Olympiad problems

2013 Peru IMO TST, 5

Determine all integers $m \geq 2$ such that every $n$ with $\frac{m}{3} \leq n \leq \frac{m}{2}$ divides the binomial coefficient $\binom{n}{m-2n}$.

2015 Belarus Team Selection Test, 3

Tags: prime number , algebra , combinatorics

Construct a tetromino by attaching two $2 \times 1$ dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them $S$- and $Z$-tetrominoes, respectively. Assume that a lattice polygon $P$ can be tiled with $S$-tetrominoes. Prove that no matter how we tile $P$ using only $S$- and $Z$-tetrominoes, we always use an even number of $Z$-tetrominoes. [i]Proposed by Tamas Fleiner and Peter Pal Pach, Hungary[/i]

2005 CHKMO, 4

Tags: number theory , function , prime number , combinatorics

Let $S=\{1,2,...,100\}$ . Find number of functions $f: S\to S$ satisfying the following conditions a)$f(1)=1$ b)$f$ is bijective c)$f(n)=f(g(n))f(h(n))\forall n\in S$, where $g(n),h(n)$ are positive integer numbers such that $g(n)\leq h(n),n=g(n)h(n)$ that minimize $h(n)-g(n)$.

2008 All-Russian Olympiad, 3

Tags: number theory , prime number

Given a finite set $ P$ of prime numbers, prove that there exists a positive integer $ x$ such that it can be written in the form $ a^p \plus{} b^p$ ($ a,b$ are positive integers), for each $ p\in P$, and cannot be written in that form for each $ p$ not in $ P$.

2014 Iran Team Selection Test, 2

Tags: prime number , number theory , function

is there a function $f:\mathbb{N}\rightarrow \mathbb{N}$ such that $i) \exists n\in \mathbb{N}:f(n)\neq n$ $ii)$ the number of divisors of $m$ is $f(n)$ if and only if the number of divisors of $f(m)$ is $n$

2021 Argentina National Olympiad, 2

Tags: prime number , positive integer , perfect square , number theory

Let $m$ be a positive integer for which there exists a positive integer $n$ such that the multiplication $mn$ is a perfect square and $m- n$ is prime. Find all $m$ for $1000\leq m \leq 2021.$

1996 Iran MO (3rd Round), 4

Tags: combinatorics , prime number , number theory

Let $n$ be a positive integer and suppose that $\phi(n)=\frac{n}{k}$, where $k$ is the greatest perfect square such that $k \mid n$. Let $a_1,a_2,\ldots,a_n$ be $n$ positive integers such that $a_i=p_1^{a_1i} \cdot p_2^{a_2i} \cdots p_n^{a_ni}$, where $p_i$ are prime numbers and $a_{ji}$ are non-negative integers, $1 \leq i \leq n, 1 \leq j \leq n$. We know that $p_i\mid \phi(a_i)$, and if $p_i\mid \phi(a_j)$, then $p_j\mid \phi(a_i)$. Prove that there exist integers $k_1,k_2,\ldots,k_m$ with $1 \leq k_1 \leq k_2 \leq \cdots \leq k_m \leq n$ such that \[\phi(a_{k_{1}} \cdot a_{k_{2}} \cdots a_{k_{m}})=p_1 \cdot p_2 \cdots p_n.\]

2012 Iran MO (3rd Round), 1

Tags: algebra , modular arithmetic , prime number , number theory , polynomial , inequalities

$P(x)$ is a nonzero polynomial with integer coefficients. Prove that there exists infinitely many prime numbers $q$ such that for some natural number $n$, $q|2^n+P(n)$. [i]Proposed by Mohammad Gharakhani[/i]

2007 Bulgarian Autumn Math Competition, Problem 12.4

Tags: prime number , linear recurrence , number theory

Let $p$ and $q$ be prime numbers and $\{a_{n}\}_{n=1}^{\infty}$ be a sequence of integers defined by: \[a_{0}=0, a_{1}=1, a_{n+2}=pa_{n+1}-qa_{n}\quad\forall n\geq 0\] Find $p$ and $q$ if there exists an integer $k$ such that $a_{3k}=-3$.

