Olympiad problems

2019 USA IMO Team Selection Test, 2

Let $\mathbb{Z}/n\mathbb{Z}$ denote the set of integers considered modulo $n$ (hence $\mathbb{Z}/n\mathbb{Z}$ has $n$ elements). Find all positive integers $n$ for which there exists a bijective function $g: \mathbb{Z}/n\mathbb{Z} \to \mathbb{Z}/n\mathbb{Z}$, such that the 101 functions \[g(x), \quad g(x) + x, \quad g(x) + 2x, \quad \dots, \quad g(x) + 100x\] are all bijections on $\mathbb{Z}/n\mathbb{Z}$. [i]Ashwin Sah and Yang Liu[/i]

2023 Ukraine National Mathematical Olympiad, 8.8

Tags: number theory

You are given a set of $m$ integers, all of which give distinct remainders modulo some integer $n$. Show that for any integer $k \le m$ you can split this set into $k$ nonempty groups so that the sums of elements in these groups are distinct modulo $n$. [i]Proposed by Anton Trygub[/i]

2011 Iran Team Selection Test, 12

Tags: function , induction , modular arithmetic , pigeonhole principle , number theory

Suppose that $f : \mathbb{N} \rightarrow \mathbb{N}$ is a function for which the expression $af(a)+bf(b)+2ab$ for all $a,b \in \mathbb{N}$ is always a perfect square. Prove that $f(a)=a$ for all $a \in \mathbb{N}$.

2022 Durer Math Competition (First Round), 4

Tags: number theory , coprime , combinatorics

We want to partition the integers $1, 2, 3, . . . , 100$ into several groups such that within each group either any two numbers are coprime or any two are not coprime. At least how many groups are needed for such a partition? [i]We call two integers coprime if they have no common divisor greater than $1$.[/i]

1987 Mexico National Olympiad, 6

Tags: number theory , divisor

Prove that for every positive integer n the number $(n^3 -n)(5^{8n+4} +3^{4n+2})$ is a multiple of $3804$.

1999 Greece JBMO TST, 3

Tags: decimal representation , digit , number theory

Find digits $a,b,c,x$ ($a>0$) such that $\overline{abc}+\overline{acb}=\overline{199x}$

1992 Rioplatense Mathematical Olympiad, Level 3, 6

Tags: number theory

Definition: A natural number is [i]abundant [/i] if the sum of its positive divisors is greater than its double. Find an odd abundant number and prove that there are infinitely many odd abundant numbers.

1969 IMO Shortlist, 43

Tags: number theory , divisibility , modular arithmetic

$(MON 4)$ Let $p$ and $q$ be two prime numbers greater than $3.$ Prove that if their difference is $2^n$, then for any two integers $m$ and $n,$ the number $S = p^{2m+1} + q^{2m+1}$ is divisible by $3.$

1993 Tournament Of Towns, (369) 1

Tags: power of 2 , digit , number theory

Find all integers of the form $2^n$ (where $n$ is a natural number) such that after deleting the first digit of its decimal representation we again get a power of $2$.

2015 Stars Of Mathematics, 4

Tags: number theory , combinatorics

Let $n\ge 5$ be a positive integer and let $\{a_1,a_2,...,a_n\}=\{1,2,...,n\}$.Prove that at least $\lfloor \sqrt{n}\rfloor +1$ numbers from $a_1,a_1+a_2,...,a_1+a_2+...+a_n$ leave different residues when divided by $n$.

2018 Nepal National Olympiad, 1a

Tags: number theory , algebra

[b]Problem Section #1 a) A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are $189, 320, 287, 264, x$, and y. Find the greatest possible value of: $x + y$. [color=red]NOTE: There is a high chance that this problems was copied.[/color]

2004 Estonia National Olympiad, 3

Tags: number theory , digit , perfect square

The teacher had written on the board a positive integer consisting of a number of $4$s followed by the same number of $8$s followed . During the break, Juku stepped up to the board and added to the number one more $4$ at the start and a $9$ at the end. Prove that the resulting number is an a square. of an integer.

2025 Chile TST IMO-Cono, 5

Tags: number theory

Let $ u_n $ be the $ n $-th term of the Fibonacci sequence (where $ u_1 = u_2 = 1 $ and $ u_{n+1} = u_n + u_{n-1} $ for $ n \geq 2 $). For each prime $ p $, let $ n(p) $ be the smallest integer $ n $ such that $ u_n $ is divisible by $ p $. Find the smallest possible value of $ p - n(p) $.

