Olympiad problems

2012 IMO Shortlist, N8

Prove that for every prime $p>100$ and every integer $r$, there exist two integers $a$ and $b$ such that $p$ divides $a^2+b^5-r$.

2012 APMO, 3

Tags: modular arithmetic , prime , divisibility , number theory

Determine all the pairs $ (p , n )$ of a prime number $ p$ and a positive integer $ n$ for which $ \frac{ n^p + 1 }{p^n + 1} $ is an integer.

2017 Peru IMO TST, 10

Tags: squarefree , polynomial , integer , number theory , prime

Let $P (n)$ and $Q (n)$ be two polynomials (not constant) whose coefficients are integers not negative. For each positive integer $n$, define $x_n = 2016^{P (n)} + Q (n)$. Prove that there exist infinite primes $p$ for which there is a positive integer $m$, squarefree, such that $p | x_m$. Clarification: A positive integer is squarefree if it is not divisible by the square of any prime number.

2010 Thailand Mathematical Olympiad, 10

Tags: number theory , binomial , divisible , prime

Find all primes $p$ such that ${100 \choose p} + 7$ is divisible by $p$.

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? $\textbf{(A) } 11 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 21 \qquad \textbf{(E) } 27$

2016 Romanian Master of Mathematics Shortlist, A2

Tags: ceiling function , algebra , polynomial , prime

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 $P$.

Mathley 2014-15, 7

Tags: number theory , diophantine equation , diophantine , prime

Find all primes $p,q, r$ such that $\frac{p^{2q}+q^{2p}}{p^3-pq+q^3} = r$. Titu Andreescu, Mathematics Department, College of Texas, USA

2012 Tournament of Towns, 2

Tags: number theory , prime , divisor

Let $C(n)$ be the number of prime divisors of a positive integer n. (For example, $C(10) = 2,C(11) = 1, C(12) = 2$). 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 set $S$ finite or infinite?

2019 India PRMO, 14

Tags: prime , perfect square , relation

Find the smallest positive integer $n \geq 10$ such that $n + 6$ is a prime and $9n + 7$ is a perfect square.

2013 Dutch BxMO/EGMO TST, 3

Tags: number theory , prime , diophantine equation

Find all triples $(x,n,p)$ of positive integers $x$ and $n$ and primes $p$ for which the following holds $x^3 + 3x + 14 = 2 p^n$

1987 Bundeswettbewerb Mathematik, 1

Tags: number theory , prime , power , digit

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.

2007 Estonia Team Selection Test, 3

Tags: number theory , power of prime , prime

Let $n$ be a natural number, $n > 2$. Prove that if $\frac{b^n-1}{b-1}$ is a prime power for some positive integer $b$ then $n$ is prime.

2022 Mexico National Olympiad, 1

Tags: reciprocal , prime

A number $x$ is "Tlahuica" if there exist prime numbers $p_1,\ p_2,\ \dots,\ p_k$ such that \[x=\frac{1}{p_1}+\frac{1}{p_2}+\dots+\frac{1}{p_k}.\] Find the largest Tlahuica number $x$ such that $0<x<1$ and there exists a positive integer $m\leq 2022$ such that $mx$ is an integer.

2002 VJIMC, Problem 2

Tags: number theory , divisibility , prime

Let $p>3$ be a prime number and $n=\frac{2^{2p}-1}3$. Show that $n$ divides $2^n-2$.

2018 Malaysia National Olympiad, A6

Tags: number theory , prime

How many integers $n$ are there such that $n^4 + 2n^3 + 2n^2 + 2n + 1$ is a prime number?

1984 All Soviet Union Mathematical Olympiad, 386

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

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.

2016 Mediterranean Mathematics Olympiad, 4

Tags: number theory , prime

Determine all integers $n\ge1$ for which the number $n^8+n^6+n^4+4$ is prime. (Proposed by Gerhard Woeginger, Austria)

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)