Olympiad problems

2001 China National Olympiad, 1

Let $a,b,c$ be positive integers such that $a,b,c,a+b-c,a+c-b,b+c-a,a+b+c$ are $7$ distinct primes. The sum of two of $a,b,c$ is $800$. If $d$ be the difference of the largest prime and the least prime among those $7$ primes, find the maximum value of $d$.

2009 Romania Team Selection Test, 2

Tags: modular arithmetic , number theory proposed , number theory

Let $a$ and $n$ be two integers greater than $1$. Prove that if $n$ divides $(a-1)^k$ for some integer $k\geq 2$, then $n$ also divides $a^{n-1}+a^{n-2}+\cdots+a+1$.

1992 Brazil National Olympiad, 2

Tags: logarithms , number theory proposed , number theory

Show that there is a positive integer n such that the first 1992 digits of $n^{1992}$ are 1.

1998 IberoAmerican Olympiad For University Students, 3

Tags: number theory proposed , number theory

The positive divisors of a positive integer $n$ are written in increasing order starting with 1. \[1=d_1<d_2<d_3<\cdots<n\] Find $n$ if it is known that: [b]i[/b]. $\, n=d_{13}+d_{14}+d_{15}$ [b]ii[/b]. $\,(d_5+1)^3=d_{15}+1$

2007 Macedonia National Olympiad, 3

Tags: number theory , prime numbers , number theory proposed

Natural numbers $a, b$ and $c$ are pairwise distinct and satisfy \[a | b+c+bc, b | c+a+ca, c | a+b+ab.\] Prove that at least one of the numbers $a, b, c$ is not prime.

2010 Contests, 3

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

Prove that for every given positive integer $n$, there exists a prime $p$ and an integer $m$ such that $(a)$ $p \equiv 5 \pmod 6$ $(b)$ $p \nmid n$ $(c)$ $n \equiv m^3 \pmod p$

2014 Contests, 1

Tags: number theory proposed , number theory

Find with proof all positive $3$ digit integers $\overline{abc}$ satisfying \[ b\cdot \overline{ac}=c \cdot \overline{ab} +10 \]

2009 Poland - Second Round, 2

Tags: number theory proposed , number theory

Given are two integers $a>b>1$ such that $a+b \mid ab+1$ and $a-b \mid ab-1$. Prove that $a<\sqrt{3}b$.

2007 Pre-Preparation Course Examination, 3

Tags: number theory , graph theory , number theory proposed

This question is both combinatorics and Number Theory : a ) Prove that we can color edges of $K_{p}$ with $p$ colors which is proper, ($p$ is an odd prime) and $K_{p}$ can be partitioned to $\frac{p-1}2$ rainbow Hamiltonian cycles. (A Hamiltonian cycle is a cycle that passes from all of verteces, and a rainbow is a subgraph that all of its edges have different colors.) b) Find all answers of $x^{2}+y^{2}+z^{2}=1$ is $\mathbb Z_{p}$

2008 Irish Math Olympiad, 1

Tags: number theory , prime numbers , number theory proposed

Let $ p_1, p_2, p_3$ and $ p_4$ be four different prime numbers satisying the equations $ 2p_1 \plus{} 3p_2 \plus{} 5p_3 \plus{} 7p_4 \equal{} 162$ $ 11p_1 \plus{} 7p_2 \plus{} 5p_3 \plus{} 4p_4 \equal{} 162$ Find all possible values of the product $ p_1p_2p_3p_4$

2014 Peru IMO TST, 13

Tags: number theory proposed , number theory

Let $r$ be a positive integer and let $N$ be the smallest positive integer such that the numbers $\frac{N}{n+r}\binom{2n}{n}$, $n=0,1,2,\ldots $, are all integer. Show that $N=\frac{r}{2}\binom{2r}{r}$.

2006 India IMO Training Camp, 1

Tags: number theory , greatest common divisor , modular arithmetic , symmetry , number theory proposed

Find all triples $(a,b,c)$ such that $a,b,c$ are integers in the set $\{2000,2001,\ldots,3000\}$ satisfying $a^2+b^2=c^2$ and $\text{gcd}(a,b,c)=1$.

