Olympiad problems

2010 Saudi Arabia Pre-TST, 2.2

Find all $n$ for which there are $n$ consecutive integers whose sum of squares is a prime.

2012 Albania National Olympiad, 1

Tags: modular arithmetic , number theory , prime

Find all primes $p$ such that $p+2$ and $p^2+2p-8$ are also primes.

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 Chile National Olympiad, 3

Tags: number theory , prime , digit

The $19$ numbers $472$ , $473$ , $...$ , $490$ are juxtaposed in some order to form a $57$-digit number. Can any of the numbers thus obtained be prime?

2023 Germany Team Selection Test, 1

Tags: divisibility , number theory , prime

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

2007 Thailand Mathematical Olympiad, 18

Tags: prime , remainder , number theory , prime number

Let $p_k$ be the $k$-th prime number. Find the remainder when $\sum_{k=2}^{2550}p_k^{p_k^4-1}$ is divided by $2550$.

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 $P$.

2014 Hanoi Open Mathematics Competitions, 4

Tags: number theory , diophantine equation , diophantine , prime

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.

2006 Singapore Senior Math Olympiad, 1

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

Let $a, d$ be integers such that $a,a + d, a+ 2d$ are all prime numbers larger than $3$. Prove that $d$ is a multiple of $6$.

2000 Singapore Team Selection Test, 2

Tags: number theory , prime , perfect square

Find all prime numbers $p$ such that $5^p + 12^p$ is a perfect square

2004 Mexico National Olympiad, 1

Tags: perfect square , prime , 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.

2012 India Regional Mathematical Olympiad, 5

Tags: positive integer , number theory , diophantine equation , diophantine , prime

Determine with proof all triples $(a, b, c)$ of positive integers satisfying $\frac{1}{a}+ \frac{2}{b} +\frac{3}{c} = 1$, where $a$ is a prime number and $a \le b \le c$.

2013 Balkan MO Shortlist, N5

Tags: number theory , diophantine equation , diophantine , prime

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

1949-56 Chisinau City MO, 7

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

Prove that if the product $1\cdot 2\cdot ...\cdot n$ ($n> 3$) is not divisible by $n + 1$, then $n + 1$ is prime.

2010 Thailand Mathematical Olympiad, 6

Tags: divide , number theory , prime

Show that no triples of primes $p, q, r$ satisfy $p > r, q > r$, and $pq | r^p + r^q$

VII Soros Olympiad 2000 - 01, 9.3

Tags: number theory , sum , prime

Write $102$ as the sum of the largest number of distinct primes.

2023 Thailand TST, 1

Tags: number theory , divisibility , prime

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

2014 IFYM, Sozopol, 2

Tags: point , circles , prime , number theory

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 $r$.

2013 Taiwan TST Round 1, 1

Tags: number theory , prime , prime number

Starting from 37, adding 5 before each previous term, forms the following sequence: \[37,537,5537,55537,555537,...\] How many prime numbers are there in this sequence?

2014 IMO Shortlist, N5

Tags: prime , number theory

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]

2003 France Team Selection Test, 3

Tags: algebra , number theory , divisor , 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.

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.

2012 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: prime , number theory

Let $a$, $b$, $c$, $d$, $e$, $f$ and $g$ be seven distinct positive integers not bigger than $7$. Find all primes which can be expressed as $abcd+efg$

2022 IMO Shortlist, N6

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

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 $p(42)=3$, $q(42)=2$, $p(63)=3$, $q(63)=3$, $p(2022)=3$, $q(2022)=1$.)

1994 All-Russian Olympiad Regional Round, 10.5

Tags: number theory , prime , difference

Find all primes that can be written both as a sum and as a difference of two primes (note that $ 1$ is not a prime).