Olympiad problems

2016 Switzerland - Final Round, 9

Let $n \ge 2$ be a natural number. For an $n$-element subset $F$ of $\{1, . . . , 2n\}$ we define $m(F)$ as the minimum of all $lcm \,\, (x, y)$ , where $x$ and $y$ are two distinct elements of $F$. Find the maximum value of $m(F)$.

2025 Kosovo National Mathematical Olympiad`, P3

Tags: number theory , decimal representation , power

Find all pairs of natural numbers $(m,n)$ such that the number $5^m+6^n$ has all same digits when written in decimal representation.

1989 Romania Team Selection Test, 2

Tags: system of equations , algebra , number theory , coprime

Let $a,b,c$ be coprime nonzero integers. Prove that for any coprime integers $u,v,w$ with $au+bv+cw = 0$ there exist integers $m,n, p$ such that $$\begin{cases} a = nw- pv \\ b = pu-mw \\ c = mv-nu \end{cases}$$

2008 Alexandru Myller, 1

Tags: number theory , quadratic , algebra

How many solutions does the equation $ \frac{[x]}{\{ x\}} =\frac{2007x}{2008} $ have? [i]Mihail Bălună[/i]

2021 Thailand Mathematical Olympiad, 5

Tags: number theory

Determine all triples $(p,m,k)$ of positive integers such that $p$ is a prime number, $m$ and $k$ are odd integers, and $m^4+4^kp^4$ divides $m^2(m^4-4^kp^4)$.

2024 Ukraine National Mathematical Olympiad, Problem 2

Tags: number theory , algebra , power of 2

You are given a positive integer $n$. Find the smallest positive integer $k$, for which there exist integers $a_1, a_2, \ldots, a_k$, for which the following equality holds: $$2^{a_1} + 2^{a_2} + \ldots + 2^{a_k} = 2^n - n + k$$ [i]Proposed by Mykhailo Shtandenko[/i]

2017 Germany, Landesrunde - Grade 11/12, 3

Tags: number theory , prime number , exponent , power , catalan

Find the smallest prime number that can not be written in the form $\left| 2^a-3^b \right|$ with non-negative integers $a,b$.

2019 IFYM, Sozopol, 6

Tags: function , number theory

Does there exist a function $f: \mathbb N \to \mathbb N$ such that for all integers $n \geq 2$, \[ f(f(n-1)) = f (n+1) - f(n)\, ?\]

2004 Italy TST, 2

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

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.

PEN C Problems, 4

Tags: floor function , quadratic , inequalities , number theory , prime number , quadratic residue

Let $M$ be an integer, and let $p$ be a prime with $p>25$. Show that the set $\{M, M+1, \cdots, M+ 3\lfloor \sqrt{p} \rfloor -1\}$ contains a quadratic non-residue to modulus $p$.

2022 Macedonian Team Selection Test, Problem 5

Tags: number theory

Given is an arithmetic progression {$a_n$} of positive integers. Prove that there exist infinitely many $k$, such that $\omega (a_k)$ is even and $\omega (a_{k+1})$ is odd ($\omega (n)$ is the number of distinct prime factors of $n$). $\textit {Proposed by Viktor Simjanoski and Nikola Velov}$

2004 Cuba MO, 4

Tags: number theory , gcd

Determine all pairs of natural numbers $ (x, y)$ for which it holds that $$x^2 = 4y + 3gcd (x, y).$$

2020 Serbian Mathematical Olympiad, Problem 5

Tags: number theory , functional equation , function , algebra

For a natural number $n$, with $v_2(n)$ we denote the largest integer $k\geq0$ such that $2^k|n$. Let us assume that the function $f\colon\mathbb{N}\to\mathbb{N}$ meets the conditions: $(i)$ $f(x)\leq3x$ for all natural numbers $x\in\mathbb{N}$. $(ii)$ $v_2(f(x)+f(y))=v_2(x+y)$ for all natural numbers $x,y\in\mathbb{N}$. Prove that for every natural number $a$ there exists exactly one natural number $x$ such that $f(x)=3a$.