2013 Tuymaada Olympiad, 7

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

Solve the equation $p^2-pq-q^3=1$ in prime numbers. [i]A. Golovanov[/i]

2013 Macedonia National Olympiad, 1

Tags: modular arithmetic , number theory proposed , number theory

Let $ p,q,r $ be prime numbers. Solve the equation $ p^{2q}+q^{2p}=r $

1985 IMO Longlists, 74

Tags: number theory proposed , number theory

Find all triples of positive integers $x, y, z$ satisfying \[\frac{1}{x} +\frac{1}{y} + \frac{1}{z} = \frac{4}{5} .\]

2004 ITAMO, 5

Tags: number theory proposed , number theory

Decide if the following statement is true or false: For every sequence $\{x_n\}_{n\in \mathbb{N}}$ of non-negative real numbers, there exist sequences $\{a_n\}_{n\in\mathbb{N}}$ and $\{b_n\}_{n\in\mathbb{N}}$ of non-negative real numbers such that: (a) $x_n = a_n + b_n$ for all $n$; (b) $a_1 + \cdots + a_n \le n$ for infinitely many values of $n$; (c) $b_1 + \cdots + b_n \le n$ for infinitely many values of $n$.

2024 Francophone Mathematical Olympiad, 4

Tags: number theory , number theory proposed , Divisibility , Sequence

Find all integers $n \ge 2$ for which there exists $n$ integers $a_1,a_2,\dots,a_n \ge 2$ such that for all indices $i \ne j$, we have $a_i \mid a_j^2+1$.

2014 Switzerland - Final Round, 2

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

Let $a,b\in\mathbb{N}$ such that : \[ ab(a-b)\mid a^3+b^3+ab \] Then show that $\operatorname{lcm}(a,b)$ is a perfect square.

2011 Iran MO (2nd Round), 3

Tags: induction , number theory proposed , number theory

Find all increasing sequences $a_1,a_2,a_3,...$ of natural numbers such that for each $i,j\in \mathbb N$, number of the divisors of $i+j$ and $a_i+a_j$ is equal. (an increasing sequence is a sequence that if $i\le j$, then $a_i\le a_j$.)

2017 Bulgaria JBMO TST, 1

Tags: number theory proposed , number theory

Find all positive integers $ a, b, c, d $ so that $ a^2+b^2+c^2+d^2=13 \cdot 4^n $

2004 Italy TST, 2

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.

1996 Turkey MO (2nd round), 1

Tags: number theory proposed , number theory

Let $({{A}_{n}})_{n=1}^{\infty }$ and $({{a}_{n}})_{n=1}^{\infty }$ be sequences of positive integers. Assume that for each positive integer $x$, there is a unique positive integer $N$ and a unique $N-tuple$ $({{x}_{1}},...,{{x}_{N}})$ such that $0\le {{x}_{k}}\le {{a}_{k}}$ for $k=1,2,...N$, ${{x}_{N}}\ne 0$, and $x=\sum\limits_{k=1}^{N}{{{A}_{k}}{{x}_{k}}}$. (a) Prove that ${{A}_{k}}=1$ for some $k$; (b) Prove that ${{A}_{k}}={{A}_{j}}\Leftrightarrow k=j$; (c) Prove that if ${{A}_{k}}\le {{A}_{j}}$, then $\left. {{A}_{k}} \right|{{A}_{j}}$.

2009 CentroAmerican, 6

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

Find all prime numbers $ p$ and $ q$ such that $ p^3 \minus{} q^5 \equal{} (p \plus{} q)^2$.

2004 China Western Mathematical Olympiad, 1

Tags: number theory proposed , number theory

Find all integers $n$, such that the following number is a perfect square \[N= n^4 + 6n^3 + 11n^2 +3n+31. \]

2008 Saint Petersburg Mathematical Olympiad, 4

Tags: number theory proposed , number theory

A wizard thinks of a number from $1$ to $n$. You can ask the wizard any number of yes/no questions about the number. The wizard must answer all those questions, but not necessarily in the respective order. What is the least number of questions that must be asked in order to know what the number is for sure. (In terms of $n$.) Fresh translation.