Olympiad problems

2015 Ukraine Team Selection Test, 3

Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$. [i]Proposed by Belgium[/i]

2015 Belarus Team Selection Test, 1

Tags: number theory , primes

Find all positive integers $n$ such that $n=q(q^2-q-1)=r(2r+1)$ for some primes $q$ and $r$. B.Gilevich

2017 Saudi Arabia BMO TST, 1

Tags: Product , integer sidelengths , geometry , number theory , right triangle , primes

Let $n = p_1p_2... p_{2017}$ be the positive integer where $p_1, p_2, ..., p_{2017}$ are $2017$ distinct odd primes. A triangle is called [i]nice [/i] if it is a right triangle with integer side lengths and the inradius is $n$. Find the number of nice triangles (two triangles are consider different if their tuples of length of sides are different)

2013 IMAR Test, 1

Tags: primes , fermat's little theorem , number theory

Given a prime $p \geq 5$ , show that there exist at least two distinct primes $q$ and $r$ in the range $2, 3, \ldots p-2$ such that $q^{p-1} \not\equiv 1 \pmod{p^2}$ and $r^{p-1} \not\equiv 1 \pmod{p^2}$.

2021 Baltic Way, 17

Tags: number theory , number theory proposed , primes

Distinct positive integers $a, b, c, d$ satisfy $$\begin{cases} a \mid b^2 + c^2 + d^2,\\ b\mid a^2 + c^2 + d^2,\\ c \mid a^2 + b^2 + d^2,\\ d \mid a^2 + b^2 + c^2,\end{cases}$$ and none of them is larger than the product of the three others. What is the largest possible number of primes among them?

2014 Danube Mathematical Competition, 1

Tags: number theory , Sum , primes , natural

Determine the natural number $a =\frac{p+q}{r}+\frac{q+r}{p}+\frac{r+p}{q}$ where $p, q$ and $r$ are prime positive numbers.

1997 Israel Grosman Mathematical Olympiad, 1

Tags: number theory , Digits , primes

Prove that there are at most three primes between $10$ and $10^{10}$ all of whose decimal digits are $1$.

2012 EGMO, 5

Tags: number theory , modular arithmetic , equation , EGMO , primes , GCD

The numbers $p$ and $q$ are prime and satisfy \[\frac{p}{{p + 1}} + \frac{{q + 1}}{q} = \frac{{2n}}{{n + 2}}\] for some positive integer $n$. Find all possible values of $q-p$. [i]Luxembourg (Pierre Haas)[/i]

2021 South Africa National Olympiad, 3

Tags: number theory , Perfect Squares , Sum of Squares , primes , power of 2

Determine the smallest integer $k > 1$ such that there exist $k$ distinct primes whose squares sum to a power of $2$.

2019 Peru EGMO TST, 1

Tags: Diophantine equation , diophantine , primes , number theory

Find all the prime numbers $p, q$ and $r$ such that $p^2 + 1 = 74 (q^2 + r^2)$.

2023 Switzerland - Final Round, 6

Tags: number theory , divsibility , primes

Find all positive integers $n>2$ such that $$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$

2025 VJIMC, 1

Tags: number theory , primes , Divisibility

Let $a\geq 2$ be an integer. Prove that there exists a positive integer $b$ with the following property: For each positive integer $n$, there is a prime number $p$ (possibly depending on $a,b,n$) such that $a^n + b$ is divisible by $p$, but not divisible by $p^2$.

2013 Balkan MO Shortlist, N5

Tags: number theory , Diophantine equation , diophantine , primes

Prove that there do not exist distinct prime numbers $p$ and $q$ and a positive integer $n$ satisfying the equation $p^{q-1}- q^{p-1}=4n^2$

2012 NZMOC Camp Selection Problems, 4

Tags: number theory , primes , multiple , divides , divisible

A pair of numbers are [i]twin primes[/i] if they differ by two, and both are prime. Prove that, except for the pair $\{3, 5\}$, the sum of any pair of twin primes is a multiple of $ 12$.

