Olympiad problems

2023 Korea National Olympiad, 3

For a given positive integer $n(\ge 2)$, find maximum positive integer $A$ such that there exists $P \in \mathbb{Z}[x]$ with degree $n$ that satisfies the following two conditions. [list] [*] For any $1 \le k \le A$, it satisfies that $A \mid P(k)$, and [*] $P(0)= 0$ and the coefficient of the first term of $P$ is $1$, which means that $P(x)$ is in the following form where $c_2, c_3, \cdots, c_n$ are all integers and $c_n \neq 0$. $$P(x) = c_nx^n + c_{n-1}x^{n-1}+\dots+c_2x^2+x$$ [/list]

2014 Contests, 2

Tags: floor function , number theory

Let $M$ be the set of palindromic integers of the form $5n+4$ where $n\ge 0$ is an integer. [list=a] [*]If we write the elements of $M$ in increasing order, what is the $50^{\text{th}}$ number? [*]Among all numbers in $M$ with nonzero digits which sum up to $2014$ which is the largest and smallest one?[/list]

2009 Middle European Mathematical Olympiad, 4

Tags: quadratic , modular arithmetic , number theory

Determine all integers $ k\ge 2$ such that for all pairs $ (m$, $ n)$ of different positive integers not greater than $ k$, the number $ n^{n\minus{}1}\minus{}m^{m\minus{}1}$ is not divisible by $ k$.

2024 Korea Junior Math Olympiad (First Round), 14.

Tags: number theory , algebra , number

Find the number of positive integer $x$ that has $ {a}_{1},{a}_{2},\cdot \cdot \cdot {a}_{20} $ which follows the following ($x \ge 1000$) 1) $ {a}_{1}=2, {a}_{2}=1, {a}_{3}=x $ 2) for positive integer $n$, ($ 4 \le n \le 20 $), $ {a}_{n}={a}_{n-3}+\frac{(-2)^n}{{a}_{n-1}{a}_{n-2}} $

2017 Auckland Mathematical Olympiad, 3

Tags: number theory , divisible , digit , divide

The positive integer $N = 11...11$, whose decimal representation contains only ones, is divisible by $7$. Prove that this positive integer is also divisible by $13$.

2014 Saudi Arabia BMO TST, 2

Tags: function , number theory

Let $\mathbb{N}$ denote the set of positive integers, and let $S$ be a set. There exists a function $f :\mathbb{N} \rightarrow S$ such that if $x$ and $y$ are a pair of positive integers with their difference being a prime number, then $f(x) \neq f(y)$. Determine the minimum number of elements in $S$.

2025 Taiwan TST Round 2, N

Tags: number theory

Find all prime number pairs $(p, q)$ such that \[p^q+q^p+p+q-5pq\] is a perfect square. [i]Proposed by chengbilly[/i]

1999 Switzerland Team Selection Test, 9

Tags: algebra , polynomial , number theory

Suppose that $P(x)$ is a polynomial with degree $10$ and integer coefficients. Prove that, there is an infinite arithmetic progression (open to bothside) not contain value of $P(k)$ with $k\in\mathbb{Z}$

1931 Eotvos Mathematical Competition, 1

Tags: number theory

Let $p$ be a prime greater than $2$. Prove that $\frac{2}{p}$ can be expressed in exactly one way in the form $$\frac{1}{x}+\frac{1}{y}$$ where $x$ and $y$ are positive integers with $x > y$.

2018 CMIMC Number Theory, 8

Tags: number theory

It is given that there exists a unique triple of positive primes $(p,q,r)$ such that $p<q<r$ and \[\dfrac{p^3+q^3+r^3}{p+q+r} = 249.\] Find $r$.

2023 Thailand Mathematical Olympiad, 10

Tags: combinatorics , number theory , algebra

To celebrate the 20th Thailand Mathematical Olympiad (TMO), Ratchasima Witthayalai School put up flags around the Thao Suranari Monument so that [list=i] [*] Each flag is painted in exactly one color, and at least $2$ distinct colors are used. [*] The number of flags are odd. [*] Every flags are on a regular polygon such that each vertex has one flag. [*] Every flags with the same color are on a regular polygon. [/list] Prove that there are at least $3$ colors with the same amount of flags.

2005 Argentina National Olympiad, 4

Tags: number theory , perfect cube , perfect square , geometry , 3d geometry

We will say that a positive integer is a [i]winner [/i] if it can be written as the sum of a perfect square plus a perfect cube. For example, $33$ is a winner because $33=5^2+2^3$ . Gabriel chooses two positive integers, r and s, and Germán must find $2005$ positive integers $n$ such that for each $n$, the numbers $r+n$ and $s+n$ are winners. Prove that Germán can always achieve his goal.

