Olympiad problems

2003 Junior Balkan Team Selection Tests - Romania, 2

Consider the prime numbers $n_1< n_2 <...< n_{31}$. Prove that if $30$ divides $n_1^4 + n_2^4+...+n_{31}^4$, then among these numbers one can find three consecutive primes.

1987 Polish MO Finals, 5

Tags: product , number theory , min , prime

Find the smallest $n$ such that $n^2 -n+11$ is the product of four primes (not necessarily distinct).

2016 Croatia Team Selection Test, Problem 4

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

Find all pairs $(p,q)$ of prime numbers such that $$ p(p^2 - p - 1) = q(2q + 3) .$$

2017 JBMO Shortlist, NT5

Tags: number theory , power , positive integer , prime

Find all positive integers $n$ such that there exists a prime number $p$, such that $p^n-(p-1)^n$ is a power of $3$. Note. A power of $3$ is a number of the form $3^a$ where $a$ is a positive integer.

2020 Dutch Mathematical Olympiad, 4

Tags: number theory , perfect square , prime

Determine all pairs of integers $(x, y)$ such that $2xy$ is a perfect square and $x^2 + y^2$ is a prime number.

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).

2014 Hanoi Open Mathematics Competitions, 4

Tags: diophantine equation , diophantine , prime , 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.

2014 Finnish National High School Mathematics, 4

Tags: number theory , point , circles , prime

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 $r$.

1955 Moscow Mathematical Olympiad, 299

Tags: number theory , arithmetic progression , sequence , divisible , prime

Suppose that primes $a_1, a_2, . . . , a_p$ form an increasing arithmetic progression and $a_1 > p$. Prove that if $p$ is a prime, then the difference of the progression is divisible by $p$.

2015 Caucasus Mathematical Olympiad, 1

Tags: number theory , prime , natural number , product , prime number

Find some four different natural numbers with the following property: if you add to the product of any two of them the product of the two remaining numbers. you get a prime number.

2017 Brazil National Olympiad, 6.

Tags: number theory , divisibility , prime , group theory

[b]6.[/b] Let $a$ be a positive integer and $p$ a prime divisor of $a^3-3a+1$, with $p \neq 3$. Prove that $p$ is of the form $9k+1$ or $9k-1$, where $k$ is integer.

2024 Singapore MO Open, Q5

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

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]

2022 Cyprus JBMO TST, 2

Tags: number theory , prime number , prime

Determine all pairs of prime numbers $(p, q)$ which satisfy the equation \[ p^3+q^3+1=p^2q^2 \]

2011 Belarus Team Selection Test, 1

Tags: number theory , odd , divisible , prime

Given natural number $a>1$ and different odd prime numbers $p_1,p_2,...,p_n$ with $a^{p_1}\equiv 1$ (mod $p_2$), $a^{p_2}\equiv 1$ (mod $p_3$), ..., $a^{p_n}\equiv 1$(mod $p_1$). Prove that a) $(a-1)\vdots p_i$ for some $i=1,..,n$ b) Can $(a-1)$ be divisible by $p_i $for exactly one $i$ of $i=1,...,n$? I. Bliznets