Olympiad problems

1995 India National Olympiad, 6

Find all primes $p$ for which the quotient \[ \dfrac{2^{p-1} - 1 }{p} \] is a square.

2019 AMC 10, 2

Tags: prime

Consider the statement, "If $n$ is not prime, then $n-2$ is prime." Which of the following values of $n$ is a counterexample to this statement? $\textbf{(A) } 11 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 21 \qquad \textbf{(E) } 27$

2013 Balkan MO Shortlist, N5

Tags: diophantine equation , prime , diophantine , number theory

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

2020 New Zealand MO, 4

Tags: prime number , prime , number theory

Determine all prime numbers $p$ such that $p^2 - 6$ and $p^2 + 6$ are both prime numbers.

2024 SG Originals, Q5

Tags: prime number , grid , combinatorics , prime , number theory

Let $p$ be a prime number. Determine the largest possible $n$ such that the following holds: it is possible to fill an $n\times n$ table with integers $a_{ik}$ in the $i$th row and $k$th column, for $1\le i,k\le n$, such that for any quadruple $i,j,k,l$ with $1\le i<j\le n$ and $1\le k<l\le n$, the number $a_{ik}a_{jl}-a_{il}a_{jk}$ is not divisible by $p$. [i]Proposed by oneplusone[/i]

2003 BAMO, 4

Tags: divisor , prime , number theory

An integer $n > 1$ has the following property: for every (positive) divisor $d$ of $n, d + 1$ is a divisor of $n + 1$. Prove that $n$ is prime.

2002 IMO Shortlist, 3

Tags: divisor , number theory , algebra , prime

Let $p_1,p_2,\ldots,p_n$ be distinct primes greater than $3$. Show that $2^{p_1p_2\cdots p_n}+1$ has at least $4^n$ divisors.

1987 Bundeswettbewerb Mathematik, 1

Tags: power , digit , prime , number theory

Let $p>3$ be a prime and $n$ a positive integer such that $p^n$ has $20$ digits. Prove that at least one digit appears more than twice in this number.

2014 Contests, 4

Tags: point , prime , number theory , circles

The radius $r$ of a circle with center at the origin is an odd integer. There is a point ($p^m, q^n$) on the circle, with $p,q$ prime numbers and $m,n$ positive integers. Determine $r$.

2017 Saudi Arabia BMO TST, 1

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

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)

2009 Cuba MO, 1

Tags: remainder , prime , number theory

Show that when a prime number is divided by $30$, the remainder is $1$ or a prime number. Shows that if it is divided by $60$ or $90$ the same thing does not happen.

1978 Bundeswettbewerb Mathematik, 4

Tags: decimal representation , permutation , prime , number theory

A prime number has the property that however its decimal digits are permuted, the obtained number is also prime. Prove that this number has at most three different digits. Also prove a stronger statement.

2013 Balkan MO Shortlist, N6

Tags: diophantine equation , diophantine , number theory , prime

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

1969 Czech and Slovak Olympiad III A, 3

Tags: fraction , sequence , prime , number theory

Let $p$ be a prime. How many different (infinite) sequences $\left(a_k\right)_{k\ge0}$ exist such that for every positive integer $n$ \[\frac{a_0}{a_1}+\frac{a_0}{a_2}+\cdots+\frac{a_0}{a_n}+\frac{p}{a_{n+1}}=1?\]

1984 Tournament Of Towns, (060) A5

Tags: number theory , prime , consecutive

The two pairs of consecutive natural numbers $(8, 9)$ and $(288, 289)$ have the following property: in each pair, each number contains each of its prime factors to a power not less than $2$. Prove that there are infinitely many such pairs. (A Andjans, Riga)

2017 Junior Regional Olympiad - FBH, 4

Tags: statements , prime , divisible

Let $n$ and $k$ be positive integers for which we have $4$ statements: $i)$ $n+1$ is divisible with $k$ $ii)$ $n=2k+5$ $iii)$ $n+k$ is divisible with $3$ $iv)$ $n+7k$ is prime Determine all possible values for $n$ and $k$, if out of the $4$ statements, three of them are true and one is false

2013 Abels Math Contest (Norwegian MO) Final, 3

Tags: divide , prime , number theory

A prime number $p \ge 5$ is given. Write $\frac13+\frac24+... +\frac{p -3}{p - 1}=\frac{a}{b}$ for natural numbers $a$ and $b$. Show that $p$ divides $a$.

2022 JBMO Shortlist, N5

Tags: remainder , prime , shortlist , number theory

Find all pairs $(a, p)$ of positive integers, where $p$ is a prime, such that for any pair of positive integers $m$ and $n$ the remainder obtained when $a^{2^n}$ is divided by $p^n$ is non-zero and equals the remainder obtained when $a^{2^m}$ is divided by $p^m$.

2016 Costa Rica - Final Round, N2

Tags: prime , number theory

Determine all positive integers $a$ and $b$ for which $a^4 + 4b^4$ be a prime number.

2020 Nordic, 1

Tags: number theory , arithmetic progression , min , 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]

2014 Estonia Team Selection Test, 1

Tags: prime , combinatorics

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.