(n) $ is the product of all positive divisors of $n$ that are divisible by $k$ (the empty product is equal to $1$). Show that $P_1(n)P_2(n)\cdots P_n(n)$ is a perfect square, for any positive integer $n$.

2009 Brazil National Olympiad, 2

Tags: quadratics , number theory , number theory unsolved

Let $ q \equal{} 2p\plus{}1$, $ p, q > 0$ primes. Prove that there exists a multiple of $ q$ whose digits sum in decimal base is positive and at most $ 3$.

2007 Tournament Of Towns, 5

Tags: number theory , prime numbers , number theory unsolved

Find all (finite) increasing arithmetic progressions, consisting only of prime numbers, such that the number of terms is larger than the common difference.

2011 Mongolia Team Selection Test, 2

Tags: Putnam , function , number theory unsolved , number theory

Mongolia TST 2011 Test 1 #2 Let $p$ be a prime number. Prove that: $\sum_{k=0}^p (-1)^k \dbinom{p}{k} \dbinom{p+k}{k} \equiv -1 (\mod p^3)$ (proposed by B. Batbayasgalan, inspired by Putnam olympiad problem) Note: I believe they meant to say $p>2$ as well.

2003 France Team Selection Test, 2

Tags: analytic geometry , modular arithmetic , number theory unsolved , number theory

A lattice point in the coordinate plane with origin $O$ is called invisible if the segment $OA$ contains a lattice point other than $O,A$. Let $L$ be a positive integer. Show that there exists a square with side length $L$ and sides parallel to the coordinate axes, such that all points in the square are invisible.

1988 China Team Selection Test, 1

Tags: number theory unsolved , number theory

Let $f(x) = 3x + 2.$ Prove that there exists $m \in \mathbb{N}$ such that $f^{100}(m)$ is divisible by $1988$.

2014 Contests, 3

Tags: modular arithmetic , number theory unsolved , Zsigmondy

Find all nonnegative integer numbers such that $7^x- 2 \cdot 5^y = -1$

2010 Albania National Olympiad, 4

Tags: algorithm , algebra , system of equations , number theory unsolved , number theory

The sequence of Fibonnaci's numbers if defined from the two first digits $f_1=f_2=1$ and the formula $f_{n+2}=f_{n+1}+f_n$, $\forall n \in N$. [b](a)[/b] Prove that $f_{2010} $ is divisible by $10$. [b](b)[/b] Is $f_{1005}$ divisible by $4$? Albanian National Mathematical Olympiad 2010---12 GRADE Question 4.

2011 International Zhautykov Olympiad, 2

Tags: number theory , greatest common divisor , Euler , function , modular arithmetic , number theory unsolved

Let $n$ be integer, $n>1.$ An element of the set $M=\{ 1,2,3,\ldots,n^2-1\}$ is called [i]good[/i] if there exists some element $b$ of $M$ such that $ab-b$ is divisible by $n^2.$ Furthermore, an element $a$ is called [i]very good[/i] if $a^2-a$ is divisible by $n^2.$ Let $g$ denote the number of [i]good[/i] elements in $M$ and $v$ denote the number of [i]very good[/i] elements in $M.$ Prove that \[v^2+v \leq g \leq n^2-n.\]

2003 China Team Selection Test, 3

Tags: number theory unsolved , number theory

Sequence $\{ a_n \}$ satisfies: $a_1=3$, $a_2=7$, $a_n^2+5=a_{n-1}a_{n+1}$, $n \geq 2$. If $a_n+(-1)^n$ is prime, prove that there exists a nonnegative integer $m$ such that $n=3^m$.