Olympiad problems

2019 Durer Math Competition Finals, 1

a) Is it possible to colour the positive rational numbers with blue and red in a way that there are numbers of both colours and the sum of any two numbers of the same colour is the same colour as them? b) Is it possible to colour the positive rational numbers with blue and red in a way that there are numbers of both colours and the product of any two numbers of the same colour is the same colour as them? Note: When forming a sum or a product, it is allowed to pick the same number twice.

2000 AMC 10, 11

Tags: number theory , prime number

Two different prime numbers between $ 4$ and $ 18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained? $ \textbf{(A)}\ 21 \qquad \textbf{(B)}\ 60\qquad \textbf{(C)}\ 119 \qquad \textbf{(D)}\ 180\qquad \textbf{(E)}\ 231$

2017 AMC 10, 14

Tags: number theory

An integer $N$ is selected at random in the range $1\le N \le 2020.$ What is the probability that the remainder when $N^{16}$ is divided by $5$ is $1$? $\textbf{(A)} \text{ }\frac{1}{5} \qquad \textbf{(B)} \text{ }\frac{2}{5} \qquad \textbf{(C)} \text{ }\frac{3}{5} \qquad \textbf{(D)} \text{ }\frac{4}{5} \qquad \textbf{(E)} \text{ 1}$

2021 Serbia Team Selection Test, P6

Tags: number theory , algebra

Let $S=\{1,2, \ldots ,10^{10}\}$. Find all functions $f:S \rightarrow S$, such that $$f(x+1)=f(f(x))+1 \pmod {10^{10}}$$ for each $x \in S$ (assume $f(10^{10}+1)=f(1)$).

1997 Czech and Slovak Match, 5

Tags: number theory , integer , sum of powers

The sum of several integers (not necessarily distinct) equals $1492$. Decide whether the sum of their seventh powers can equal (a) $1996$; (b) $1998$.

1998 Romania Team Selection Test, 3

Tags: number theory

Let $m\ge 2$ be an integer. Find the smallest positive integer $n>m$ such that for any partition with two classes of the set $\{ m,m+1,\ldots ,n \}$ at least one of these classes contains three numbers $a,b,c$ (not necessarily different) such that $a^b=c$. [i]Ciprian Manolescu[/i]

1989 Canada National Olympiad, 3

Tags: modular arithmetic , algebra , binomial theorem , number theory , divisibility tests , logarithm

Define $ \{ a_n \}_{n\equal{}1}$ as follows: $ a_1 \equal{} 1989^{1989}; \ a_n, n > 1,$ is the sum of the digits of $ a_{n\minus{}1}$. What is the value of $ a_5$?

2004 Mexico National Olympiad, 2

Tags: integer , number theory , inequality

Find the maximum number of positive integers such that any two of them $a, b$ (with $a \ne b$) satisfy that$ |a - b| \ge \frac{ab}{100} .$

2006 Alexandru Myller, 1

Tags: binomial coefficient , modular arithmetic , number theory

For an odd prime $ p, $ show that $ \sum_{k=1}^{p-1} \frac{k^p-k}{p}\equiv \frac{1+p}{2}\pmod p . $

2002 Junior Balkan Team Selection Tests - Moldova, 1

Tags: number theory , greatest common divisor

For any integer $n$ we define the numbers $a = n^5 + 6n^3 + 8n$ ¸ $b = n^4 + 4n^2 + 3$. Prove that the numbers $a$ and $b$ are relatively prime or have the greatest common factor of $3$.

