Olympiad problems

2005 Kyiv Mathematical Festival, 4

Prove that there exist infinitely many collections of positive integers $ (a,b,c,d,e,f)$ such that $ a < b < c$ and the equalities $ ab \minus{} c \equal{} de,$ $ bc \minus{} a \equal{} ef$ and $ ac \minus{} b \equal{} df$ hold.

2005 Germany Team Selection Test, 3

Tags: number theory proposed , number theory

Let $a$, $b$, $c$, $d$ and $n$ be positive integers such that $7\cdot 4^n = a^2+b^2+c^2+d^2$. Prove that the numbers $a$, $b$, $c$, $d$ are all $\geq 2^{n-1}$.

2003 Romania Team Selection Test, 15

Tags: number theory proposed , number theory

In a plane we choose a cartesian system of coordinates. A point $A(x,y)$ in the plane is called an integer point if and only if both $x$ and $y$ are integers. An integer point $A$ is called invisible if on the segment $(OA)$ there is at least one integer point. Prove that for each positive integer $n$ there exists a square of side $n$ in which all the interior integer points are invisible.

2007 Kyiv Mathematical Festival, 1

Tags: number theory proposed , number theory

Find all pairs of positive integers $(a,b)$ such that $\sqrt{a-1}+\sqrt{b-1}=\sqrt{ab-1}.$

1990 Flanders Math Olympiad, 2

Tags: number theory proposed , number theory

Let $a$ and $b$ be two primes having at least two digits, such that $a > b$. Show that \[240|\left(a^4-b^4\right)\] and show that 240 is the greatest positive integer having this property.

2003 Bulgaria Team Selection Test, 4

Tags: number theory proposed , number theory

Is it true that for any permulation $a_1,a_2.....,a_{2002}$ of $1,2....,2002$ there are positive integers $m,n$ of the same parity such that $0<m<n<2003$ and $a_m+a_n=2a_{\frac {m+n}{2}}$

2010 Baltic Way, 19

Tags: inequalities , number theory proposed , number theory

For which $k$ do there exist $k$ pairwise distinct primes $p_1,p_2,\ldots ,p_k$ such that \[p_1^2+p_2^2+\ldots +p_k^2=2010? \]

2002 Romania Team Selection Test, 1

Tags: number theory proposed , number theory

Let $(a_n)_{n\ge 1}$ be a sequence of positive integers defined as $a_1,a_2>0$ and $a_{n+1}$ is the least prime divisor of $a_{n-1}+a_{n}$, for all $n\ge 2$. Prove that a real number $x$ whose decimals are digits of the numbers $a_1,a_2,\ldots a_n,\ldots $ written in order, is a rational number. [i]Laurentiu Panaitopol[/i]

2001 JBMO ShortLists, 1

Tags: geometry , 3D geometry , number theory , Diophantine equation , number theory proposed

Find the positive integers $n$ that are not divisible by $3$ if the number $2^{n^2-10}+2133$ is a perfect cube. [hide="Note"] [color=#BF0000]The wording of this problem is perhaps not the best English. As far as I am aware, just solve the diophantine equation $x^3=2^{n^2-10}+2133$ where $x,n \in \mathbb{N}$ and $3\nmid n$.[/color][/hide]

2010 Contests, 1

Tags: modular arithmetic , number theory proposed , number theory

Denote by $S(n)$ the sum of the digits of the positive integer $n$. Find all the solutions of the equation $n(S(n)-1)=2010.$

2006 Taiwan TST Round 1, 1

Tags: number theory proposed , number theory

Find the largest integer that is a factor of $(a-b)(b-c)(c-d)(d-a)(a-c)(b-d)$ for all integers $a,b,c,d$.

2010 Kazakhstan National Olympiad, 6

Tags: number theory proposed , number theory

Call $A \in \mathbb{N}^0$ be $number of year$ if all digits of $A$ equals $0$, $1$ or $2$ (in decimal representation). Prove that exist infinity $N \in \mathbb{N}$, such that $N$ can't presented as $A^2+B$ where $A \in \mathbb{N}^0 ; B$- $number of year$.

2015 India Regional MathematicaI Olympiad, 4

Tags: geometric sequence , number theory , algebra , number theory proposed

Find all three digit natural numbers of the form $(abc)_{10}$ such that $(abc)_{10}$, $(bca)_{10}$ and $(cab)_{10}$ are in geometric progression. (Here $(abc)_{10}$ is representation in base $10$.)

