Olympiad problems

2011 China Team Selection Test, 3

A positive integer $n$ is known as an [i]interesting[/i] number if $n$ satisfies \[{\ \{\frac{n}{10^k}} \} > \frac{n}{10^{10}} \] for all $k=1,2,\ldots 9$. Find the number of interesting numbers.

2011 CentroAmerican, 4

Tags: number theory , prime numbers , number theory proposed

Find all positive integers $p$, $q$, $r$ such that $p$ and $q$ are prime numbers and $\frac{1}{p+1}+\frac{1}{q+1}-\frac{1}{(p+1)(q+1)} = \frac{1}{r}.$

2004 Iran MO (2nd round), 4

Tags: function , induction , number theory , prime numbers , number theory proposed

$\mathbb{N}$ is the set of positive integers. Determine all functions $f:\mathbb{N}\to\mathbb{N}$ such that for every pair $(m,n)\in\mathbb{N}^2$ we have that: \[f(m)+f(n) \ | \ m+n .\]

2004 Bulgaria Team Selection Test, 2

Tags: floor function , number theory proposed , number theory

Find all primes $p \ge 3$ such that $p- \lfloor p/q \rfloor q$ is a square-free integer for any prime $q<p$.

1996 Baltic Way, 10

Tags: number theory , relatively prime , number theory proposed

Denote by $d(n)$ the number of distinct positive divisors of a positive integer $n$ (including $1$ and $n$). Let $a>1$ and $n>0$ be integers such that $a^n+1$ is a prime. Prove that $d(a^n-1)\ge n$.

2012 IberoAmerican, 2

Tags: number theory proposed , number theory

A positive integer is called [i]shiny[/i] if it can be written as the sum of two not necessarily distinct integers $a$ and $b$ which have the same sum of their digits. For instance, $2012$ is [i]shiny[/i], because $2012 = 2005 + 7$, and both $2005$ and $7$ have the same sum of their digits. Find all positive integers which are [b]not[/b] [i]shiny[/i] (the dark integers).

2006 Czech and Slovak Olympiad III A, 1

Tags: number theory proposed , number theory

Define a sequence of positive integers $\{a_n\}$ through the recursive formula: $a_{n+1}=a_n+b_n(n\ge 1)$,where $b_n$ is obtained by rearranging the digits of $a_n$ (in decimal representation) in reverse order (for example,if $a_1=250$,then $b_1=52,a_2=302$,and so on). Can $a_7$ be a prime?

1998 Baltic Way, 4

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

Let $P$ be a polynomial with integer coefficients. Suppose that for $n=1,2,3,\ldots ,1998$ the number $P(n)$ is a three-digit positive integer. Prove that the polynomial $P$ has no integer roots.

2007 Danube Mathematical Competition, 3

Tags: induction , number theory proposed , number theory

For each positive integer $ n$, define $ f(n)$ as the exponent of the $ 2$ in the decomposition in prime factors of the number $ n!$. Prove that the equation $ n\minus{}f(n)\equal{}a$ has infinitely many solutions for any positive integer $ a$.

2004 Iran Team Selection Test, 1

Tags: modular arithmetic , quadratics , special factorizations , number theory proposed , number theory

Suppose that $ p$ is a prime number. Prove that for each $ k$, there exists an $ n$ such that: \[ \left(\begin{array}{c}n\\ \hline p\end{array}\right)\equal{}\left(\begin{array}{c}n\plus{}k\\ \hline p\end{array}\right)\]

2009 Moldova Team Selection Test, 4

Tags: linear algebra , matrix , number theory proposed , number theory

[color=darkred]Let $ m$ and $ n$ be two nonzero natural numbers. In every cell $ 1 \times 1$ of the rectangular table $ 2m \times 2n$ are put signs $ \plus{}$ or $ \minus{}$. We call [i]cross[/i] an union of all cells which are situated in a line and in a column of the table. Cell, which is situated at the intersection of these line and column is called [i]center of the cross[/i]. A transformation is defined in the following way: firstly we mark all points with the sign $ \minus{}$. Then consecutively, for every marked cell we change the signs in the cross, whose center is the choosen cell. We call a table [i]accesible[/i] if it can be obtained from another table after one transformation. Find the number of all [i]accesible[/i] tables.[/color]

1972 Bundeswettbewerb Mathematik, 3

Tags: number theory proposed , number theory

The arithmetic mean of two different positive integers $x,y$ is a two digit integer. If one interchanges the digits, the geometric mean of these numbers is archieved. a) Find $x,y$. b) Show that a)'s solution is unique up to permutation if we work in base $g=10$, but that there is no solution in base $g=12$. c) Give more numbers $g$ such that a) can be solved; give more of them such that a) can't be solved, too.

2014 Contests, 3

Tags: number theory proposed , number theory

Find all pairs $(m, n)$ of positive integers satsifying $m^6+5n^2=m+n^3$.

