Olympiad problems

2002 Abels Math Contest (Norwegian MO), 1b

Find all integers $c$ such that the equation $(2a+b) (2b+a) =5^c$ has integer solutions.

2016 Lusophon Mathematical Olympiad, 6

Tags: diophantine equation , power of 2 , number theory

Source: Lusophon MO 2016 Prove that any positive power of $2$ can be written as: $$5xy-x^2-2y^2$$ where $x$ and $y$ are odd numbers.

2013 Czech And Slovak Olympiad IIIA, 1

Tags: diophantine equation , diophantine , number theory

Find all pairs of integers $a, b$ for which equality holds $\frac{a^2+1}{2b^2-3}=\frac{a-1}{2b-1}$

2020 Thailand TSTST, 3

Tags: diophantine equation , number theory

Find all pairs of positive integers $(m, n)$ satisfying the equation $$m!+n!=m^n+1.$$

PEN H Problems, 91

Tags: diophantine equation , rectangle , perimeter , geometry

If $R$ and $S$ are two rectangles with integer sides such that the perimeter of $R$ equals the area of $S$ and the perimeter of $S$ equals the area of $R$, then we call $R$ and $S$ a friendly pair of rectangles. Find all friendly pairs of rectangles.

1982 IMO, 1

Tags: diophantine equation , divisibility , equation , number theory

Prove that if $n$ is a positive integer such that the equation \[ x^3-3xy^2+y^3=n \] has a solution in integers $x,y$, then it has at least three such solutions. Show that the equation has no solutions in integers for $n=2891$.

2022 Korea Winter Program Practice Test, 1

Tags: diophantine equation , number theory

Prove that equation $y^2=x^3+7$ doesn't have any solution on integers.

1997 Brazil Team Selection Test, Problem 2

Tags: diophantine equation , number theory

We say that a subset $A$ of $\mathbb N$ is good if for some positive integer $n$, the equation $x-y=n$ admits infinitely many solutions with $x,y\in A$. If $A_1,A_2,\ldots,A_{100}$ are sets whose union is $\mathbb N$, prove that at least one of the $A_i$s is good.

1985 Brazil National Olympiad, 4

Tags: diophantine equation , trinomial , integer , number theory

$a, b, c, d$ are integers. Show that $x^2 + ax + b = y^2 + cy + d$ has infinitely many integer solutions iff $a^2 - 4b = c^2 - 4d$.

2013 Balkan MO Shortlist, N5

Tags: diophantine equation , prime , diophantine , number theory

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

1988 All Soviet Union Mathematical Olympiad, 471

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

Find all positive integers $n$ satisfying $\left(1 +\frac{1}{n}\right)^{n+1} = \left(1 + \frac{1}{1998}\right)^{1998}$.

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

Tags: diophantine equation , diophantine , factorial , number theory

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

2019 Thailand TSTST, 2

Tags: diophantine equation , number theory

Find all nonnegative integers $x, y, z$ satisfying the equation $$2^x+31^y=z^2.$$

2014 NIMO Summer Contest, 15

Tags: diophantine equation

Let $A = (0,0)$, $B=(-1,-1)$, $C=(x,y)$, and $D=(x+1,y)$, where $x > y$ are positive integers. Suppose points $A$, $B$, $C$, $D$ lie on a circle with radius $r$. Denote by $r_1$ and $r_2$ the smallest and second smallest possible values of $r$. Compute $r_1^2 + r_2^2$. [i]Proposed by Lewis Chen[/i]

2016 Junior Balkan Team Selection Tests - Moldova, 6

Tags: diophantine equation , diophantine , number theory

Determine all pairs $(x, y)$ of natural numbers satisfying the equation $5^x=y^4+4y+1$.

2017 Hanoi Open Mathematics Competitions, 2

Tags: diophantine equation , number theory , diophantine

How many pairs of positive integers $(x, y)$ are there, those satisfy the identity $2^x - y^2 = 1$? (A): $1$ (B): $2$ (C): $3$ (D): $4$ (E): None of the above.

PEN H Problems, 40

Tags: diophantine equation

Determine all pairs of rational numbers $(x, y)$ such that \[x^{3}+y^{3}= x^{2}+y^{2}.\]

2020 Czech-Austrian-Polish-Slovak Match, 5

Tags: diophantine equation , number theory

Let $n$ be a positive integer and let $d(n)$ denote the number of ordered pairs of positive integers $(x,y)$ such that $(x+1)^2-xy(2x-xy+2y)+(y+1)^2=n$. Find the smallest positive integer $n$ satisfying $d(n) = 61$. (Patrik Bak, Slovakia)

2016 Ecuador NMO (OMEC), 1

Tags: diophantine equation , diophantine

Prove that there are no positive integers $x, y$ such that: $(x + 1)^2 + (x + 2)^2 +...+ (x + 9)^2 = y^2$

1999 Singapore Team Selection Test, 1

Tags: diophantine equation , diophantine , number theory

Find all integers $m$ for which the equation $$x^3 - mx^2 + mx - (m^2 + 1) = 0$$ has an integer solution.

PEN H Problems, 90

Tags: diophantine equation

Find all triples of positive integers $(x, y, z)$ such that \[(x+y)(1+xy)= 2^{z}.\]

PEN H Problems, 12

Tags: diophantine equation , number theory , relatively prime

Find all $(x,y,z) \in {\mathbb{N}}^3$ such that $x^{4}-y^{4}=z^{2}$.

1967 Poland - Second Round, 4

Tags: diophantine , diophantine equation , number theory

Solve the equation in natural numbers $$ xy+yz+zx = xyz + 2. $$

1917 Eotvos Mathematical Competition, 1

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

If $a$ and $b$ are integers and if the solutions of the system of equations $$y - 2x - a = 0$$ $$y^2 - xy + x^2 - b = 0$$ are rational, prove that the solutions are integers.

1989 IMO Shortlist, 5

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

Find the roots $ r_i \in \mathbb{R}$ of the polynomial \[ p(x) \equal{} x^n \plus{} n \cdot x^{n\minus{}1} \plus{} a_2 \cdot x^{n\minus{}2} \plus{} \ldots \plus{} a_n\] satisfying \[ \sum^{16}_{k\equal{}1} r^{16}_k \equal{} n.\]