Olympiad problems

2010 Regional Olympiad of Mexico Center Zone, 2

Let $p>5$ be a prime number. Show that $p-4$ cannot be the fourth power of a prime number.

2017 Balkan MO Shortlist, N5

Tags: quadratic residue , number theory , fractional part , arithmetic mean , odd , prime

Given a positive odd integer $n$, show that the arithmetic mean of fractional parts $\{\frac{k^{2n}}{p}\}, k=1,..., \frac{p-1}{2}$ is the same for infinitely many primes $p$ .

2012 NZMOC Camp Selection Problems, 4

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

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$.

1984 All Soviet Union Mathematical Olympiad, 386

Tags: 3-digit , digit , prime number , prime , number theory

Let us call "absolutely prime" the prime number, if having transposed its digits in an arbitrary order, we obtain prime number again. Prove that its notation cannot contain more than three different digits.

2014 Hanoi Open Mathematics Competitions, 4

Tags: diophantine equation , prime , diophantine , number theory

If $p$ is a prime number such that there exist positive integers $a$ and $b$ such that $\frac{1}{p}=\frac{1}{a^2}+\frac{1}{b^2}$ then $p$ is (A): $3$, (B): $5$, (C): $11$, (D): $7$, (E) None of the above.

1999 IMO Shortlist, 1

Tags: number theory , divisibility , prime

Find all the pairs of positive integers $(x,p)$ such that p is a prime, $x \leq 2p$ and $x^{p-1}$ is a divisor of $ (p-1)^{x}+1$.

1962 Czech and Slovak Olympiad III A, 1

Tags: quadratic polynomial , perfect square , prime , algebra

Determine all integers $x$ such that $2x^2-x-36$ is a perfect square of a prime.

2015 Gulf Math Olympiad, 1

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

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 Czech-Polish-Slovak Junior Match, 1

Tags: infinitely many , prime , number theory

Decide whether there are infinitely many primes $p$ having a multiple in the form $n^2 + n + 1$ for some natural number $n$

2025 Macedonian Balkan MO TST, 4

Tags: sum of squares , prime , number theory

Let $n$ be a positive integer. Prove that for every odd prime $p$ dividing $n^2 + n + 2$, there exist integers $a, b$ such that $p = a^2 + 7b^2$.

2003 France Team Selection Test, 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.

2023 Thailand TST, 1

Tags: divisibility , number theory , prime

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

2022 IMO Shortlist, N6

Tags: prime , prime number , number theory , prime factorization , function

Let $Q$ be a set of prime numbers, not necessarily finite. For a positive integer $n$ consider its prime factorization: define $p(n)$ to be the sum of all the exponents and $q(n)$ to be the sum of the exponents corresponding only to primes in $Q$. A positive integer $n$ is called [i]special[/i] if $p(n)+p(n+1)$ and $q(n)+q(n+1)$ are both even integers. Prove that there is a constant $c>0$ independent of the set $Q$ such that for any positive integer $N>100$, the number of special integers in $[1,N]$ is at least $cN$. (For example, if $Q=\{3,7\}$, then $p(42)=3$, $q(42)=2$, $p(63)=3$, $q(63)=3$, $p(2022)=3$, $q(2022)=1$.)

2010 Gheorghe Vranceanu, 3

Tags: prime , combinatorics

Prove that however we choose the majority of numbers among an even number of the first consecutive natural numbers, there will be two numbers among this choosing whose sum is a prime.

2016 Korea Summer Program Practice Test, 3

Tags: primitive root , prime number , decimal representation , number theory , prime

Let $p > 10^9$ be a prime number such that $4p + 1$ is also prime. Prove that the decimal expansion of $\frac{1}{4p+1}$ contains all the digits $0,1, \ldots, 9$.

2022 Bulgarian Spring Math Competition, Problem 9.3

Tags: prime , algebra , system of equations

Find all primes $p$, such that there exist positive integers $x$, $y$ which satisfy $$\begin{cases} p + 49 = 2x^2\\ p^2 + 49 = 2y^2\\ \end{cases}$$

2019 Indonesia MO, 1

Tags: prime , number theory

Given that $n$ and $r$ are positive integers. Suppose that \[ 1 + 2 + \dots + (n - 1) = (n + 1) + (n + 2) + \dots + (n + r) \] Prove that $n$ is a composite number.

2017 Romania National Olympiad, 4

Tags: digit , prime number , prime , number theory

Find all prime numbers with $n \ge 3$ digits, having the property: for every $k \in \{1, 2, . . . , n -2\}$, deleting any $k$ of its digits leaves a prime number.

2015 IFYM, Sozopol, 5

Tags: divisibility , prime , number theory

Let $p>3$ be a prime number. The natural numbers $a,b,c, d$ are such that $a+b+c+d$ and $a^3+b^3+c^3+d^3$ are divisible by $p$. Prove that for all odd $n$, $a^n+b^n+c^n+d^n$ is divisible by $p$.

2012 Tournament of Towns, 2

Tags: table , rectangle table , prime , combinatorics

The cells of a $1\times 2n$ board are labelled $1,2,...,, n, -n,..., -2, -1$ from left to right. A marker is placed on an arbitrary cell. If the label of the cell is positive, the marker moves to the right a number of cells equal to the value of the label. If the label is negative, the marker moves to the left a number of cells equal to the absolute value of the label. Prove that if the marker can always visit all cells of the board, then $2n + 1$ is prime.

2019 Bundeswettbewerb Mathematik, 4

Tags: prime , number theory

Prove that for no integer $k \ge 2$, between $10k$ and $10k + 100$ there are more than $23$ prime numbers.

2022 AMC 12/AHSME, 3

Tags: prime

How many of the first ten numbers of the sequence $121$, $11211$, $1112111$, ... are prime numbers? $\textbf{(A) } 0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3 \qquad \textbf{(E) }4$

2012 Tournament of Towns, 4

Tags: prime , divisor , number theory

Let $C(n)$ be the number of prime divisors of a positive integer $n$. (a) Consider set $S$ of all pairs of positive integers $(a, b)$ such that $a \ne b$ and $C(a + b) = C(a) + C(b)$. Is $S$ finite or infinite? (b) Define $S'$ as a subset of S consisting of the pairs $(a, b)$ such that $C(a+b) > 1000$. Is $S'$ finite or infinite?

2016 Romanian Master of Mathematics Shortlist, A2

Tags: polynomial , prime , ceiling function , algebra

Let $p > 3$ be a prime number, and let $F_p$ denote the (finite) set of residue classes modulo $p$. Let $S_d$ denote the set of $2$-variable polynomials $P(x, y)$ with coefficients in $F_p$, total degree $\le d$, and satisfying $P(x, y) = P(y,- x -y)$. Show that $$|S_d| = p^{\lceil (d+1)(d+2)/6 \rceil}$$. [i]The total degree of a $2$-variable polynomial $P(x, y)$ is the largest value of $i + j$ among monomials $x^iy^j$ [/i] appearing in $P$.

2004 Mexico National Olympiad, 1

Tags: prime , perfect square , number theory

Find all the prime number $p, q$ and r with $p < q < r$, such that $25pq + r = 2004$ and $pqr + 1 $ is a perfect square.