A grasshopper can make jumps of 8, 9 and 10 cells in any direction on a strip of n cells. We will call the natural number n jumpable if the grasshopper can, starting from some cell, bypass the entire strip, having visited each cell exactly once. Find at least one n $>$ 50 that is not jumpable.
On every cell of a $9 \times 9$ board there is a beetle. At the sound of a whistle, every beetle crawls onto one of the diagonally neighbouring cells. Note that, in some cells, there may be more than one beetle, and some cells will be unoccupied.
Prove that there will be at least 9 unoccupied cells.
Ten pairwise distinct non-zero numbers are such that for each two of them either the sum of these numbers or their product is a rational number.
Prove that the squares of all numbers are rational.