Olympiad problems

1984 IMO Shortlist, 11

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

2006 QEDMO 2nd, 1

Tags: number theory , diophantine equation , diophantine

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

1984 IMO Shortlist, 2

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

Prove: (a) There are infinitely many triples of positive integers $m, n, p$ such that $4mn - m- n = p^2 - 1.$ (b) There are no positive integers $m, n, p$ such that $4mn - m- n = p^2.$

1995 Canada National Olympiad, 4

Tags: diophantine equation , number theory

Let $n$ be a constant positive integer. Show that for only non-negative integers $k$, the Diophantine equation $\sum_{i=1 }^{n}{ x_i ^3}=y^{3k+2}$ has infinitely many solutions in the positive integers $x_i, y$.

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.

PEN H Problems, 87

Tags: diophantine equation

What is the smallest perfect square that ends in $9009$?

2015 Indonesia MO Shortlist, N8

Tags: diophantine equation , number theory , infinitely many solutions

The natural number $n$ is said to be good if there are natural numbers $a$ and $b$ that satisfy $a + b = n$ and $ab | n^2 + n + 1$. (a) Show that there are infinitely many good numbers. (b) Show that if $n$ is a good number, then $7 \nmid n$.

PEN H Problems, 45

Tags: diophantine equation

Show that there cannot be four squares in arithmetical progression.

2020 Malaysia IMONST 1, 16

Tags: number theory , diophantine equation

Find the number of positive integer solutions $(a,b,c,d)$ to the equation \[(a^2+b^2)(c^2-d^2)=2020.\] Note: The solutions $(10,1,6,4)$ and $(1,10,6,4)$ are considered different.

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 $7pq^2 + p = q^3 + 43p^3 + 1$

PEN P Problems, 16

Tags: search , diophantine equation , additive number theory

Prove that there exist infinitely many integers $n$ such that $n, n+1, n+2$ are each the sum of the squares of two integers.

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

2017 USA TSTST, 4

Tags: factorial , number theory , diophantine equation

Find all nonnegative integer solutions to $2^a + 3^b + 5^c = n!$. [i]Proposed by Mark Sellke[/i]

1989 IMO Shortlist, 4

Tags: algebra , equation , polynomial , diophantine equation , rational roots

Prove that $ \forall n > 1, n \in \mathbb{N}$ the equation \[ \sum^n_{k\equal{}1} \frac{x^k}{k!} \plus{} 1 \equal{} 0\] has no rational roots.

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.

2014 Contests, 1

Tags: diophantine equation , modular arithmetic

Find all triples of primes $(p,q,r)$ satisfying $3p^{4}-5q^{4}-4r^{2}=26$.

1979 IMO, 2

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

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}.$

2024 Girls in Mathematics Tournament, 2

Tags: number theory , diophantine equation

Show that there are no triples of positive integers $(x,y,z)$ satisfying the equation \[x^2= 5^y+3^z\]

1997 Tuymaada Olympiad, 2

Tags: diophantine equation , algebra , system of equations , natural number

Solve in natural numbers the system of equations $3x^2+6y^2+5z^2=1997$ and $3x+6y+5z=161$ .

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: $$p^m - q^3 = n^3$$

2023 Olympic Revenge, 2

Tags: number theory , diophantine equation

Find all triples ($a$,$b$,$n$) of positive integers such that $$a^3=b^2+2^n$$

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 $12^x + 13^y - 14^z = 2013^t$.