Author: A.A. Egorov
Calculate the square root of the number \(0.111 \dots 111\) (100 ones) to within a) 100; b) 101; c)* 200 decimal places.
In some country there are 101 cities, and some of them are connected by roads. However, every two cities are connected by exactly one path.
How many roads are there in this country?
There are \(16\) cubes, each face of every cube is coloured yellow, black, or red (different cubes can be coloured differently). After looking at their colouring pattern, Pinoccio said that he could put all the cubes on the table in such a way that only the yellow color would be visible, on the next turn he could put the cubes in such a way that only the black color would be visible, and also he could put them in such a way that only the red color would be visible. Is there a colouring of the cubes such that he could tell the truth?