2023 Korea National Olympiad, 5

Tags: number theory

Find all positive integers $n$ such that $$\phi(n) + \sigma(n) = 2n + 8.$$

2000 Manhattan Mathematical Olympiad, 2

Tags: least common multiple , number theory , modular arithmetic

How many zeroes are there at the end the number $9^{999} + 1$?

2018 Belarusian National Olympiad, 11.7

Tags: inequalities , number theory

Consider the expression $M(n, m)=|n\sqrt{n^2+a}-bm|$, where $n$ and $m$ are arbitrary positive integers and the numbers $a$ and $b$ are fixed, moreover $a$ is an odd positive integer and $b$ is a rational number with an odd denominator of its representation as an irreducible fraction. Prove that there is [b]a)[/b] no more than a finite number of pairs $(n, m)$ for which $M(n, m)=0$; [b]b)[/b] a positive constant $C$ such that the inequality $M(n, m)\geqslant0$ holds for all pairs $(n, m)$ with $M(n, m)\ne 0$.

2017 Romania Team Selection Test, P1

Tags: number theory

Let m be a positive interger, let $p$ be a prime, let $a_1=8p^m$, and let $a_n=(n+1)^{\frac{a_{n-1}}{n}}$, $n=2,3...$. Determine the primes $p$ for which the products $a_n(1-\frac{1}{a_1})(1-\frac{1}{a_2})...(1-\frac{1}{a_n})$, $n=1,2,3...$ are all integral.

2021 Kyiv City MO Round 1, 8.5

Tags: number theory , prime number

For a prime number $p > 3$, define the following irreducible fraction: $$\frac{m}{n} = \frac{p-1}{2} + \frac{p-2}{3} + \ldots + \frac{2}{p-1} - 1$$ Prove that $m$ is divisible by $p$. [i]Proposed by Oleksii Masalitin[/i]

2022 Latvia Baltic Way TST, P16

Tags: number theory

Find all triples of positive integers $(a,b,p)$, where $p$ is a prime, such that both $a+b$ and $ab+1$ are some powers of $p$ (not necessarily the same).

2020 SJMO, 6

Tags: combinatorics , number theory

We say a positive integer $n$ is [i]$k$-tasty[/i] for some positive integer $k$ if there exists a permutation $(a_0, a_1, a_2, \ldots , a_n)$ of $(0,1,2, \ldots, n)$ such that $|a_{i+1} - a_i| \in \{k, k+1\}$ for all $0 \le i \le n-1$. Prove that for all positive integers $k$, there exists a constant $N$ such that all integers $n \geq N$ are $k$-tasty. [i]Proposed by Anthony Wang[/i]

1966 IMO Shortlist, 34

Tags: number theory , equation

Find all pairs of positive integers $\left( x;\;y\right) $ satisfying the equation $2^{x}=3^{y}+5.$

2024 Nigerian MO Round 3, Problem 2

Tags: number theory

Prove that there exist infinitely many distinct positive integers, $x$ and $y$, such that $$x^3+y^2|x^2+y^3$$

2019 Girls in Mathematics Tournament, 1

Tags: combinatorics , perfect square , number theory

During the factoring class, Esmeralda observed that $1$, $3$ and $5$ can be written as the difference of two perfect squares, as can be seen: $1 = 1^2 - 0^2$ $3 = 2^2 - 1^2$ $5 = 3^2 - 2^2$ a) Show that all numbers written in the form $2 * m + 1$ can be written as a difference of two perfect squares. b) Show how to calculate the value of the expression $E = 1 + 3 + 5 + ... + (2m + 1)$. c) Esmeralda, happy with what she discovered, decided to look for other ways to write $2019$ as the difference of two perfect squares of positive integers. Determine how many ways it can do what you want.

1960 Kurschak Competition, 2

Tags: number theory , integer , sum

Let $a_1 = 1, a_2, a_3,...$: be a sequence of positive integers such that $$a_k < 1 + a_1 + a_2 +... + a_{k-1}$$ for all $k > 1$. Prove that every positive integer can be expressed as a sum of $a_i$s.

2007 India IMO Training Camp, 2

Tags: number theory

Find all integer solutions of the equation \[\frac {x^{7} \minus{} 1}{x \minus{} 1} \equal{} y^{5} \minus{} 1.\]