Olympiad problems

2009 Danube Mathematical Competition, 4

Let be $ a,b,c $ positive integers.Prove that $ |a-b\sqrt{c}|<\frac{1}{2b} $ is true if and only if $ |a^{2}-b^{2}c|<\sqrt{c} $.

2012 Baltic Way, 19

Tags: number theory unsolved , number theory

Show that $n^n + (n + 1)^{n + 1}$ is composite for infinitely many positive integers $n$.

2014 Contests, 2

Tags: inequalities , induction , number theory , number theory unsolved

Define a [i]beautiful number[/i] to be an integer of the form $a^n$, where $a\in\{3,4,5,6\}$ and $n$ is a positive integer. Prove that each integer greater than $2$ can be expressed as the sum of pairwise distinct beautiful numbers. [i]Proposed by Matthew Babbitt[/i]

1983 Vietnam National Olympiad, 1

Tags: number theory unsolved , number theory

Show that it is possible to express $1$ as a sum of $6$, and as a sum of $9$ reciprocals of odd positive integers. Generalize the problem.

1996 Bundeswettbewerb Mathematik, 4

Tags: inequalities , IMO Shortlist , number theory unsolved , number theory

Let $p$ be an odd prime. Determine the positive integers $x$ and $y$ with $x\leq y$ for which the number $\sqrt{2p}-\sqrt{x}-\sqrt{y}$ is non-negative and as small as possible.

2005 France Team Selection Test, 1

Tags: number theory unsolved , number theory

Let $x$, $y$ be two positive integers such that $\displaystyle 3x^2+x=4y^2+y$. Prove that $x-y$ is a perfect square.

2011 Moldova Team Selection Test, 1

Tags: arithmetic sequence , number theory unsolved , number theory

Natural numbers have been divided in groups as follow: $(1), (2, 4), (3, 5, 7), (6, 8, 10, 12), (9, 11, 13, 15, 17), \ldots$. Let $S_n$ be the sum of the elements of the $n$th group. Prove that $\frac{S_{2n+1}}{2n+1}-\frac{S_{2n}}{2n}$ is even.

2010 BMO TST, 1

Tags: modular arithmetic , number theory unsolved , number theory

[b]a) [/b]Is the number $ 1111\cdots11$ (with $ 2010$ ones) a prime number? [b]b)[/b] Prove that every prime factor of $ 1111\cdots11$ (with $ 2011$ ones) is of the form $ 4022j\plus{}1$ where $ j$ is a natural number.

2011 Philippine MO, 3

Tags: number theory , prime numbers , combinatorial geometry , number theory unsolved

The $2011$th prime number is $17483$ and the next prime is $17489$. Does there exist a sequence of $2011^{2011}$ consecutive positive integers that contain exactly $2011$ prime numbers?

2010 Contests, 1

Tags: modular arithmetic , number theory unsolved , number theory

[b]a) [/b]Is the number $ 1111\cdots11$ (with $ 2010$ ones) a prime number? [b]b)[/b] Prove that every prime factor of $ 1111\cdots11$ (with $ 2011$ ones) is of the form $ 4022j\plus{}1$ where $ j$ is a natural number.

2010 Contests, 3

Tags: induction , Diophantine equation , number theory unsolved , number theory

Determine all $(m,n) \in \mathbb{Z}^+ \times \mathbb{Z}^+$ which satisfy $3^m-7^n=2.$

2006 Pan African, 2

Tags: Vieta , algebra , polynomial , Rational Root Theorem , number theory unsolved , number theory

Let $a, b, c$ be three non-zero integers. It is known that the sums $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ and $\frac{b}{a}+\frac{c}{b}+\frac{a}{c}$ are integers. Find these sums.

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 $2^{m} -1$, $m > 1 \in \mathbb{N}$

2010 Tuymaada Olympiad, 2

Tags: number theory unsolved , number theory

For a given positive integer $n$, it's known that there exist $2010$ consecutive positive integers such that none of them is divisible by $n$ but their product is divisible by $n$. Prove that there exist $2004$ consecutive positive integers such that none of them is divisible by $n$ but their product is divisible by $n$.

