Olympiad problems

1997 All-Russian Olympiad Regional Round, 8.7

Find all pairs of prime numbers $p$ and $q$ such that $p^3-q^5 = (p+q)^2$.

2010 Grand Duchy of Lithuania, 2

Tags: diophantine , number theory , diophantine equation

Find all positive integers $n$ for which there are distinct integer numbers $a_1, a_2, ... , a_n$ such that $$\frac{1}{a_1}+\frac{2}{a_2}+...+\frac{n}{a_n}=\frac{a_1 + a_2 + ... + a_n}{2}$$

1980 All Soviet Union Mathematical Olympiad, 288

Tags: number theory , diophantine , diophantine equation

Are there three integers $x,y,z$, such that $x^2 + y^3 = z^4$?

2021 Olympic Revenge, 5

Tags: arithmetic sequence , diophantine equation , perfect square , number theory

Prove there aren't positive integers $a, b, c, d$ forming an arithmetic progression such that $ ab + 1, ac + 1, ad + 1, bc + 1, bd + 1, cd + 1 $ are all perfect squares.

1974 Czech and Slovak Olympiad III A, 3

Tags: number theory , digit , permutation , equation , diophantine equation , sum of digits

Let $m\ge10$ be any positive integer such that all its decimal digits are distinct. Denote $f(m)$ sum of positive integers created by all non-identical permutations of digits of $m,$ e.g. \[f(302)=320+023+032+230+203=808.\] Determine all positive integers $x$ such that \[f(x)=138\,012.\]

2011 NZMOC Camp Selection Problems, 6

Tags: number theory , diophantine equation , diophantine

Find all pairs of non-negative integers $m$ and $n$ that satisfy $$3 \cdot 2^m + 1 = n^2.$$

PEN H Problems, 9

Tags: polynomial , algebra , vieta , absolute value , diophantine equation

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

2014 Saudi Arabia GMO TST, 1

Tags: diophantine , number theory , diophantine equation

Find all ordered triples $(a,b, c)$ of positive integers which satisfy $5^a + 3^b - 2^c = 32$

2020 OMpD, 1

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

Determine all pairs of positive integers $(x, y)$ such that: $$x^4 - 6x^2 + 1 = 7\cdot 2^y$$

1975 Chisinau City MO, 92

Tags: number theory , diophantine equation , diophantine

Solve in natural numbers the equation $x^2-y^2=105$.

2017 German National Olympiad, 6

Tags: number theory , diophantine equation

Prove that there exist infinitely many positive integers $m$ such that there exist $m$ consecutive perfect squares with sum $m^3$. Specify one solution with $m>1$.

1991 Greece National Olympiad, 4

Tags: number theory , diophantine , diophantine equation

Find all positive intger solutions of $3^x+29=2^y$.

2007 Swedish Mathematical Competition, 1

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

Solve the following system \[ \left\{ \begin{array}{l} xyzu-x^3=9 \\ x+yz=\dfrac{3}{2}u \\ \end{array} \right. \] in positive integers $x$, $y$, $z$ and $u$.

1986 IMO Longlists, 18

Tags: equation , number theory , algebra , diophantine equation

Provided the equation $xyz = p^n(x + y + z)$ where $p \geq 3$ is a prime and $n \in \mathbb{N}$. Prove that the equation has at least $3n + 3$ different solutions $(x,y,z)$ with natural numbers $x,y,z$ and $x < y < z$. Prove the same for $p > 3$ being an odd integer.

2000 Putnam, 2

Tags: polynomial , algebra , quadratic , diophantine equation , induction

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. [Example: $0=0^2+0^2$, $1=0^2+1^2$, $2=1^2+1^2$.]

1990 India National Olympiad, 2

Tags: calculus , integration , quadratic , diophantine equation , number theory

Determine all non-negative integral pairs $ (x, y)$ for which \[ (xy \minus{} 7)^2 \equal{} x^2 \plus{} y^2.\]

2014 Belarusian National Olympiad, 2

Tags: number theory , diophantine , diophantine equation

Pairwise distinct prime numbers $p, q, r$ satisfy the equality $$rp^3 + p^2 + p = 2rq^2 +q^2 + q.$$ Determine all possible values of the product $pqr$.

2016 Costa Rica - Final Round, A2

Tags: diophantine , diophantine equation , number theory

Find all integer solutions of the equation $p (x + y) = xy$, where $p$ is a prime number.

2021 Israel TST, 4

Tags: number theory , diophantine equation

Let $r$ be a positive integer and let $a_r$ be the number of solutions to the equation $3^x-2^y=r$ ,such that $0\leq x,y\leq 5781$ are integers. What is the maximal value of $a_r$?