Olympiad problems

2015 Azerbaijan JBMO TST, 4

Prove that there are not intgers $a$ and $b$ with conditions, i) $16a-9b$ is a prime number. ii) $ab$ is a perfect square. iii) $a+b$ is also perfect square.

KoMaL A Problems 2017/2018, A. 717

Tags: number theory , prime number , perfect square

Let's call a positive integer $n$ special, if there exist two nonnegativ integers ($a, b$), such that $n=2^a\times 3^b$. Prove that if $k$ is a positive integer, then there are at most two special numbers greater then $k^2$ and less than $k^2+2k+1$.

2017 Argentina National Math Olympiad Level 2, 4

Tags: number theory , prime number

Find all positive integers $a$ such that $4x^2 + a$ is prime for all $x = 0, 1, \dots, a - 1$.

2023 Moldova EGMO TST, 4

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

Find all triplets of prime numbers $(m, n, p)$, that satisfy the system of equations: $$\left\{\begin{matrix} 2m-n+13p=2072,\\3m+11n+13p=2961.\end{matrix}\right.$$

2015 Romania Masters in Mathematics, 5

Tags: number theory , prime number

Let $p \ge 5$ be a prime number. For a positive integer $k$, let $R(k)$ be the remainder when $k$ is divided by $p$, with $0 \le R(k) \le p-1$. Determine all positive integers $a < p$ such that, for every $m = 1, 2, \cdots, p-1$, $$ m + R(ma) > a. $$

2005 China Western Mathematical Olympiad, 3

Tags: number theory , prime number , relatively prime

Set $S = \{1, 2, 3, ..., 2005\}$. If among any $n$ pairwise coprime numbers in $S$ there exists at least a prime number, find the minimum of $n$.

2015 Portugal MO, 3

Tags: number theory , prime number

The numbers from $1$ to $2015$ are written on sheets so that if if $n-m$ is a prime, then $n$ and $m$ are on different sheets. What is the minimum number of sheets required?

2010 Iran Team Selection Test, 1

Tags: function , prime number , arithmetic sequence , number theory , logarithm

Let $f:\mathbb N\rightarrow\mathbb N$ be a non-decreasing function and let $n$ be an arbitrary natural number. Suppose that there are prime numbers $p_1,p_2,\dots,p_n$ and natural numbers $s_1,s_2,\dots,s_n$ such that for each $1\leq i\leq n$ the set $\{f(p_ir+s_i)|r=1,2,\dots\}$ is an infinite arithmetic progression. Prove that there is a natural number $a$ such that \[f(a+1), f(a+2), \dots, f(a+n)\] form an arithmetic progression.

2013 JBMO TST - Turkey, 2

Tags: modular arithmetic , prime number , number theory

[b]a)[/b] Find all prime numbers $p, q, r$ satisfying $3 \nmid p+q+r$ and $p+q+r$ and $pq+qr+rp+3$ are both perfect squares. [b]b)[/b] Do there exist prime numbers $p, q, r$ such that $3 \mid p+q+r$ and $p+q+r$ and $pq+qr+rp+3$ are both perfect squares?

1993 All-Russian Olympiad, 1

Tags: geometry , perimeter , inequalities , number theory , prime number , area of a triangle , heron's formula

The lengths of the sides of a triangle are prime numbers of centimeters. Prove that its area cannot be an integer number of square centimeters.

2023 VN Math Olympiad For High School Students, Problem 6

Tags: algebra , polynomial , number theory , prime number

Prove that these polynomials are irreducible in $\mathbb{Q}[x]:$ a) $\frac{{{x^p}}}{{p!}} + \frac{{{x^{p - 1}}}}{{(p - 1)!}} + ... + \frac{{{x^2}}}{2} + x + 1,$ with $p$ is a prime number. b) $x^{2^n}+1,$ with $n$ is a positive integer.

2017 Dutch BxMO TST, 2

Tags: function , number theory , prime number

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 $f(n)=n$

2007 South East Mathematical Olympiad, 3

Tags: ratio , prime number , number theory

Find all triples $(a,b,c)$ satisfying the following conditions: (i) $a,b,c$ are prime numbers, where $a<b<c<100$. (ii) $a+1,b+1,c+1$ form a geometric sequence.

2019 BMT Spring, 6

Tags: number theory , prime number

Define $ f(n) = \dfrac{n^2 + n}{2} $. Compute the number of positive integers $ n $ such that $ f(n) \leq 1000 $ and $ f(n) $ is the product of two prime numbers.