2011 Spain Mathematical Olympiad, 2

Tags: function , algorithm , group theory , abstract algebra , induction , euclidean algorithm , number theory

Each rational number is painted either white or red. Call such a coloring of the rationals [i]sanferminera[/i] if for any distinct rationals numbers $x$ and $y$ satisfying one of the following three conditions: [list=1][*]$xy=1$, [*]$x+y=0$, [*]$x+y=1$,[/list]we have $x$ and $y$ painted different colors. How many sanferminera colorings are there?

1999 Italy TST, 1

Tags: modular arithmetic , number theory

Prove that for any prime number $p$ the equation $2^p+3^p=a^n$ has no solution $(a,n)$ in integers greater than $1$.

2006 Silk Road, 3

Tags: quadratic , number theory

A subset $S$ of the set $M=\{1,2,.....,p-1\}$,where $p$ is a prime number of the kind $12n+11$,is [i]essential[/i],if the product ${\Pi}_s$ of all elements of the subset is not less than the product $\bar{{\Pi}_s}$ of all other elements of the set.The [b]difference[/b] $\bigtriangleup_s=\Pi_s-\bar{{\Pi}_s}$ is called [i]the deviation[/i] of the subset $S$.Define the least possible remainder of division by $p$ of the deviation of an essential subset,containing $\frac{p-1}{2}$ elements.

1995 Yugoslav Team Selection Test, Problem 2

Tags: number theory

A natural number $n$ has exactly $1995$ units in its binary representation. Show that $n!$ is divisible by $2^{n-1995}$.

2024 Malaysia IMONST 2, 4

Tags: number theory

Pingu is given two positive integers $m$ and $n$ without any common factors greater than $1$. a) Help Pingu find positive integers $p, q$ such that $$\operatorname{gcd}(pm+q, n) \cdot \operatorname{gcd}(m, pn+q) = mn$$ b) Prove to Pingu that he can never find positive integers $r, s$ such that $$\operatorname{lcm}(rm+s, n) \cdot \operatorname{lcm}(m, rn+s) = mn$$ regardless of the choice of $m$ and $n$.

2002 Czech and Slovak Olympiad III A, 1

Tags: algebra , function , domain , number theory

Solve the system \[(4x)_5+7y=14 \\ (2y)_5 -(3x)_7=74\] in the domain of integers, where $(n)_k$ stands for the multiple of the number $k$ closest to the number $n$.

2021 Romania National Olympiad, 3

Tags: number theory , prime number

Let $n\ge 2$ be a positive integer such that the set of $n$th roots of unity has less than $2^{\lfloor\sqrt n\rfloor}-1$ subsets with the sum $0$. Show that $n$ is a prime number. [i]Cristi Săvescu[/i]

2017 CMIMC Number Theory, 8

Tags: number theory

Let $N$ be the number of ordered triples $(a,b,c) \in \{1, \ldots, 2016\}^{3}$ such that $a^{2} + b^{2} + c^{2} \equiv 0 \pmod{2017}$. What are the last three digits of $N$?

2014 Cuba MO, 7

Tags: diophantine equation , diophantine , number theory

Find all pairs of integers $(a, b)$ that satisfy the equation $$(a + 1)(b- 1) = a^2b^2.$$

2014 Junior Balkan Team Selection Tests - Moldova, 2

Tags: diophantine equation , diophantine , number theory

Determine all pairs of integers $(x, y)$ that satisfy equation $(y - 2) x^2 + (y^2 - 6y + 8) x = y^2 - 5y + 62$.

2003 Bulgaria National Olympiad, 2

Tags: inequalities , number theory

Let $a,b,c$ be rational numbers such that $a+b+c$ and $a^2+b^2+c^2$ are [b]equal[/b] integers. Prove that the number $abc$ can be written as the ratio of a perfect cube and a perfect square which are relatively prime.

1993 All-Russian Olympiad Regional Round, 11.2

Tags: number theory , divide , divisible

Prove that, for every integer $n > 2$, the number $$\left[\left( \sqrt[3]{n}+\sqrt[3]{n+2}\right)^3\right]+1$$ is divisible by $8$.

2013 Ukraine Team Selection Test, 4

Tags: quadratic , number theory

Call admissible a set $A$ of integers that has the following property: If $x,y \in A$ (possibly $x=y$) then $x^2+kxy+y^2 \in A$ for every integer $k$. Determine all pairs $m,n$ of nonzero integers such that the only admissible set containing both $m$ and $n$ is the set of all integers. [i]Proposed by Warut Suksompong, Thailand[/i]