Olympiad problems

2004 Argentina National Olympiad, 2

Determine all positive integers $a,b,c,d$ such that$$\begin{cases} a<b \\ a^2c =b^2d \\ ab+cd =2^{99}+2^{101} \end{cases}$$

PEN H Problems, 68

Tags: ratio , calculus , Diophantine Equations

Consider the system \[x+y=z+u,\] \[2xy=zu.\] Find the greatest value of the real constant $m$ such that $m \le \frac{x}{y}$ for any positive integer solution $(x, y, z, u)$ of the system, with $x \ge y$.

PEN H Problems, 40

Tags: Diophantine Equations , pen

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

2015 Greece National Olympiad, 1

Tags: number theory , prime numbers , Diophantine Equations

Find all triplets $(x,y,p)$ of positive integers such that $p$ be a prime number and $\frac{xy^3}{x+y}=p$

2016 Tuymaada Olympiad, 4

Tags: number theory proposed , Diophantine Equations , number theory , Tuymaada 2016

For each positive integer $k$ find the number of solutions in nonnegative integers $x,y,z$ with $x\le y \le z$ of the equation $$8^k=x^3+y^3+z^3-3xyz$$

PEN H Problems, 52

Tags: Diophantine Equations

Do there exist two right-angled triangles with integer length sides that have the lengths of exactly two sides in common?

PEN H Problems, 87

Tags: Diophantine Equations

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

PEN H Problems, 19

Tags: Diophantine Equations

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

PEN H Problems, 12

Tags: number theory , relatively prime , Diophantine Equations

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

PEN H Problems, 9

Tags: algebra , polynomial , Vieta , LaTeX , absolute value , Diophantine Equations

Determine all integers $a$ for which the equation \[x^{2}+axy+y^{2}=1\] has infinitely many distinct integer solutions $x, \;y$.

1994 Tuymaada Olympiad, 7

Tags: algebra , system of equations , relatively prime , Diophantine Equations

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 $a^2+b^2+c^2=u^2+v^2$

2023 Bangladesh Mathematical Olympiad, P3

Tags: number theory , Diophantine Equations , factorization , Diophantine equation

Solve the equation for the positive integers: $$(x+2y)^2+2x+5y+9=(y+z)^2$$

PEN H Problems, 79

Tags: modular arithmetic , Diophantine Equations

Find all positive integers $m$ and $n$ for which \[1!+2!+3!+\cdots+n!=m^{2}\]

PEN H Problems, 77

Tags: ratio , number theory , relatively prime , Diophantine Equations

Find all pairwise relatively prime positive integers $l, m, n$ such that \[(l+m+n)\left( \frac{1}{l}+\frac{1}{m}+\frac{1}{n}\right)\] is an integer.

PEN H Problems, 8

Tags: calculus , integration , Diophantine Equations

Show that the equation \[x^{3}+y^{3}+z^{3}+t^{3}=1999\] has infinitely many integral solutions.

2005 Swedish Mathematical Competition, 1

Tags: number theory , Diophantine Equations , Sweden

Find all integer solutions $x$,$y$ of the equation $(x+y^2)(x^2+y)=(x+y)^3$.

PEN H Problems, 50

Tags: modular arithmetic , Diophantine Equations

Show that the equation $y^{2}=x^{3}+2a^{3}-3b^2$ has no solution in integers if $ab \neq 0$, $a \not\equiv 1 \; \pmod{3}$, $3$ does not divide $b$, $a$ is odd if $b$ is even, and $p=t^2 +27u^2$ has a solution in integers $t,u$ if $p \vert a$ and $p \equiv 1 \; \pmod{3}$.

PEN H Problems, 89

Tags: Diophantine Equations

Prove that the number $99999+111111\sqrt{3}$ cannot be written in the form $(A+B\sqrt{3})^2$, where $A$ and $B$ are integers.

PEN H Problems, 66

Tags: inequalities , Diophantine Equations

Let $b$ be a positive integer. Determine all $2002$-tuples of non-negative integers $(a_{1}, a_{2}, \cdots, a_{2002})$ satisfying \[\sum^{2002}_{j=1}{a_{j}}^{a_{j}}=2002{b}^{b}.\]

2023 Costa Rica - Final Round, 3.6

Tags: number theory , Digits , Diophantine Equations

Given a positive integer $N$, define $u(N)$ as the number obtained by making the ones digit the left-most digit of $N$, that is, taking the last, right-most digit (the ones digit) and moving it leftwards through the digits of $N$ until it becomes the first (left-most) digit; for example, $u(2023) = 3202$. [b](1)[/b] Find a $6$-digit positive integer $N$ such that \[\frac{u(N)}{N} = \frac{23}{35}.\] [b](2)[/b] Prove that there is no positive integer $N$ with less than $6$ digits such that \[\frac{u(N)}{N} = \frac{23}{35}.\]

PEN H Problems, 70

Tags: Diophantine Equations

Show that the equation $\{x^3\}+\{y^3\}=\{z^3\}$ has infinitely many rational non-integer solutions.

2016 Saudi Arabia IMO TST, 1

Tags: SAU , Diophantine Equations

Let $k$ be a positive integer. Prove that there exist integers $x$ and $y$, neither of which divisible by $7$, such that \begin{align*} x^2 + 6y^2 = 7^k. \end{align*}

PEN H Problems, 53

Tags: modular arithmetic , Diophantine Equations

Suppose that $a, b$, and $p$ are integers such that $b \equiv 1 \; \pmod{4}$, $p \equiv 3 \; \pmod{4}$, $p$ is prime, and if $q$ is any prime divisor of $a$ such that $q \equiv 3 \; \pmod{4}$, then $q^{p}\vert a^{2}$ and $p$ does not divide $q-1$ (if $q=p$, then also $q \vert b$). Show that the equation \[x^{2}+4a^{2}= y^{p}-b^{p}\] has no solutions in integers.

PEN H Problems, 17

Tags: Diophantine Equations , Vieta Jumping

Find all positive integers $n$ for which the equation \[a+b+c+d=n \sqrt{abcd}\] has a solution in positive integers.

2014 Contests, 1

Tags: modular arithmetic , Diophantine equation , Diophantine Equations

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