Olympiad problems

1976 Miklós Schweitzer, 1

Assume that $ R$, a recursive, binary relation on $ \mathbb{N}$ (the set of natural numbers), orders $ \mathbb{N}$ into type $ \omega$. Show that if $ f(n)$ is the $ n$th element of this order, then $ f$ is not necessarily recursive. [i]L. Posa[/i]

2012 Miklós Schweitzer, 1

Tags: recursive , computable analysis , discrete mathematics , real analysis

Is there any real number $\alpha$ for which there exist two functions $f,g: \mathbb{N} \to \mathbb{N}$ such that $$\alpha=\lim_{n \to \infty} \frac{f(n)}{g(n)},$$ but the function which associates to $n$ the $n$-th decimal digit of $\alpha$ is not recursive?

2010 N.N. Mihăileanu Individual, 4

Tags: pigeonhole principle , discrete mathematics , square grid , locusts , combinatorics , modular arithmetic

A square grid is composed of $ n^2\equiv 1\pmod 4 $ unit cells that contained each a locust that jumped the same amount of cells in the direccion of columns or lines, without leaving the grid. Prove that, as a result of this, at least two locusts landed on the same cell. [i]Marius Cavachi[/i]

2019 VJIMC, 1

Tags: discrete mathematics

a)Is it true that for every non-empty set $A$ and every associative operation $*:A \times A \to A$ the conditions $$x*x*y=y \;\;\; \text{and}\; \;\; y*x*x=y$$ for every $x,y\in A$ imply commutativity of $*$? b)a)Is it true that for every non-empty set $A$ and every associative operation $*:A \times A \to A$ the condition$$x*x*y=y $$ for every $x,y\in A$ implies commutativity of $*$? [i]Proposed by Paulius Drungilas, Arturas Dubickas (Vilnius University). [/i]