2009 Indonesia TST, 4

Tags: number theory proposed , number theory

Let $ n>1$ be an odd integer and define: \[ N\equal{}\{\minus{}n,\minus{}(n\minus{}1),\dots,\minus{}1,0,1,\dots,(n\minus{}1),n\}.\] A subset $ P$ of $ N$ is called [i]basis[/i] if we can express every element of $ N$ as the sum of $ n$ different elements of $ P$. Find the smallest positive integer $ k$ such that every $ k\minus{}$elements subset of $ N$ is basis.

1987 Iran MO (2nd round), 1

Tags: algebra , system of equations , number theory proposed , number theory

Solve the following system of equations in positive integers \[\left\{\begin{array}{cc}a^3-b^3-c^3=3abc\\ \\ a^2=2(b+c)\end{array}\right.\]

2014 Middle European Mathematical Olympiad, 8

Tags: modular arithmetic , quadratics , inequalities , geometry , 3D geometry , number theory proposed , number theory

Determine all quadruples $(x,y,z,t)$ of positive integers such that \[ 20^x + 14^{2y} = (x + 2y + z)^{zt}.\]

2010 Indonesia TST, 2

Tags: modular arithmetic , number theory , number theory proposed

Let $ A\equal{}\{n: 1 \le n \le 2009^{2009},n \in \mathbb{N} \}$ and let $ S\equal{}\{n: n \in A,\gcd \left(n,2009^{2009}\right)\equal{}1\}$. Let $ P$ be the product of all elements of $ S$. Prove that \[ P \equiv 1 \pmod{2009^{2009}}.\] [i]Nanang Susyanto, Jogjakarta[/i]

2010 Kazakhstan National Olympiad, 3

Tags: number theory proposed , number theory

Call $A \in \mathbb{N}^0$ be $number of year$ if all digits of $A$ equals $0$, $1$ or $2$ (in decimal representation). Prove that exist infinity $N \in \mathbb{N}$, such that $N$ can't presented as $A^2+B$ where $A \in \mathbb{N}^0 ; B$- $number of year$.

2002 India IMO Training Camp, 21

Tags: number theory proposed , number theory

Given a prime $p$, show that there exists a positive integer $n$ such that the decimal representation of $p^n$ has a block of $2002$ consecutive zeros.

2003 Tournament Of Towns, 4

Tags: number theory proposed , number theory

Each term of a sequence of positive integers is obtained from the previous term by adding to it its largest digit. What is the maximal number of successive odd terms in such a sequence?

2011 Vietnam Team Selection Test, 4

Tags: floor function , induction , number theory proposed , number theory

Let $\langle a_n\rangle_{n\ge 0}$ be a sequence of integers satisfying $a_0=1, a_1=3$ and $a_{n+2}=1+\left\lfloor \frac{a_{n+1}^2}{a_n}\right\rfloor \ \ \forall n\ge0.$ Prove that $a_n\cdot a_{n+2}-a_{n+1}^2=2^n$ for every natural number $n.$

2012 Singapore MO Open, 4

Tags: modular arithmetic , floor function , number theory , relatively prime , number theory proposed

Let $p$ be an odd prime. Prove that \[1^{p-2}+2^{p-2}+\cdots+\left(\frac{p-1}{2}\right)^{p-2}\equiv\frac{2-2^p}{p}\pmod p.\]

2014 Contests, 2

Tags: modular arithmetic , number theory proposed , number theory

A pair of positive integers $(a,b)$ is called [i]charrua[/i] if there is a positive integer $c$ such that $a+b+c$ and $a\times b\times c$ are both square numbers; if there is no such number $c$, then the pair is called [i]non-charrua[/i]. a) Prove that there are infinite [i]non-charrua[/i] pairs. b) Prove that there are infinite positive integers $n$ such that $(2,n)$ is [i]charrua[/i].

2003 Tournament Of Towns, 1

Tags: number theory proposed , number theory

For any integer $n+1,\ldots, 2n$ ($n$ is a natural number) consider its greatest odd divisor. Prove that the sum of all these divisors equals $n^2.$

2008 Greece National Olympiad, 2

Tags: number theory , number theory proposed

Find all integers $x$ and prime numbers $p$ satisfying $x^8 + 2^{2^x+2} = p$.