2011 Romania Team Selection Test, 3

Tags: ratio , floor function , modular arithmetic , number theory proposed , number theory

Let $S$ be a finite set of positive integers which has the following property:if $x$ is a member of $S$,then so are all positive divisors of $x$. A non-empty subset $T$ of $S$ is [i]good[/i] if whenever $x,y\in T$ and $x<y$, the ratio $y/x$ is a power of a prime number. A non-empty subset $T$ of $S$ is [i]bad[/i] if whenever $x,y\in T$ and $x<y$, the ratio $y/x$ is not a power of a prime number. A set of an element is considered both [i]good[/i] and [i]bad[/i]. Let $k$ be the largest possible size of a [i]good[/i] subset of $S$. Prove that $k$ is also the smallest number of pairwise-disjoint [i]bad[/i] subsets whose union is $S$.

1997 Baltic Way, 10

Tags: number theory proposed , number theory

Prove that in every sequence of $79$ consecutive positive integers written in the decimal system, there is a positive integer whose sum of digits is divisible by $13$.

2003 Iran MO (3rd Round), 8

Tags: induction , number theory , least common multiple , arithmetic sequence , number theory proposed

A positive integer $n$ is said to be a [i]perfect power[/i] if $n=a^b$ for some integers $a,b$ with $b>1$. $(\text{a})$ Find $2004$ perfect powers in arithmetic progression. $(\text{b})$ Prove that perfect powers cannot form an infinite arithmetic progression.

2012 All-Russian Olympiad, 1

Tags: modular arithmetic , number theory proposed , number theory

Let $a_1,\ldots a_{11}$ be distinct positive integers, all at least $2$ and with sum $407$. Does there exist an integer $n$ such that the sum of the $22$ remainders after the division of $n$ by $a_1,a_2,\ldots ,a_{11},4a_1,4a_2,\ldots ,4a_{11}$ is $2012$?

2009 Middle European Mathematical Olympiad, 4

Tags: quadratics , modular arithmetic , number theory , number theory proposed

Determine all integers $ k\ge 2$ such that for all pairs $ (m$, $ n)$ of different positive integers not greater than $ k$, the number $ n^{n\minus{}1}\minus{}m^{m\minus{}1}$ is not divisible by $ k$.

2008 Iran MO (3rd Round), 2

Tags: arithmetic sequence , number theory proposed , number theory

Prove that there exists infinitely many primes $ p$ such that: \[ 13|p^3\plus{}1\]

2010 ELMO Shortlist, 2

Tags: modular arithmetic , algebra , difference of squares , special factorizations , number theory proposed , number theory

Given a prime $p$, show that \[\left(1+p\sum_{k=1}^{p-1}k^{-1}\right)^2 \equiv 1-p^2\sum_{k=1}^{p-1}k^{-2} \pmod{p^4}.\] [i]Timothy Chu.[/i]

1997 Korea - Final Round, 6

Tags: number theory proposed , number theory

Let $ p_1,p_2,\dots,p_r$ be distinct primes, and let $ n_1,n_2,\dots,n_r$ be arbitrary natural numbers. Prove that the number of pairs of integers $ (x, y)$ such that \[ x^3 \plus{} y^3 \equal{} p_1^{n_1}p_2^{n_2}\cdots p_r^{n_r}\] does not exceed $ 2^{r\plus{}1}$.

2001 China National Olympiad, 3

Tags: modular arithmetic , limit , number theory proposed , number theory

Let $a=2001$. Consider the set $A$ of all pairs of integers $(m,n)$ with $n\neq0$ such that (i) $m<2a$; (ii) $2n|(2am-m^2+n^2)$; (iii) $n^2-m^2+2mn\leq2a(n-m)$. For $(m, n)\in A$, let \[f(m,n)=\frac{2am-m^2-mn}{n}.\] Determine the maximum and minimum values of $f$.

2008 Korea - Final Round, 4

Tags: modular arithmetic , number theory proposed , number theory

For any positive integer $m\ge2$ define $A_m=\{m+1, 3m+2, 5m+3, 7m+4, \ldots, (2k-1)m + k, \ldots\}$. (1) For every $m\ge2$, prove that there exists a positive integer $a$ that satisfies $1\le a<m$ and $2^a\in A_m$ or $2^a+1\in A_m$. (2) For a certain $m\ge2$, let $a, b$ be positive integers that satisfy $2^a\in A_m$, $2^b+1\in A_m$. Let $a_0, b_0$ be the least such pair $a, b$. Find the relation between $a_0$ and $b_0$.

1996 Hungary-Israel Binational, 1

Tags: number theory proposed , number theory

Find all integer sequences of the form $ x_i, 1 \le i \le 1997$, that satisfy $ \sum_{k\equal{}1}^{1997} 2^{k\minus{}1} x_{k}^{1997}\equal{}1996\prod_{k\equal{}1}^{1997}x_k$.

2006 Bulgaria National Olympiad, 1

Tags: modular arithmetic , number theory proposed , number theory

Let $p$ be a prime such that $p^2$ divides $2^{p-1}-1$. Prove that for all positive integers $n$ the number $\left(p-1\right)\left(p!+2^n\right)$ has at least $3$ different prime divisors. [i]Aleksandar Ivanov[/i]