Math Abacus · Hard

What is the largest number of $1\times2\times2$ tiles that fit into a $3\times3\times3$ box?

Anya draws a row of little houses, each new one placed against the one before it (the picture shows 3 of them). Altogether she drew 10 houses. How many triangles can be seen in Anya's drawing?

There is a pit 1 km deep, a rope 900 m long and a knife. The rope can be fastened at the top of the pit and at its middle — a point 500 m down. How can you get to the bottom unharmed?

Prove that in your class there are two people with the same number of friends in that class.

A monkey has 2 nuts and a 15-storey building. It may drop a nut from any floor and see whether the nut breaks — in which case that nut is gone — or not. Can it find the lowest floor a nut breaks from in 5 drops?

Note that when a sheet with digits written on it is turned upside down, 0 and 8 stay as they are, 6 becomes 9 and 9 becomes 6, while every other digit loses its meaning. How many five-digit numbers are left unchanged by turning the sheet?

Prove that every natural number can be written as the ratio of a square to a cube of natural numbers:

$$n = \frac{a^2}{b^3}, \qquad a,\, b \in \mathbb{N}.$$

All the natural numbers from 1 to 20 are written on a board. What is the smallest count of them one has to erase so that no number left is twice another?

Put a digit from 0 to 9 in every cell so that within each row all the numbers are different. Down a column, neighbouring numbers must not be equal, and the sum of each column is given underneath.