Olympiad problems

2019 Spain Mathematical Olympiad, 4

Find all pairs of integers $(x,y)$ that satisfy the equation $3^4 2^3(x^2+y^2)=x^3y^3$

2016 Belarus Team Selection Test, 3

Tags: diophantine equation , number theory

Solve the equation $2^a-5^b=3$ in positive integers $a,b$.

2013 Balkan MO Shortlist, N6

Tags: diophantine equation , diophantine , number theory , prime

Prove that there do not exist distinct prime numbers $p$ and $q$ and a positive integer $n$ satisfying the equation $p^{q-1}- q^{p-1}=4n^3$

1988 IMO Longlists, 14

Tags: divisibility , diophantine equation , perfect square , vieta jumping , number theory

Let $ a$ and $ b$ be two positive integers such that $ a \cdot b \plus{} 1$ divides $ a^{2} \plus{} b^{2}$. Show that $ \frac {a^{2} \plus{} b^{2}}{a \cdot b \plus{} 1}$ is a perfect square.

1979 Brazil National Olympiad, 4

Tags: number theory , diophantine equation

Show that the number of positive integer solutions to $x_1 + 2^3x_2 + 3^3x_3 + \ldots + 10^3x_{10} = 3025$ (*) equals the number of non-negative integer solutions to the equation $y_1 + 2^3y_2 + 3^3y_3 + \ldots + 10^3y_{10} = 0$. Hence show that (*) has a unique solution in positive integers and find it.

2002 Vietnam Team Selection Test, 2

Tags: diophantine equation , parameterization , derivative , calculus , polynomial , algebra

Find all polynomials $P(x)$ with integer coefficients such that the polynomial \[ Q(x)=(x^2+6x+10) \cdot P^2(x)-1 \] is the square of a polynomial with integer coefficients.

2011 India IMO Training Camp, 1

Tags: 3d geometry , diophantine equation , geometry , number theory

Find all positive integer $n$ satisfying the conditions $a)n^2=(a+1)^3-a^3$ $b)2n+119$ is a perfect square.

2014 Indonesia MO Shortlist, N2

Tags: composite number , composite , diophantine equation , number theory

Suppose that $a, b, c, k$ are natural numbers with $a, b, c \ge 3$ which fulfill the equation $abc = k^2 + 1$. Show that at least one between $a - 1, b - 1, c -1$ is composite number.

PEN H Problems, 22

Tags: symmetry , diophantine equation

Find all integers $a,b,c,x,y,z$ such that \[a+b+c=xyz, \; x+y+z=abc, \; a \ge b \ge c \ge 1, \; x \ge y \ge z \ge 1.\]

2022 Kosovo & Albania Mathematical Olympiad, 1

Tags: diophantine equation , number theory

Find all pairs of integers $(m, n)$ such that $$m+n = 3(mn+10).$$

PEN H Problems, 4

Tags: diophantine equation , ellipse , parameterization , greatest common divisor , conic

Find all pairs $(x, y)$ of positive rational numbers such that $x^{2}+3y^{2}=1$.

2013 Polish MO Finals, 1

Tags: diophantine equation , number theory

Find all solutions of the following equation in integers $x,y: x^4+ y= x^3+ y^2$

2019 Azerbaijan Senior NMO, 3

Tags: diophantine equation , number theory

Find all $x;y\in\mathbb{Z}$ satisfying the following condition: $$x^3=y^4+9x^2$$

2010 IFYM, Sozopol, 5

Tags: number theory , diophantine equation , greatest common divisor

Let n is a natural number,for which $\sqrt{1+12n^2}$ is a whole number.Prove that $2+2\sqrt{1+12n^2}$ is perfect square.

1962 Swedish Mathematical Competition, 3

Tags: diophantine equation , diophantine , number theory

Find all pairs $(m, n)$ of integers such that $n^2 - 3mn + m - n = 0$.

1996 Singapore MO Open, 3

Tags: diophantine equation , diophantine , number theory

Let $n$ be a positive integer. Prove that there is no positive integer solution to thxe equation $(x + 2)^n - x^n = 1 + 7^n$.

1996 Israel National Olympiad, 7

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

Find all positive integers $a,b,c$ such that $$\begin{cases} a^2 = 4(b+c) \\ a^3 -2b^3 -4c^3 =\frac12 abc \end {cases}$$

2020 Nigerian MO round 3, #4

Tags: diophantine equation , number theory

let $p$and $q=p+2$ be twin primes. consider the diophantine equation $(+)$ given by $n!+pq^2=(mp)^2$ $m\geq1$, $n\geq1$ i. if $m=p$,find the value of $p$. ii. how many solution quadruple $(p,q,m,n)$ does $(+)$ have ?

PEN H Problems, 29

Tags: quadratic , diophantine equation , quadratic formula , algebra

Find all pairs of integers $(x, y)$ satisfying the equality \[y(x^{2}+36)+x(y^{2}-36)+y^{2}(y-12)=0.\]

1980 IMO, 19

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

Find all pairs of solutions $(x,y)$: \[ x^3 + x^2y + xy^2 + y^3 = 8(x^2 + xy + y^2 + 1). \]

PEN H Problems, 42

Tags: diophantine equation

Find all integers $a$ for which $x^3 -x+a$ has three integer roots.

2016 Belarus Team Selection Test, 3

Tags: diophantine equation , number theory

Solve the equation $p^3-q^3=pq^3-1$ in primes $p,q$.

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

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

Find all positive integers $a, b, c, d$ with $a \le b$ and $c \le d$ such that $\begin{cases} a + b = cd \\ c + d = ab \end{cases}$ .

1979 Czech And Slovak Olympiad IIIA, 1

Tags: diophantine equation , diophantine , number theory

Let $n$ be a given natural number. Determine the number of all orderer triples $(x, y, z)$ of non-negative integers $x, y, z$ that satisfy the equation $$x + 2y + 5z=10n.$$

2023 Durer Math Competition Finals, 2

Tags: diophantine equation , number theory

[b]a)[/b] Find all solutions of the equation $p^2+q^2+r^2=pqr$, where $p,q,r$ are positive primes.\\ [b]b)[/b] Show that for every positive integer $N$, there exist three integers $a,b,c\geq N$ with $a^2+b^2+c^2=abc$.