2006 Silk Road, 3

Tags: quadratics , number theory unsolved , number theory

A subset $S$ of the set $M=\{1,2,.....,p-1\}$,where $p$ is a prime number of the kind $12n+11$,is [i]essential[/i],if the product ${\Pi}_s$ of all elements of the subset is not less than the product $\bar{{\Pi}_s}$ of all other elements of the set.The [b]difference[/b] $\bigtriangleup_s=\Pi_s-\bar{{\Pi}_s}$ is called [i]the deviation[/i] of the subset $S$.Define the least possible remainder of division by $p$ of the deviation of an essential subset,containing $\frac{p-1}{2}$ elements.

1989 China Team Selection Test, 2

Tags: LaTeX , number theory unsolved , number theory

Let $v_0 = 0, v_1 = 1$ and $v_{n+1} = 8 \cdot v_n - v_{n-1},$ $n = 1,2, ...$. Prove that in the sequence $\{v_n\}$ there aren't terms of the form $3^{\alpha} \cdot 5^{\beta}$ with $\alpha, \beta \in \mathbb{N}.$

2003 China Team Selection Test, 3

Tags: analytic geometry , inequalities , graph theory , number theory unsolved , number theory

Given $S$ be the finite lattice (with integer coordinate) set in the $xy$-plane. $A$ is the subset of $S$ with most elements such that the line connecting any two points in $A$ is not parallel to $x$-axis or $y$-axis. $B$ is the subset of integer with least elements such that for any $(x,y)\in S$, $x \in B$ or $y \in B$ holds. Prove that $|A| \geq |B|$.

2014 Contests, 1

Tags: modular arithmetic , number theory unsolved , number theory

Let $p$ be a prime such that $p\mid 2a^2-1$ for some integer $a$. Show that there exist integers $b,c$ such that $p=2b^2-c^2$.

2000 Cono Sur Olympiad, 1

Tags: number theory unsolved , number theory

Call a positive integer [i]descending[/i] if, reading left to right, each of its digits (other than its leftmost) is less than or equal to the previous digit. For example, $4221$ and $751$ are descending while $476$ and $455$ are not descending. Determine whether there exists a positive integer $n$ for which $16^n$ is descending.

2005 International Zhautykov Olympiad, 3

Tags: number theory , relatively prime , number theory unsolved

Find all prime numbers $ p,q$ less than 2005 and such that $ q|p^2 \plus{} 4$, $ p|q^2 \plus{} 4$.

2003 Canada National Olympiad, 2

Tags: modular arithmetic , Euler , function , number theory unsolved , number theory

Find the last three digits of the number $2003^{{2002}^{2001}}$.

2005 MOP Homework, 2

Tags: number theory unsolved , number theory

Suppose that $n$ is s positive integer. Determine all the possible values of the first digit after the decimal point in the decimal expression of the number $\sqrt{n^3+2n^2+n}$

2000 Vietnam National Olympiad, 3

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

Consider the polynomial $ P(x) \equal{} x^3 \plus{} 153x^2 \minus{} 111x \plus{} 38$. (a) Prove that there are at least nine integers $ a$ in the interval $ [1, 3^{2000}]$ for which $ P(a)$ is divisible by $ 3^{2000}$. (b) Find the number of integers $ a$ in $ [1, 3^{2000}]$ with the property from (a).

2005 All-Russian Olympiad, 3

Tags: number theory unsolved , number theory

Positive integers $x>1$ and $y$ satisfy an equation $2x^2-1=y^{15}$. Prove that 5 divides $x$.

2013 Miklós Schweitzer, 2

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

Prove there exists a constant $k_0$ such that for any $k\ge k_0$, the equation \[a^{2n}+b^{4n}+2013=ka^nb^{2n}\] has no positive integer solutions $a,b,n$. [i]Proposed by István Pink.[/i]