2017 Pan-African Shortlist, A2

Tags: number theory

Find all integers $a,b,c $ such that $a+b+c=abc$

2015 Israel National Olympiad, 7

Tags: fibonacci , fibonacci sequence , prime number , divisibility , number theory , combinatorics

The Fibonacci sequence $F_n$ is defined by $F_0=0,F_1=1$ and the recurrence relation $F_n=F_{n-1}+F_{n-2}$ for all integers $n\geq2$. Let $p\geq3$ be a prime number. [list=a] [*] Prove that $F_{p-1}+F_{p+1}-1$ is divisible by $p$. [*] Prove that $F_{p^{k+1}-1}+F_{p^{k+1}+1}-\left(F_{p^k-1}+F_{p^k+1}\right)$ is divisible by $p^{k+1}$ for any positive integer $k$. [/list]

2021 Indonesia TST, N

Tags: number theory

A positive integer $n$ is said to be $interesting$ if there exist some coprime positive integers $a$ and $b$ such that $n = a^2 - ab + b^2$. Show that if $n^2$ is $interesting$, then $n$ or $3n$ is $interesting$.

2005 India IMO Training Camp, 1

Tags: number theory

Let $0 <a <b$ be two rational numbers. Let $M$ be a set of positive real numbers with the properties: (i) $a \in M$ and $b \in M$; (ii) if $x$ $\in M$ and $y \in M$, then $\sqrt{xy} \in M$. Let $M^*$denote the set of all irrational numbers in $M$. prove that every $c,d$ such that $a <c <d<b$, $M^*$ contains an element $m$ with property $c<m<d$

2006 MOP Homework, 3

Tags: inequalities , number theory , odd , divisor , greatest

For positive integer $k$, let $p(k)$ denote the greatest odd divisor of $k$. Prove that for every positive integer $n$, $$\frac{2n}{3} < \frac{p(1)}{1}+ \frac{p(2)}{2}+... +\frac{ p(n)}{n}<\frac{2(n + 1)}{3}$$

2011 Argentina Team Selection Test, 4

Tags: function , number theory

Determine all positive integers $n$ such that the number $n(n+2)(n+4)$ has at most $15$ positive divisors.

2022 Malaysia IMONST 2, 3

Tags: number theory

Given an integer $n$. We rearrange the digits of $n$ to get another number $m$. Prove that it is impossible to get $m+n = 999999999$.

1967 Poland - Second Round, 4

Tags: diophantine , diophantine equation , number theory

Solve the equation in natural numbers $$ xy+yz+zx = xyz + 2. $$

2008 Argentina National Olympiad, 5

Tags: geometry , 3d geometry , number theory

Find all perfect powers whose last $ 4$ digits are $ 2,0,0,8$, in that order.

1996 USAMO, 2

Tags: ratio , number theory , inequalities

For any nonempty set $S$ of real numbers, let $\sigma(S)$ denote the sum of the elements of $S$. Given a set $A$ of $n$ positive integers, consider the collection of all distinct sums $\sigma(S)$ as $S$ ranges over the nonempty subsets of $A$. Prove that this collection of sums can be partitioned into $n$ classes so that in each class, the ratio of the largest sum to the smallest sum does not exceed 2.

2021 Austrian MO National Competition, 3

Tags: number theory , diophantine equation , diophantine

Find all triples $(a, b, c)$ of natural numbers $a, b$ and $c$, for which $a^{b + 20} (c-1) = c^{b + 21} - 1$ is satisfied. (Walther Janous)

1995 All-Russian Olympiad, 5

Tags: number theory

We call natural numbers [i]similar[/i] if they are written with the same (decimal) digits. For example, 112, 121, 211 are similar numbers having the digits 1, 1, 2. Show that there exist three similar 1995-digit numbers with no zero digits, such that the sum of two of them equals the third. [i]S. Dvoryaninov[/i]

2017 Estonia Team Selection Test, 11

Tags: number theory , polynomial , functional equation , sum of digits

For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$ [i]Proposed by Warut Suksompong, Thailand[/i]

2017 Macedonia JBMO TST, 1

Tags: number theory , prime number , divisibility

Let $p$ be a prime number such that $3p+10$ is a sum of squares of six consecutive positive integers. Prove that $p-7$ is divisible by $36$.

2017 Saudi Arabia Pre-TST + Training Tests, 7

Tags: number theory , diophantine equation , diophantine

Find all pairs of integers $(x, y)$ such that $y^3 = 8x^6 + 2x^3 y -y^2$.