2008 IMAR Test, 2

Tags: analytic geometry , prime number , number theory

A point $ P$ of integer coordinates in the Cartesian plane is said [i]visible[/i] if the segment $ OP$ does not contain any other points with integer coordinates (except its ends). Prove that for any $ n\in\mathbb{N}^*$ there exists a visible point $ P_{n}$, at distance larger than $ n$ from any other visible point. [b]Dan Schwarz[/b]

2012 Middle European Mathematical Olympiad, 4

Tags: prime number , number theory

The sequence $ \{ a_n \} _ { n \ge 0 } $ is defined by $ a_0 = 2 , a_1 = 4 $ and \[ a_{n+1} = \frac{a_n a_{n-1}}{2} + a_n + a_{n-1} \] for all positive integers $ n $. Determine all prime numbers $ p $ for which there exists a positive integer $ m $ such that $ p $ divides the number $ a_m - 1 $.

2010 All-Russian Olympiad, 3

Tags: prime number , number theory , logarithm

Given $n \geq 3$ pairwise different prime numbers $p_1, p_2, ....,p_n$. Given, that for any $k \in \{ 1,2,....,n \}$ residue by division of $ \prod_{i \neq k} p_i$ by $p_k$ equals one number $r$. Prove, that $r \leq n-2 $.

2001 National Olympiad First Round, 23

Tags: modular arithmetic , number theory , prime number

Which of the followings is false for the sequence $9,99,999,\dots$? $\textbf{(A)}$ The primes which do not divide any term of the sequence are finite. $\textbf{(B)}$ Infinitely many primes divide infinitely many terms of the sequence. $\textbf{(C)}$ For every positive integer $n$, there is a term which is divisible by at least $n$ distinct prime numbers. $\textbf{(D)}$ There is an inteter $n$ such that every prime number greater than $n$ divides infinitely many terms of the sequence. $\textbf{(E)}$ None of above

2021 IMC, 6

Tags: group theory , number theory , prime number , abstract algebra

For a prime number $p$, let $GL_2(\mathbb{Z}/p\mathbb{Z})$ be the group of invertible $2 \times 2$ matrices of residues modulo $p$, and let $S_p$ be the symmetric group (the group of all permutations) on $p$ elements. Show that there is no injective group homomorphism $\phi : GL_2(\mathbb{Z}/p\mathbb{Z}) \rightarrow S_p$.

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 France Team Selection Test, 3

Tags: induction , prime number , number theory

Let $P$ be the set of prime numbers. Consider a subset $M$ of $P$ with at least three elements. We assume that, for each non empty and finite subset $A$ of $M$, with $A \neq M$, the prime divisors of the integer $( \prod_{p \in A} ) - 1$ belong to $M$. Prove that $M = P$.

2013 National Olympiad First Round, 2

Tags: modular arithmetic , number theory , prime number

How many triples $(p,q,n)$ are there such that $1/p+2013/q = n/5$ where $p$, $q$ are prime numbers and $n$ is a positive integer? $ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 4 $

2016 Bosnia and Herzegovina Team Selection Test, 3

Tags: number theory , prime number , integer sequence

For an infinite sequence $a_1<a_2<a_3<...$ of positive integers we say that it is [i]nice[/i] if for every positive integer $n$ holds $a_{2n}=2a_n$. Prove the following statements: $a)$ If there is given a [i]nice[/i] sequence and prime number $p>a_1$, there exist some term of the sequence which is divisible by $p$. $b)$ For every prime number $p>2$, there exist a [i]nice[/i] sequence such that no terms of the sequence are divisible by $p$.

1999 Baltic Way, 20

Tags: prime number , number theory

Let $a,b,c$ and $d$ be prime numbers such that $a>3b>6c>12d$ and $a^2-b^2+c^2-d^2=1749$. Determine all possible values of $a^2+b^2+c^2+d^2$ .

1977 IMO Longlists, 27

Tags: number theory , prime number , prime factorization , dirichlet s theorem

Let $n$ be a given number greater than 2. We consider the set $V_n$ of all the integers of the form $1 + kn$ with $k = 1, 2, \ldots$ A number $m$ from $V_n$ is called indecomposable in $V_n$ if there are not two numbers $p$ and $q$ from $V_n$ so that $m = pq.$ Prove that there exist a number $r \in V_n$ that can be expressed as the product of elements indecomposable in $V_n$ in more than one way. (Expressions which differ only in order of the elements of $V_n$ will be considered the same.)