Olympiad problems

2014 Junior Balkan Team Selection Tests - Romania, 2

Determine the prime numbers $p$ and $q$ that satisfy the equality: $p^3 + 107 = 2q (17q + 24)$ .

1968 All Soviet Union Mathematical Olympiad, 107

Tags: number theory , diophantine

Prove that the equation $x^2 + x + 1 = py$ has solution $(x,y)$ for the infinite number of simple $p$.

2018 Ecuador NMO (OMEC), 1

Tags: number theory , diophantine equation , diophantine

Let $a, b$ be integers. Show that the equation $a^2 + b^2 = 26a$ has at least $12$ solutions.

2019 Romania Team Selection Test, 4

Tags: number theory , diophantine

Four positive integers $x,y,z$ and $t$ satisfy the relations \[ xy - zt = x + y = z + t. \] Is it possible that both $xy$ and $zt$ are perfect squares?

2010 Abels Math Contest (Norwegian MO) Final, 4a

Tags: diophantine equation , diophantine , number theory

Find all positive integers $k$ and $\ell$ such that $k^2 -\ell^2 = 1005$.

2019 Brazil Team Selection Test, 5

Tags: number theory , diophantine

Four positive integers $x,y,z$ and $t$ satisfy the relations \[ xy - zt = x + y = z + t. \] Is it possible that both $xy$ and $zt$ are perfect squares?

2010 NZMOC Camp Selection Problems, 3

Tags: diophantine , diophantine equation , number theory

Let $p$ be a prime number. Find all pairs $(x, y)$ of positive integers such that $x^3 + y^3 - 3xy = p -1$.

2011 Junior Balkan Team Selection Tests - Moldova, 6

Tags: number theory , diophantine , diophantine equation

Find the sum of the numbers written with two digits $\overline{ab}$ for which the equation $3^{x + y} =3^x + 3^y + \overline{ab}$ has at least one solution $(x, y)$ in natural numbers.

2017 QEDMO 15th, 6

Tags: number theory , diophantine , diophantine equation

Find all integers $x,y$ satisfy the $x^3 + y^3 = 3xy$.

2012 Mathcenter Contest + Longlist, 1

Tags: number theory , diophantine equation , diophantine

Prove without using modulo that there are no integers $a,b,c$ such that $$a^2+b^2-8c = 6$$ [i](Metamorphosis)[/i]

1942 Eotvos Mathematical Competition, 2

Tags: number theory , system of equations , system , diophantine

Let $a, b, c $and $d$ be integers such that for all integers m and n, there exist integers $x$ and $y$ such that $ax + by = m$, and $cx + dy = n$. Prove that $ad - bc = \pm 1$.

2015 Dutch IMO TST, 2

Tags: diophantine equation , system of equations , diophantine , number theory

Determine all positive integers $n$ for which there exist positive integers $a_1,a_2, ..., a_n$ with $a_1 + 2a_2 + 3a_3 +... + na_n = 6n$ and $\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+ ... +\frac{n}{a_n}= 2 + \frac1n$

2006 QEDMO 2nd, 1

Tags: number theory , diophantine equation , diophantine

Solve the equation $x^{2}+y^{2}=10xy$ for integers $x$ and $y$

2001 Croatia Team Selection Test, 3

Tags: diophantine equation , diophantine , number theory

Find all solutions of the equation $(a^a)^5 = b^b$ in positive integers.

1991 Greece National Olympiad, 4

Tags: number theory , diophantine equation , diophantine

Find all positive intger solutions of $3^x+29=2^y$.

2015 Dutch Mathematical Olympiad, 4

Tags: number theory , prime , diophantine equation , diophantine

Find all pairs of prime numbers $(p, q)$ for which $7pq^2 + p = q^3 + 43p^3 + 1$

2017 Saudi Arabia JBMO TST, 3

Tags: number theory , diophantine equation , diophantine

Find all pairs of primes $(p, q)$ such that $p^3 - q^5 = (p + q)^2$ .

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

1984 Tournament Of Towns, (069) T3

Tags: algebra , number theory , diophantine equation , diophantine , equation

Find all solutions of $2^n + 7 = x^2$ in which n and x are both integers . Prove that there are no other solutions.

2013 Switzerland - Final Round, 9

Tags: number theory , diophantine , diophantine equation

Find all quadruples $(p, q, m, n)$ of natural numbers such that $p$ and $q$ are prime and the the following equation is fulfilled: $$p^m - q^3 = n^3$$

2020 LIMIT Category 2, 15

Tags: number theory , limit , diophantine , diophantine equation

How many integer pairs $(x,y)$ satisfies $x^2+y^2=9999(x-y)$?

2013 Balkan MO Shortlist, N3

Tags: diophantine equation , diophantine , number theory

Determine all quadruplets ($x, y, z, t$) of positive integers, such that $12^x + 13^y - 14^z = 2013^t$.

2022 Federal Competition For Advanced Students, P1, 4

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

Find all triples $(p, q, r)$ of prime numbers for which $4q - 1$ is a prime number and $$\frac{p + q}{p + r} = r - p$$ holds. [i](Walther Janous)[/i]

1997 Singapore MO Open, 3

Tags: factor , number theory , diophantine equation , diophantine , divisor

Find all the natural numbers $N$ which satisfy the following properties: (i) $N$ has exactly $6$ distinct factors $1, d_1, d_2, d_3, d_4, N$ and (ii) $1 + N = 5(d_1 + d_2+d_3 + d_4)$. Justify your answers.

2013 Swedish Mathematical Competition, 3

Tags: diophantine equation , diophantine , number theory

Determine all primes $p$ and all non-negative integers $m$ and $n$, such that $$1 + p^n = m^3. $$