2017 Dutch BxMO TST, 2

Tags: function , prime number , number theory

Let define a function $f: \mathbb{N} \rightarrow \mathbb{Z}$ such that : $i)$$f(p)=1$ for all prime numbers $p$. $ii)$$f(xy)=xf(y)+yf(x)$ for all positive integers $x,y$ find the smallest $n \geq 2016$ such that $f(n)=n$

2003 AMC 10, 14

Tags: prime number , number theory

Let $ n$ be the largest integer that is the product of exactly $ 3$ distinct prime numbers, $ d$, $ e$, and $ 10d\plus{}e$, where $ d$ and $ e$ are single digits. What is the sum of the digits of $ n$? $ \textbf{(A)}\ 12 \qquad \textbf{(B)}\ 15 \qquad \textbf{(C)}\ 18 \qquad \textbf{(D)}\ 21 \qquad \textbf{(E)}\ 24$

2016 Turkmenistan Regional Math Olympiad, Problem 3

Tags: number theory , prime number

Find all distinct prime numbers $p,q,r,s$ such that $1-\frac{1}{p} - \frac{1}{q} -\frac{1}{r} - \frac{1}{s} =\frac{1}{pqrs}$

2012 India National Olympiad, 2

Tags: prime number , divisibility tests , number theory

Let $p_1<p_2<p_3<p_4$ and $q_1<q_2<q_3<q_4$ be two sets of prime numbers, such that $p_4 - p_1 = 8$ and $q_4 - q_1= 8$. Suppose $p_1 > 5$ and $q_1>5$. Prove that $30$ divides $p_1 - q_1$.

2021 Turkey Team Selection Test, 1

Tags: legendre symbol , prime number , divisibility , number theory

Let $n$ be a positive integer. Prove that \[\frac{20 \cdot 5^n-2}{3^n+47}\] is not an integer.

2004 Iran MO (3rd Round), 1

Tags: combinatorics , prime number , number theory

We say $m \circ n$ for natural m,n $\Longleftrightarrow$ nth number of binary representation of m is 1 or mth number of binary representation of n is 1. and we say $m \bullet n$ if and only if $m,n$ doesn't have the relation $\circ$ We say $A \subset \mathbb{N}$ is golden $\Longleftrightarrow$ $\forall U,V \subset A$ that are finite and arenot empty and $U \cap V = \emptyset$,There exist $z \in A$ that $\forall x \in U,y \in V$ we have $z \circ x ,z \bullet y$ Suppose $\mathbb{P}$ is set of prime numbers.Prove if $\mathbb{P}=P_1 \cup ... \cup P_k$ and $P_i \cap P_j = \emptyset$ then one of $P_1,...,P_k$ is golden.

2014 All-Russian Olympiad, 1

Tags: prime number , number theory

Call a natural number $n$ [i]good[/i] if for any natural divisor $a$ of $n$, we have that $a+1$ is also divisor of $n+1$. Find all good natural numbers. [i]S. Berlov[/i]

2024 Mozambican National MO Selection Test, P3

Tags: number theory , nt , diophantine equation , factorization , prime number , positive integer

Find all triples of positive integers $(a,b,c)$ such that: $a^2bc-2ab^2c-2abc^2+b^3c+bc^3+2b^2c^2=11$

1994 Bulgaria National Olympiad, 3

Tags: number theory , prime number

Let $p$ be a prime number, determine all positive integers $(x, y, z)$ such that: $x^p + y^p = p^z$

1997 Bundeswettbewerb Mathematik, 2

Tags: number theory , prime number

Find a prime number $p$ such that $\frac{p+1}{2}$ and $\frac{p^2+1}{2}$ are perfect square

2017 Turkey MO (2nd round), 4

Tags: number theory , prime number

Let $d(n)$ be number of prime divisors of $n$. Prove that one can find $k,m$ positive integers for any positive integer $n$ such that $k-m=n$ and $d(k)-d(m)=1$

2023 IMAR Test, P3

Tags: permutation , prime number , number theory

Let $p{}$ be an odd prime number. Determine whether there exists a permutation $a_1,\ldots,a_p$ of $1,\ldots,p$ satisfying \[(i-j)a_k+(j-k)a_i+(k-i)a_j\neq 0,\] for all pairwise distinct $i,j,k.$

2005 Flanders Junior Olympiad, 3

Tags: geometry , complex number , prime number , number theory

Prove that $2005^2$ can be written in at least $4$ ways as the sum of 2 perfect (non-zero) squares.

2009 China Team Selection Test, 2

Tags: algebra , modular arithmetic , polynomial , prime number , number theory

Find all integers $ n\ge 2$ having the following property: for any $ k$ integers $ a_{1},a_{2},\cdots,a_{k}$ which aren't congruent to each other (modulo $ n$), there exists an integer polynomial $ f(x)$ such that congruence equation $ f(x)\equiv 0 (mod n)$ exactly has $ k$ roots $ x\equiv a_{1},a_{2},\cdots,a_{k} (mod n).$

2014 Poland - Second Round, 6.

Tags: prime number , number theory

Call a positive number $n$ [i]fine[/i], if there exists a prime number $p$ such that $p|n$ and $p^2\nmid n$. Prove that at least 99% of numbers $1, 2, 3, \ldots, 10^{12}$ are fine numbers.

2016 Hong Kong TST, 1

Tags: number theory , prime number

Find all prime numbers $p$ and $q$ such that $p^2|q^3+1$ and $q^2|p^6-1$