Olympiad problems

2004 Junior Tuymaada Olympiad, 6

Tags: number theory , sum , reciprocal sum , prime , positive integer

We call a positive integer [i] good[/i] if the sum of the reciprocals of all its natural divisors are integers. Prove that if $ m $ is a [i]good [/i] number, and $ p> m $ is a prime number, then $ pm $ is not [i]good[/i].

2017 Saudi Arabia BMO TST, 1

Tags: product , integer sidelengths , geometry , number theory , right triangle , prime

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)

2020 Nordic, 1

Tags: number theory , min , arithmetic progression , prime

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]

2009 Silk Road, 4

Tags: number theory , perfect square , prime

Prove that for any prime number $p$ there are infinitely many fours $(x, y, z, t)$ pairwise distinct natural numbers such that the number $(x^2+p t^2)(y^2+p t^2)(z^2+p t^2)$ is a perfect square.

2021 239 Open Mathematical Olympiad, 1

Tags: integer , polynomial , integer polynomial , algebra , prime

You are given $n$ different primes $p_1, p_2,..., p_n$. Consider the polynomial $$x^n + a_1x^{n -1} + a_2x^{n - 2} + ...+ a_{n - 1}x + a_n$$, where $a_i$ is the product of the first $i$ given prime numbers. For what $n$ can it have an integer root?

1992 Mexico National Olympiad, 2

Tags: number theory , prime

Given a prime number $p$, how many $4$-tuples $(a, b, c, d)$ of positive integers with $0 \le a, b, c, d \le p-1$ satisfy $ad = bc$ mod $p$?

2012 Switzerland - Final Round, 4

Tags: number theory , recurrence relation , prime

Show that there is no infinite sequence of primes $p_1, p_2, p_3, . . .$ there any for each $ k$: $p_{k+1} = 2p_k - 1$ or $p_{k+1} = 2p_k + 1$ is fulfilled. Note that not the same formula for every $k$.

2015 India IMO Training Camp, 2

Tags: number theory , prime

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]

2010 Philippine MO, 1

Tags: number theory , prime

Find all primes that can be written both as a sum of two primes and as a difference of two primes.

2019 Austrian Junior Regional Competition, 4

Tags: number theory , divisible , divide , prime

Let $p, q, r$ and $s$ be four prime numbers such that $$5 <p <q <r <s <p + 10.$$ Prove that the sum of the four prime numbers is divisible by $60$. (Walther Janous)

1978 Chisinau City MO, 159

Tags: number theory , prime , divide , divisible , factorial

Prove that the product of numbers $1, 2, ..., n$ ($n \ge 2$) is divisible by their sum if and only if the number $n + 1$ is not prime.

1994 Chile National Olympiad, 3

Tags: prime , number theory , digit

Let $x$ be an integer of $n$ digits, all equal to $ 1$. Show that if $x$ is prime, then $n$ is also prime.

2025 VJIMC, 1

Tags: number theory , divisibility , prime

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 $p^2$.

2020 Dutch BxMO TST, 5

Tags: number theory , product , prime

A set S consisting of $2019$ (different) positive integers has the following property: [i]the product of every 100 elements of $S$ is a divisor of the product of the remaining $1919$ elements[/i]. What is the maximum number of prime numbers that $S$ can contain?

1994 Poland - Second Round, 6

Tags: number theory , divide , prime

Let $p$ be a prime number. Prove that there exists $n \in Z$ such that $p | n^2 -n+3$ if and only if there exists $m \in Z$ such that $p | m^2 -m+25$.

1970 Czech and Slovak Olympiad III A, 1

Tags: number theory , prime , fraction , harmonic , prime divisor , number

Let $p>2$ be a prime and $a,b$ positive integers such that \[\frac ab=1+\frac12+\frac13+\cdots+\frac{1}{p-1}.\] Show that $p$ is a divisor of $a.$

2019 Argentina National Olympiad, 5

Tags: number theory , arithmetic sequence , prime

There is an arithmetic progression of $7$ terms in which all the terms are different prime numbers. Determine the smallest possible value of the last term of such a progression. Clarification: In an arithmetic progression of difference $d$ each term is equal to the previous one plus $d$.

2018 All-Russian Olympiad, 1

Tags: algebra , prime , prime number , number theory

Suppose $a_1,a_2, \dots$ is an infinite strictly increasing sequence of positive integers and $p_1, p_2, \dots$ is a sequence of distinct primes such that $p_n \mid a_n$ for all $n \ge 1$. It turned out that $a_n-a_k=p_n-p_k$ for all $n,k \ge 1$. Prove that the sequence $(a_n)_n$ consists only of prime numbers.

2001 Estonia Team Selection Test, 5

Tags: product , power , number theory , digit , prime

Find the exponent of $37$ in the representation of the number $111...... 11$ with $3\cdot 37^{2000}$ digits equals to $1$, as product of prime powers

2012 Czech And Slovak Olympiad IIIA, 1

Tags: number theory , prime number , prime

Find all integers for which $n$ is $n^4 -3n^2 + 9$ prime

2014 Estonia Team Selection Test, 1

Tags: combinatorics , prime

In Wonderland, the government of each country consists of exactly $a$ men and $b$ women, where $a$ and $b$ are fixed natural numbers and $b > 1$. For improving of relationships between countries, all possible working groups consisting of exactly one government member from each country, at least $n$ among whom are women, are formed (where $n$ is a fixed non-negative integer). The same person may belong to many working groups. Find all possibilities how many countries can be in Wonderland, given that the number of all working groups is prime.

1996 Estonia National Olympiad, 4

Tags: decimal representation , power , prime

Prove that for each prime number $p > 5$ there exists a positive integer n such that $p^n$ ends in $001$ in decimal representation.

2022 Olimphíada, 1

Tags: divisibility , prime

Let $p,q$ prime numbers such that $$p+q \mid p^3-q^3$$ Show that $p=q$.

2018 Federal Competition For Advanced Students, P2, 3

Tags: number theory , combinatorics , prime

There are $n$ children in a room. Each child has at least one piece of candy. In Round $1$, Round $2$, etc., additional pieces of candy are distributed among the children according to the following rule: In Round $k$, each child whose number of pieces of candy is relatively prime to $k$ receives an additional piece. Show that after a sufficient number of rounds the children in the room have at most two different numbers of pieces of candy. [i](Proposed by Theresia Eisenkölbl)[/i]

2018 Costa Rica - Final Round, N4

Tags: prime , number theory

Let $p$ be a prime number such that $p = 10^{d -1} + 10^{d-2} + ...+ 10 + 1$. Show that $d$ is a prime.