Olympiad problems

2009 China Team Selection Test, 1

Let $ a > b > 1, b$ is an odd number, let $ n$ be a positive integer. If $ b^n|a^n\minus{}1,$ then $ a^b > \frac {3^n}{n}.$

2010 Contests, 1

Tags: modular arithmetic , number theory proposed , number theory

Solve the equation \[ x^3+2y^3-4x-5y+z^2=2012, \] in the set of integers.

2009 Paraguay Mathematical Olympiad, 3

Tags: number theory proposed , number theory

Find out how many positive integers $n$ not larger than $2009$ exist such that the last digit of $n^{20}$ is $1$.

2010 Canadian Mathematical Olympiad Qualification Repechage, 5

Tags: number theory , number theory proposed

The Fibonacci sequence is dened by $f_1=f_2=1$ and $f_n=f_{n-1}+f_{n-2}$ for $n\ge 3$. A Pythagorean triangle is a right-angled triangle with integer side lengths. Prove that $f_{2k+1}$ is the hypotenuse of a Pythagorean triangle for every positive integer $k$ with $k\ge 2$

1989 Turkey Team Selection Test, 2

Tags: number theory proposed , number theory

A positive integer is called a "double number" if its decimal representation consists of a block of digits, not commencing with $0$, followed immediately by an identical block. So, for instance, $360360$ is a double number, but $36036$ is not. Show that there are infinitely many double numbers which are perfect squares.

2013 ELMO Shortlist, 4

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]

2008 Stars Of Mathematics, 2

Tags: inequalities , number theory proposed , number theory

Let $\sqrt{23}>\frac{m}{n}$ where $ m,n$ are positive integers. i) Prove that $ \sqrt{23}>\frac{m}{n}\plus{}\frac{3}{mn}.$ ii) Prove that $ \sqrt{23}<\frac{m}{n}\plus{}\frac{4}{mn}$ occurs infinitely often, and give at least three such examples. [i]Dan Schwarz[/i]

1993 Austrian-Polish Competition, 4

Tags: number theory proposed , number theory

The Fibonacci numbers are defined by $ F_0 \equal{} 1, F_1 \equal{} 1, F_{n\plus{}2} \equal{} F_{n\plus{}1} \plus{} F_n$. The positive integers $ A, B$ are such that $ A^{19}$ divides $ B^{93}$ and $ B^{19}$ divides $ A^{93}$. Show that if $ h < k$ are consecutive Fibonacci numbers then $ (AB)^h$ divides $ (A^4 \plus{} B^8)^k$

2004 Canada National Olympiad, 5

Tags: analytic geometry , pigeonhole principle , number theory proposed , number theory

Let $ T$ be the set of all positive integer divisors of $ 2004^{100}$. What is the largest possible number of elements of a subset $ S$ of $ T$ such that no element in $ S$ divides any other element in $ S$?

2006 IberoAmerican Olympiad For University Students, 1

Tags: modular arithmetic , number theory proposed , number theory

Let $m,n$ be positive integers greater than $1$. We define the sets $P_m=\left\{\frac{1}{m},\frac{2}{m},\cdots,\frac{m-1}{m}\right\}$ and $P_n=\left\{\frac{1}{n},\frac{2}{n},\cdots,\frac{n-1}{n}\right\}$. Find the distance between $P_m$ and $P_n$, that is defined as \[\min\{|a-b|:a\in P_m,b\in P_n\}\]

1994 Turkey Team Selection Test, 3

Tags: algebra , polynomial , Vieta , quadratics , number theory proposed , number theory

Find all integer pairs $(a,b)$ such that $a\cdot b$ divides $a^2+b^2+3$.

2007 Tuymaada Olympiad, 1

Tags: number theory proposed , number theory

Positive integers $ a<b$ are given. Prove that among every $ b$ consecutive positive integers there are two numbers whose product is divisible by $ ab$.

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.

1977 IMO Longlists, 53

Tags: number theory proposed , number theory

Find all pairs of integers $a$ and $b$ for which \[7a+14b=5a^2+5ab+5b^2\]

2007 Hong kong National Olympiad, 2

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

is there any polynomial of $deg=2007$ with integer coefficients,such that for any integer $n$,$f(n),f(f(n)),f(f(f(n))),...$ is coprime to each other?

2017 Simon Marais Mathematical Competition, B2

Tags: modular arithmetic , number theory , Diophantine equation , number theory proposed

Find all prime numbers $p,q$, for which $p^{q+1}+q^{p+1}$ is a perfect square. [i]Proposed by P. Boyvalenkov[/i]

1992 Baltic Way, 8

Tags: number theory proposed , number theory

Find all integers satisfying the equation $ 2^x\cdot(4\minus{}x)\equal{}2x\plus{}4$.

2007 Junior Balkan Team Selection Tests - Romania, 4

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

Find all integer positive numbers $n$ such that: $n=[a,b]+[b,c]+[c,a]$, where $a,b,c$ are integer positive numbers and $[p,q]$ represents the least common multiple of numbers $p,q$.

2008 Czech and Slovak Olympiad III A, 2

Tags: ratio , number theory proposed , number theory

At one moment, a kid noticed that the end of the hour hand, the end of the minute hand and one of the twelve numbers (regarded as a point) of his watch formed an equilateral triangle. He also calculated that $t$ hours would elapse for the next similar case. Suppose that the ratio of the lengths of the minute hand (whose length is equal to the distance from the center of the watch plate to any of the twelve numbers) and the hour hand is $k>1$. Find the maximal value of $t$.

2014 Canada National Olympiad, 5

Tags: number theory proposed , number theory , Operation

Fix positive integers $n$ and $k\ge 2$. A list of $n$ integers is written in a row on a blackboard. You can choose a contiguous block of integers, and I will either add $1$ to all of them or subtract $1$ from all of them. You can repeat this step as often as you like, possibly adapting your selections based on what I do. Prove that after a finite number of steps, you can reach a state where at least $n-k+2$ of the numbers on the blackboard are all simultaneously divisible by $k$.

2012 Moldova Team Selection Test, 2

Tags: number theory proposed , number theory

Positive integers $a,b$ are such that $137$ divides $a+139b$ and $139$ divides $a+137b$. Find the minimal posible value of $a+b$.

2008 IberoAmerican, 3

Tags: algebra , polynomial , quadratics , number theory proposed , number theory

Let $ P(x) \equal{} x^3 \plus{} mx \plus{} n$ be an integer polynomial satisfying that if $ P(x) \minus{} P(y)$ is divisible by 107, then $ x \minus{} y$ is divisible by 107 as well, where $ x$ and $ y$ are integers. Prove that 107 divides $ m$.

2000 JBMO ShortLists, 4

Tags: number theory proposed , number theory

Find all the integers written as $\overline{abcd}$ in decimal representation and $\overline{dcba}$ in base $7$.

2001 Iran MO (3rd Round), 1

Tags: function , induction , number theory proposed , number theory , functional equation

Find all functions $ f: \mathbb Q\longrightarrow\mathbb Q$ such that: $ f(x)+f(\frac1x)=1$ $ 2f(f(x))=f(2x)$

1998 Baltic Way, 1

Tags: function , number theory proposed , number theory

Find all functions $f$ of two variables, whose arguments $x,y$ and values $f(x,y)$ are positive integers, satisfying the following conditions (for all positive integers $x$ and $y$): \begin{align*} f(x,x)& =x,\\ f(x,y)& =f(y,x),\\ (x+y)f(x,y)& =yf(x,x+y).\end{align*}