1998 Estonia National Olympiad, 4

Tags: primes , divided , number theory , divisible

Prove that if for a positive integer $n$ is $5^n + 3^n + 1$ is prime number, then $n$ is divided by $12$.

2013 Thailand Mathematical Olympiad, 1

Tags: divides , primes , number theory

Find the largest integer that divides $p^4 - 1$ for all primes $p > 4$

2025 Israel TST, P2

Tags: number theory , primes

Prove that for all primes $ p $ such that $ p \equiv 3 \pmod{4} $ or $ p \equiv 5 \pmod{8} $, there exist integers \[ 1 \leq a_1 < a_2 < \cdots < a_{(p-1)/2} < p \] such that \[ \prod_{\substack{1 \leq i < j \leq (p-1)/2}} (a_i + a_j)^2 \equiv 1 \pmod{p}. \]

2006 Korea National Olympiad, 2

Tags: primes , Combinatorial games , combinatorics

Alice and Bob are playing "factoring game." On the paper, $270000(=2^43^35^4)$ is written and each person picks one number from the paper(call it $N$) and erase $N$ and writes integer $X,Y$ such that $N=XY$ and $\text{gcd}(X,Y)\ne1.$ Alice goes first and the person who can no longer make this factoring loses. If two people use optimal strategy, prove that Alice always win.

2015 Gulf Math Olympiad, 1

Tags: number theory , odd , primes , divides , divisible

a) Suppose that $n$ is an odd integer. Prove that $k(n-k)$ is divisible by $2$ for all positive integers $k$. b) Find an integer $k$ such that $k(100-k)$ is not divisible by $11$. c) Suppose that $p$ is an odd prime, and $n$ is an integer. Prove that there is an integer $k$ such that $k(n-k)$ is not divisible by $p$. d) Suppose that $p,q$ are two different odd primes, and $n$ is an integer. Prove that there is an integer $k$ such that $k(n-k)$ is not divisible by any of $p,q$.

2013 Balkan MO Shortlist, N6

Tags: number theory , Diophantine equation , diophantine , primes

Prove that there do not exist distinct prime numbers $p$ and $q$ and a positive integer $n$ satisfying the equation $p^{q-1}- q^{p-1}=4n^3$

1955 Moscow Mathematical Olympiad, 299

Tags: number theory , Arithmetic Progression , Sequence , primes , divisible

Suppose that primes $a_1, a_2, . . . , a_p$ form an increasing arithmetic progression and $a_1 > p$. Prove that if $p$ is a prime, then the difference of the progression is divisible by $p$.

2024 Mexican Girls' Contest, 8

Tags: primes , number theory

Find all positive integers $n$ such that among the $n$ numbers \[ 2n + 1, \, 2^2 n + 1, \, \ldots, \, 2^n n + 1 \] there are $n$, $n - 1$, or $n - 2$ primes.

2020 Nordic, 1

Tags: number theory , min , Arithmetic Progression , primes

For a positive integer $n$, denote by $g(n)$ the number of strictly ascending triples chosen from the set $\{1, 2, ..., n\}$. Find the least positive integer $n$ such that the following holds:[i] The number $g(n)$ can be written as the product of three different prime numbers which are (not necessarily consecutive) members in an arithmetic progression with common difference $336$.[/i]

2017 Czech-Polish-Slovak Junior Match, 1

Tags: perfect cube , number theory , primes

Decide if there are primes $p, q, r$ such that $(p^2 + p) (q^2 + q) (r^2 + r)$ is a square of an integer.

1993 Italy TST, 2

Tags: Integer , number theory , primes

Suppose that $p,q$ are prime numbers such that $\sqrt{p^2 +7pq+q^2}+\sqrt{p^2 +14pq+q^2}$ is an integer. Show that $p = q$.