Olympiad problems

2009 China Girls Math Olympiad, 1

Show that there are only finitely many triples $ (x,y,z)$ of positive integers satisfying the equation $ abc\equal{}2009(a\plus{}b\plus{}c).$

1989 IMO Shortlist, 15

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

Let $ a, b, c, d,m, n \in \mathbb{Z}^\plus{}$ such that \[ a^2\plus{}b^2\plus{}c^2\plus{}d^2 \equal{} 1989,\] \[ a\plus{}b\plus{}c\plus{}d \equal{} m^2,\] and the largest of $ a, b, c, d$ is $ n^2.$ Determine, with proof, the values of $m$ and $ n.$

2011 Mongolia Team Selection Test, 1

Tags: quadratic , modular arithmetic , ring theory , diophantine equation , number theory

Let $A=\{a^2+13b^2 \mid a,b \in\mathbb{Z}, b\neq0\}$. Prove that there a) exist b) exist infinitely many $x,y$ integer pairs such that $x^{13}+y^{13} \in A$ and $x+y \notin A$. (proposed by B. Bayarjargal)

2022 Czech-Polish-Slovak Junior Match, 2

Tags: diophantine equation , number theory , diophantine

Solve the following system of equations in integer numbers: $$\begin{cases} x^2 = yz + 1 \\ y^2 = zx + 1 \\ z^2 = xy + 1 \end{cases}$$

2002 Vietnam Team Selection Test, 2

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

2017 Saudi Arabia BMO TST, 2

Tags: number theory , diophantine , diophantine equation

Solve the following equation in positive integers $x, y$: $x^{2017} - 1 = (x - 1)(y^{2015}- 1)$

2021 Nigerian MO Round 3, Problem 3

Tags: diophantine equation , number theory , prime

Find all pairs of natural numbers $(p, n)$ with $p$ prime such that $p^6+p^5+n^3+n=n^5+n^2$.

PEN H Problems, 23

Tags: diophantine equation , inequalities

Find all $(x,y,z) \in {\mathbb{Z}}^3$ such that $x^{3}+y^{3}+z^{3}=x+y+z=3$.

2010 Germany Team Selection Test, 3

Tags: induction , diophantine equation , number theory

Determine all $(m,n) \in \mathbb{Z}^+ \times \mathbb{Z}^+$ which satisfy $3^m-7^n=2.$

2018 NZMOC Camp Selection Problems, 9

Tags: diophantine , diophantine equation , number theory , power

Let $x, y, p, n, k$ be positive integers such that $$x^n + y^n = p^k.$$ Prove that if $n > 1$ is odd, and $p$ is an odd prime, then $n$ is a power of $p$.

2014 Purple Comet Problems, 11

Tags: geometry , number theory , diophantine equation

Shenelle has some square tiles. Some of the tiles have side length $5\text{ cm}$ while the others have side length $3\text{ cm}$. The total area that can be covered by the tiles is exactly $2014\text{ cm}^2$. Find the least number of tiles that Shenelle can have.

2017 Kyrgyzstan Regional Olympiad, 2

Tags: diophantine equation , number theory

$x^2 + 2y^2 = 1$ solve in integers.

PEN H Problems, 52

Tags: diophantine equation

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

2014 Junior Balkan Team Selection Tests - Romania, 2

Tags: diophantine , diophantine equation , number theory

Solve, in the positive integers, the equation $5^m + n^2 = 3^p$ .

2015 Thailand TSTST, 2

Tags: number theory , diophantine equation

Find all integer solutions to the equation $y^2=2x^4+17$.

2005 Switzerland - Final Round, 7

Tags: diophantine , diophantine equation , number theory

Let $n\ge 1$ be a natural number. Determine all positive integer solutions of the equation $$7 \cdot 4^n = a^2 + b^2 + c^2 + d^2.$$

2023 Abelkonkurransen Finale, 3a

Tags: number theory , diophantine equation

Find all non-negative integers $n$, $a$, and $b$ satisfying \[2^a + 5^b + 1 = n!.\]

2022 Kosovo & Albania Mathematical Olympiad, 1

Tags: number theory , diophantine equation

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

2005 China National Olympiad, 6

Tags: number theory , diophantine equation

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

2005 Thailand Mathematical Olympiad, 13

Tags: diophantine , diophantine equation , number theory , odd

Find all odd integers $k$ for which there exists a positive integer $m$ satisfying the equation $k + (k + 5) + (k + 10) + ... + (k + 5(m - 1)) = 1372$.

2023 pOMA, 3

Tags: diophantine equation , number theory

Find all positive integers $l$ for which the equation \[ a^3+b^3+ab=(lab+1)(a+b) \] has a solution over positive integers $a,b$.

1982 IMO Longlists, 14

Tags: algebra , diophantine equation , polynomial , parametric equation , geometric progression

Determine all real values of the parameter $a$ for which the equation \[16x^4 -ax^3 + (2a + 17)x^2 -ax + 16 = 0\] has exactly four distinct real roots that form a geometric progression.

2010 China Northern MO, 3

Tags: number theory , diophantine equation , diophantine

Find all positive integer triples $(x, y, z)$ such that $1 + 2^x \cdot 3^y=5^z$ is true.

2013 German National Olympiad, 1

Tags: number theory , diophantine equation

Find all positive integers $n$ such that $n^{2}+2^{n}$ is square of an integer.

2003 Turkey MO (2nd round), 1

Tags: greatest common divisor , modular arithmetic , diophantine equation , number theory

Suppose that $2^{2n+1}+ 2^{n}+1=x^{k}$, where $k\geq2$ and $n$ are positive integers. Find all possible values of $n$.