Olympiad problems

2010 ITAMO, 2

Every non-negative integer is coloured white or red, so that: • there are at least a white number and a red number; • the sum of a white number and a red number is white; • the product of a white number and a red number is red. Prove that the product of two red numbers is always a red number, and the sum of two red numbers is always a red number.

2015 China National Olympiad, 1

Tags: logarithms , number theory proposed , number theory

Determine all integers $k$ such that there exists infinitely many positive integers $n$ [b]not[/b] satisfying \[n+k |\binom{2n}{n}\]

1996 Irish Math Olympiad, 2

Tags: number theory proposed , number theory

Let $ S(n)$ denote the sum of the digits of a natural number $ n$ (in base $ 10$). Prove that for every $ n$, $ S(2n) \le 2S(n) \le 10S(2n)$. Prove also that there is a positive integer $ n$ with $ S(n)\equal{}1996S(3n)$.

2009 All-Russian Olympiad Regional Round, 9.6

Tags: number theory proposed , number theory

Positive integer $m$ is such that the sum of decimal digits of $8^m$ equals 8. Can the last digit of $8^m$ be equal 6? (Author: V. Senderov) (compare with http://www.artofproblemsolving.com/Forum/viewtopic.php?f=57&t=431860)

2004 Iran MO (3rd Round), 30

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

Find all polynomials $ p\in\mathbb Z[x]$ such that $ (m,n)\equal{}1\Rightarrow (p(m),p(n))\equal{}1$

2007 District Olympiad, 3

Tags: number theory proposed , number theory

Let $b>a\geq 2$ be positive integers. Prove that if number $a+k$ is coprime to number $b+k$, for all $k=1,2,...,b-a$, then $a,b$ are consecutive numbers

2011 Romania Team Selection Test, 2

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

Given a prime number $p$ congruent to $1$ modulo $5$ such that $2p+1$ is also prime, show that there exists a matrix of $0$s and $1$s containing exactly $4p$ (respectively, $4p+2$) $1$s no sub-matrix of which contains exactly $2p$ (respectively, $2p+1$) $1$s.

2011 ITAMO, 5

Tags: number theory proposed , number theory

Determine all solutions $(p,n)$ of the equation \[n^3=p^2-p-1\] where $p$ is a prime number and $n$ is an integer

2000 JBMO ShortLists, 5

Tags: number theory proposed , number theory

Find all pairs of integers $(m,n)$ such that the numbers $A=n^2+2mn+3m^2+2$, $B=2n^2+3mn+m^2+2$, $C=3n^2+mn+2m^2+1$ have a common divisor greater than $1$.

2014 Baltic Way, 19

Tags: number theory , greatest common divisor , number theory proposed

Let $m$ and $n$ be relatively prime positive integers. Determine all possible values of \[\gcd(2^m - 2^n, 2^{m^2+mn+n^2}- 1).\]

2010 Contests, 2

Tags: induction , number theory proposed , number theory

Every non-negative integer is coloured white or red, so that: • there are at least a white number and a red number; • the sum of a white number and a red number is white; • the product of a white number and a red number is red. Prove that the product of two red numbers is always a red number, and the sum of two red numbers is always a red number.

2010 ELMO Shortlist, 3

Tags: quadratics , Diophantine equation , number theory proposed , number theory

Prove that there are infinitely many quadruples of integers $(a,b,c,d)$ such that \begin{align*} a^2 + b^2 + 3 &= 4ab\\ c^2 + d^2 + 3 &= 4cd\\ 4c^3 - 3c &= a \end{align*} [i]Travis Hance.[/i]

2012 Albania Team Selection Test, 4

Tags: number theory , least common multiple , greatest common divisor , relatively prime , number theory proposed

Find all couples of natural numbers $(a,b)$ not relatively prime ($\gcd(a,b)\neq\ 1$) such that \[\gcd(a,b)+9\operatorname{lcm}[a,b]+9(a+b)=7ab.\]

2011 Balkan MO Shortlist, C1

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 $S$.

