Olympiad problems

2011 JBMO Shortlist, 1

Solve in positive integers the equation $1005^x + 2011^y = 1006^z$.

1976 Vietnam National Olympiad, 1

Tags: number theory , integer sequence , integer equation , system of equations

Find all integer solutions to $m^{m+n} = n^{12}, n^{m+n} = m^3$.

1996 Czech and Slovak Match, 4

Tags: integer equation , function equation , algebra , number theory

Decide whether there exists a function $f : Z \rightarrow Z$ such that for each $k =0,1, ...,1996$ and for any integer $m$ the equation $f (x)+kx = m$ has at least one integral solution $x$.

2003 Nordic, 2

Tags: cubic equation , integer equation , number theory

Find all triples of integers ${(x, y, z)}$ satisfying ${x^3 + y^3 + z^3 - 3xyz = 2003}$

2004 Germany Team Selection Test, 4

Tags: equation , integer equation

Let the positive integers $x_1$, $x_2$, $...$, $x_{100}$ satisfy the equation \[\frac{1}{\sqrt{x_1}}+\frac{1}{\sqrt{x_2}}+...+\frac{1}{\sqrt{x_{100}}}=20.\] Show that at least two of these integers are equal to each other.

2004 Germany Team Selection Test, 4

Tags: equation , integer equation

Let the positive integers $x_1$, $x_2$, $...$, $x_{100}$ satisfy the equation \[\frac{1}{\sqrt{x_1}}+\frac{1}{\sqrt{x_2}}+...+\frac{1}{\sqrt{x_{100}}}=20.\] Show that at least two of these integers are equal to each other.

1976 Vietnam National Olympiad, 4

Tags: number theory , integer , integer equation , factorial equation

Find all three digit integers $\overline{abc} = n$, such that $\frac{2n}{3} = a! b! c!$