Problems

Several teams played a volleyball tournament amongst themselves. We will say that team $A$ is better than team $B$, if either $A$ has either beaten team $B$, or there exists such a team $C$ that was beaten by $A$, whilst $C$ beat team $B$.

a) Prove that there is a team that is better than all.

b) Prove that the team that won the tournament is the best.

Some two teams scored the same number of points in a volleyball tournament. Prove that there are teams $A$, $B$ and $C$, in which $A$ beat $B$, $B$ beat $C$ and $C$ beat $A$.

In the country called Orientation a one-way traffic system was introduced on all the roads, and each city can be reached from any other one by driving on no more than two roads. One road was closed for repairs but from every city it remained possible to get to any other. Prove that for every two cities this can still be done whilst driving on no more than 3 roads.

Prove that $\frac{1}{2}(x^2+y^2)\ge xy$ for any $x$ and $y$.

Prove that for $a,b,c>0$, the following inequality is valid: ${\left(\frac{a+b+c}{3}\right)}^2\ge \frac{ab+bc+ca}{3}$.

Prove that for $x\ge 0$ the inequality is valid: $2x+\frac{3}{8}\ge \sqrt[4]{x}$.

Will the entire population of the Earth, all buildings and structures on it, fit into a cube with a side length of 3 kilometres?

Prove that any axis of symmetry of a 45-gon passes through its vertex.

Is the number $1+2+3+\cdots +1990$ odd or even?

Every Martian has three hands. Can seven Martians join hands?