2014 ELMO Shortlist, 1

Tags: modular arithmetic , number theory proposed , number theory , mod 13

Does there exist a strictly increasing infinite sequence of perfect squares $a_1, a_2, a_3, ...$ such that for all $k\in \mathbb{Z}^+$ we have that $13^k | a_k+1$? [i]Proposed by Jesse Zhang[/i]

2013 ITAMO, 4

Tags: number theory proposed , number theory

$\overline{5654}_b$ is a power of a prime number. Find $b$ if $b > 6$.

Oliforum Contest II 2009, 1

Tags: function , modular arithmetic , number theory , prime factorization , number theory proposed

Let $ \sigma(\cdot): \mathbb{N}_0 \to \mathbb{N}_0$ be the function from every positive integer $ n$ to the sum of divisors $ \sum_{d \mid n}{d}$ (i.e. $ \sigma(6) \equal{} 6 \plus{} 3 \plus{} 2 \plus{} 1$ and $ \sigma(8) \equal{} 8 \plus{} 4 \plus{} 2 \plus{} 1$). Find all primes $ p$ such that $ p \mid \sigma(p \minus{} 1)$. [i](Salvatore Tringali)[/i]

2014 Contests, 2

Tags: quadratics , USA(J)MO , USAMO , induction , modular arithmetic , number theory proposed , Quadratic Residues

Let $a_1,a_2,a_3,\ldots$ be a sequence of integers, with the property that every consecutive group of $a_i$'s averages to a perfect square. More precisely, for every positive integers $n$ and $k$, the quantity \[\frac{a_n+a_{n+1}+\cdots+a_{n+k-1}}{k}\] is always the square of an integer. Prove that the sequence must be constant (all $a_i$ are equal to the same perfect square). [i]Evan O'Dorney and Victor Wang[/i]

1991 Romania Team Selection Test, 1

Tags: quadratics , number theory proposed , number theory

Suppose that $ a,b$ are positive integers for which $ A\equal{}\frac{a\plus{}1}{b}\plus{}\frac{b}{a}$ is an integer.Prove that $ A\equal{}3$.

2000 Irish Math Olympiad, 4

Tags: number theory proposed , number theory

Show that in each set of ten consecutive integers there is one that is coprime with each of the other integers. (For example, in the set $ \{ 114,115,...,123 \}$ there are two such numbers: $ 119$ and $ 121.)$

1999 Iran MO (2nd round), 1

Tags: modular arithmetic , number theory proposed , number theory

Does there exist a positive integer that is a power of $2$ and we get another power of $2$ by swapping its digits? Justify your answer.

2001 District Olympiad, 1

Tags: modular arithmetic , number theory proposed , number theory

A positive integer is called [i]good[/i] if it can be written as a sum of two consecutive positive integers and as a sum of three consecutive positive integers. Prove that: a)2001 is [i]good[/i], but 3001 isn't [i]good[/i]. b)the product of two [i]good[/i] numbers is a [i]good[/i] number. c)if the product of two numbers is [i]good[/i], then at least one of the numbers is [i]good[/i]. [i]Bogdan Enescu[/i]

2004 Junior Balkan Team Selection Tests - Romania, 3

Tags: number theory proposed , number theory

A finite set of positive integers is called [i]isolated [/i]if the sum of the numbers in any given proper subset is co-prime with the sum of the elements of the set. a) Prove that the set $A=\{4,9,16,25,36,49\}$ is isolated; b) Determine the composite numbers $n$ for which there exist the positive integers $a,b$ such that the set \[ A=\{(a+b)^2, (a+2b)^2,\ldots, (a+nb)^2\}\] is isolated.

2001 District Olympiad, 1

Tags: number theory proposed , number theory

a) Find all the integers $m$ and $n$ such that \[9m^2+3n=n^2+8\] b) Let $a,b\in \mathbb{N}^*$ . If $x=a^{a+b}+(a+b)^a$ and $y=a^a+(a+b)^{a+b}$ which one is bigger? [i]Florin Nicoara, Valer Pop[/i]

2010 Contests, 2

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

Positive rational number $a$ and $b$ satisfy the equality \[a^3 + 4a^2b = 4a^2 + b^4.\] Prove that the number $\sqrt{a}-1$ is a square of a rational number.