2018 Taiwan APMO Preliminary, 2

Tags: number theory

Let $k,x,y$ be postive integers. The quotients of $k$ divided by $x^2, y^2$ are $n,n+148$ respectively.($k$ is divisible by $x^2$ and $y^2$) (a) If $\gcd(x,y)=1$, then find $k$. (b) If $\gcd(x,y)=4$, then find $k$.

2017 Romania Team Selection Test, P2

Tags: irrational number , polynomial , number theory

Determine all intergers $n\geq 2$ such that $a+\sqrt{2}$ and $a^n+\sqrt{2}$ are both rational for some real number $a$ depending on $n$

2024 SG Originals, Q1

Tags: permutation , number theory , polynomial

Find all permutations $(a_1, a_2, \cdots, a_{2024})$ of $(1, 2, \cdots, 2024)$ such that there exists a polynomial $P$ with integer coefficients satisfying $P(i) = a_i$ for each $i = 1, 2, \cdots, 2024$.

1997 Tournament Of Towns, (524) 1

Tags: number theory , sum of digits , divisible

How many integers from $1$ to $1997$ have the sum of their digits divisible by $5$? (AI Galochkin)

2020 CHMMC Winter (2020-21), 7

Tags: number theory

For any positive integer $n$, let $f(n)$ denote the sum of the positive integers $k \le n$ such that $k$ and $n$ are relatively prime. Let $S$ be the sum of $\frac{1}{f(m)}$ over all positive integers $m$ that are divisible by at least one of $2$, $3$, or $5$, and whose prime factors are only $2$, $3$, or $5$. Then $S = \frac{p}{q}$ for relatively prime positive integers $p$ and $q$. Find $p+q$.

2016 Japan Mathematical Olympiad Preliminary, 1

Tags: square root , number theory

Calculate the value of $\sqrt{\dfrac{11^4+100^4+111^4}{2}}$ and answer in the form of an integer.

2020 Balkan MO, 4

Tags: number theory

Let $a_1=2$ and, for every positive integer $n$, let $a_{n+1}$ be the smallest integer strictly greater than $a_n$ that has more positive divisors than $a_n$. Prove that $2a_{n+1}=3a_n$ only for finitely many indicies $n$. [i] Proposed by Ilija Jovčevski, North Macedonia[/i]

2017-IMOC, N5

Tags: number theory , fe , functional equation

Find all functions $f:\mathbb N\to\mathbb N$ such that $$f(x)+f(y)\mid x^2-y^2$$holds for all $x,y\in\mathbb N$.

2005 Argentina National Olympiad, 1

Tags: system of equations , number theory

Let $a>b>c>d$ be positive integers satisfying $a+b+c+d=502$ and $a^2-b^2+c^2-d^2=502$ . Calculate how many possible values of $ a$ are there.

2007 Mexico National Olympiad, 1

Tags: number theory

Find all integers $N$ with the following property: for $10$ but not $11$ consecutive positive integers, each one is a divisor of $N$.

2011 Stars Of Mathematics, 2

Tags: number theory

Prove there do exist infinitely many positive integers $n$ such that if a prime $p$ divides $n(n+1)$ then $p^2$ also divides it (all primes dividing $n(n+1)$ bear exponent at least two). Exhibit (at least) two values, one even and one odd, for such numbers $n>8$. (Pál Erdös & Kurt Mahler)

2018 Iran Team Selection Test, 1

Tags: combinatorics , number theory , combinatorial number theory

Let $A_1, A_2, ... , A_k$ be the subsets of $\left\{1,2,3,...,n\right\}$ such that for all $1\leq i,j\leq k$:$A_i\cap A_j \neq \varnothing$. Prove that there are $n$ distinct positive integers $x_1,x_2,...,x_n$ such that for each $1\leq j\leq k$: $$lcm_{i \in A_j}\left\{x_i\right\}>lcm_{i \notin A_j}\left\{x_i\right\}$$ [i]Proposed by Morteza Saghafian, Mahyar Sefidgaran[/i]