Olympiad problems

For $r> 0$ denote by $B_r$ the set of points at distance at most $r$ length units from the origin. If $P_r$ is the set of the points in $B_r$ whit integer coordinates, show that the equation $$xy^3z + 2x^3z^3-3x^5y = 0$$ has an odd number of solutions $(x, y, z)$ in $P_r$.

VMEO III 2006 Shortlist, N5

Tags: number theory , diophantine equation , diophantine

Find all triples of integers $(x, y, z)$ such that $x^4 + 5y^4 = z^4$.

PEN H Problems, 69

Tags: number theory , relatively prime , diophantine equation

Determine all positive rational numbers $r \neq 1$ such that $\sqrt[r-1]{r}$ is rational.

1980 IMO Longlists, 3

Tags: number theory , equation , algebra , diophantine equation

Prove that the equation \[ x^n + 1 = y^{n+1}, \] where $n$ is a positive integer not smaller then 2, has no positive integer solutions in $x$ and $y$ for which $x$ and $n+1$ are relatively prime.

PEN H Problems, 24

Tags: modular arithmetic , geometry , 3d geometry , diophantine equation

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 when $n=2891$.

2013 Iran MO (3rd Round), 4

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

Prime $p=n^2 +1$ is given. Find the sets of solutions to the below equation: \[x^2 - (n^2 +1)y^2 = n^2.\] (25 points)

2018 Pan-African Shortlist, N5

Tags: number theory , diophantine equation

Find all quadruplets $(a, b, c, d)$ of positive integers such that \[ \left( 1 + \frac{1}{a} \right) \left( 1 + \frac{1}{b} \right) \left( 1 + \frac{1}{c} \right) \left( 1 + \frac{1}{d} \right) = 4. \]

2023 Puerto Rico Team Selection Test, 1

Tags: number theory , diophantine equation , diophantine

Determine all triples $(a, b, c)$ of positive integers such that $$a! +b! = 2^{c!} .$$

2015 Saudi Arabia Pre-TST, 2.3

Tags: diophantine , diophantine equation , number theory

Find all integer solutions of the equation $14^x - 3^y = 2015$. (Malik Talbi)

2016 Costa Rica - Final Round, N3

Tags: number theory , diophantine equation , diophantine

Find all nonnegative integers $a$ and $b$ that satisfy the equation $$3 \cdot 2^a + 1 = b^2.$$

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.

2014 China Team Selection Test, 3

Tags: number theory , diophantine equation

Show that there are no 2-tuples $ (x,y)$ of positive integers satisfying the equation $ (x+1) (x+2)\cdots (x+2014)= (y+1) (y+2)\cdots (y+4028).$

2023 Israel National Olympiad, P2

Tags: number theory , diophantine equation

The non-negative integers $x,y$ satisfy $\sqrt{x}+\sqrt{x+60}=\sqrt{y}$. Find the largest possible value for $x$.

2005 China National Olympiad, 6

Tags: number theory , diophantine equation

Find all nonnegative integer solutions $(x,y,z,w)$ of the equation\[2^x\cdot3^y-5^z\cdot7^w=1.\]

1986 ITAMO, 6

Tags: diophantine , diophantine equation , number theory

Show that for any positive integer $n$ there exists an integer $m > 1$ such that $(\sqrt2-1)^n=\sqrt{m}-\sqrt{m-1}$.

1991 Tournament Of Towns, (291) 1

Tags: number theory , diophantine , diophantine equation

Find all natural numbers $n$, and all integers $x,y$ ($x\ne y$) for which the following equation is satisfied: $$x + x^2 + x^4 + ...+ x^{2^n} = y + y^2 + y^4 + ... + y^{2^n} .$$

2025 Bangladesh Mathematical Olympiad, P4

Tags: prime number , modular arithmetic , diophantine equation , number theory , bounding

Find all prime numbers $p, q$ such that$$p(p+1)(p^2+1) = q^2(q^2+q+1) + 2025.$$ [i]Proposed by Md. Fuad Al Alam[/i]

PEN H Problems, 86

Tags: function , euler , least common multiple , diophantine equation , geometry

A triangle with integer sides is called Heronian if its area is an integer. Does there exist a Heronian triangle whose sides are the arithmetic, geometric and harmonic means of two positive integers?

1982 IMO Shortlist, 16

Tags: number theory , equation , diophantine equation , divisibility

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 $n=2891$.