Olympiad problems

2007 Argentina National Olympiad, 1

Find all the prime numbers $p$ and $q$ such that $ p^2+q=37q^2+p $. Clarification: $1$ is not a prime number.

2016 Abels Math Contest (Norwegian MO) Final, 2b

Tags: factorial , diophantine equation , diophantine , number theory

Find all non-negative integers $x, y$ and $z$ such that $x^3 + 2y^3 + 4z^3 = 9!$

2021 SG Originals, Q5

Tags: diophantine equation , number theory , algebra

Find all $a,b \in \mathbb{N}$ such that $$2049^ba^{2048}-2048^ab^{2049}=1.$$ [i]Proposed by fattypiggy123 and 61plus[/i]

2017 Hanoi Open Mathematics Competitions, 6

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

Find all pairs of integers $a, b$ such that the following system of equations has a unique integral solution $(x , y , z )$ : $\begin{cases}x + y = a - 1 \\ x(y + 1) - z^2 = b \end{cases}$

2015 Saudi Arabia Pre-TST, 1.3

Tags: diophantine equation , diophantine , number theory

Find all integer solutions of the equation $x^2y^5 - 2^x5^y = 2015 + 4xy$. (Malik Talbi)

1998 Belarus Team Selection Test, 2

Tags: number theory , perfect square , diophantine , diophantine equation

a) Given that integers $a$ and $b$ satisfy the equality $$a^2 - (b^2 - 4b + 1) a - (b^4 - 2b^3) = 0 \,\,\, (*)$$, prove that $b^2 + a$ is a square of an integer. b) Do there exist an infinitely many of pairs $(a,b)$ satisfying (*)?

2015 FYROM JBMO Team Selection Test, 1

Tags: number theory , diophantine equation , integer

Solve the equation $x^2+y^4+1=6^z$ in the set of integers.

PEN H Problems, 60

Tags: modular arithmetic , diophantine equation

Show that the equation $x^7 + y^7 = {1998}^z$ has no solution in positive integers.

2020 China Team Selection Test, 4

Tags: number theory , diophantine equation

Show that the following equation has finitely many solutions $(t,A,x,y,z)$ in positive integers $$\sqrt{t(1-A^{-2})(1-x^{-2})(1-y^{-2})(1-z^{-2})}=(1+x^{-1})(1+y^{-1})(1+z^{-1})$$

1967 IMO Shortlist, 2

Tags: algebra , polynomial , diophantine equation , parametric equation , root

The equation \[x^5 + 5 \lambda x^4 - x^3 + (\lambda \alpha - 4)x^2 - (8 \lambda + 3)x + \lambda \alpha - 2 = 0\] is given. Determine $\alpha$ so that the given equation has exactly (i) one root or (ii) two roots, respectively, independent from $\lambda.$

1984 IMO Longlists, 12

Tags: number theory , equation , polynomial , diophantine equation

Let $n$ be a positive integer and $a_1, a_2, \dots , a_{2n}$ mutually distinct integers. Find all integers $x$ satisfying \[(x - a_1) \cdot (x - a_2) \cdots (x - a_{2n}) = (-1)^n(n!)^2.\]

2011 Swedish Mathematical Competition, 1

Tags: number theory , diophantine equation , diophantine

Determine all positive integers $k$, $\ell$, $m$ and $n$, such that $$\frac{1}{k!}+\frac{1}{\ell!}+\frac{1}{m!} =\frac{1}{n!} $$

2010 Grand Duchy of Lithuania, 2

Tags: number theory , diophantine , diophantine equation

Find all positive integers $n$ for which there are distinct integer numbers $a_1, a_2, ... , a_n$ such that $$\frac{1}{a_1}+\frac{2}{a_2}+...+\frac{n}{a_n}=\frac{a_1 + a_2 + ... + a_n}{2}$$

VI Soros Olympiad 1999 - 2000 (Russia), 9.2

Tags: number theory , diophantine equation , diophantine

Find the smallest natural number n such that for all integers $m > n$ there are positive integers $x$ and $y$ for which the equality 1$7x + 23y = m$ holds

2015 District Olympiad, 2

Tags: diophantine equation , integer part , number theory

Determine the real numbers $ a,b, $ such that $$ [ax+by]+[bx+ay]=(a+b)\cdot [x+y],\quad\forall x,y\in\mathbb{R} , $$ where $ [t] $ is the greatest integer smaller than $ t. $

PEN H Problems, 78

Tags: diophantine equation

Let $x, y$, and $z$ be integers with $z>1$. Show that \[(x+1)^{2}+(x+2)^{2}+\cdots+(x+99)^{2}\neq y^{z}.\]

2022 Kosovo & Albania Mathematical Olympiad, 4

Tags: diophantine equation , algebra

Let $A$ be the set of natural numbers $n$ such that the distance of the real number $n\sqrt{2022} - \frac13$ from the nearest integer is at most $\frac1{2022}$. Show that the equation $$20x + 21y = 22z$$ has no solutions over the set $A$.

1979 IMO Shortlist, 15

Tags: algebra , system of equations , cauchy-schwarz inequality , diophantine equation

Determine all real numbers a for which there exists positive reals $x_{1}, \ldots, x_{5}$ which satisfy the relations $ \sum_{k=1}^{5} kx_{k}=a,$ $ \sum_{k=1}^{5} k^{3}x_{k}=a^{2},$ $ \sum_{k=1}^{5} k^{5}x_{k}=a^{3}.$

2020-IMOC, N2

Tags: diophantine equation , number theory

Find all positive integers $N$ such that the following holds: There exist pairwise coprime positive integers $a,b,c$ with $$\frac1a+\frac1b+\frac1c=\frac N{a+b+c}.$$

2011 Cuba MO, 2

Tags: diophantine , diophantine equation , number theory

Determine all the integer solutions of the equation $3x^4-2024y+1= 0$.

2022 China Team Selection Test, 4

Tags: diophantine equation , number theory

Find all positive integers $a,b,c$ and prime $p$ satisfying that \[ 2^a p^b=(p+2)^c+1.\]

2015 NZMOC Camp Selection Problems, 4

Tags: number theory , diophantine equation , diophantine

For which positive integers $m$ does the equation: $$(ab)^{2015} = (a^2 + b^2)^m$$ have positive integer solutions?

2005 Switzerland - Final Round, 7

Tags: diophantine , diophantine equation , number theory

Let $n\ge 1$ be a natural number. Determine all positive integer solutions of the equation $$7 \cdot 4^n = a^2 + b^2 + c^2 + d^2.$$

2021 New Zealand MO, 5

Tags: diophantine equation , diophantine , number theory

Find all pairs of integers $x, y$ such that $y^5 + 2xy = x^2 + 2y^4.$ .

1994 Tuymaada Olympiad, 7

Tags: algebra , system of equations , relatively prime , diophantine equation

Prove that there are infinitely many natural numbers $a,b,c,u$ and $v$ with greatest common divisor $1$ satisfying the system of equations: $a+b+c=u+v$ and $a^2+b^2+c^2=u^2+v^2$