﻿ Tematy na rok szkolny 2017/18

Subjects 2017/2018:

1. Distances on earth

How many points can one place on earth so that the distance between two of them is at least 10000 km ?

2. All the stones on the white face

Consider a game board with \$n\$ columns and \$m\$ rows. In each case, there is a stone with a white face and a black face. Every time a case is chosen, all the stones around except the one chosen are reversed. With that rule and starting from a situation where all the stones are on the black face, is it possible to reverse all the stones on the white face ?

3. Divisibility and last digits.

You are familiar with the divisibility rule for 2: a natural number is divisible by 2 if and only if its last digit is one of: 0, 2, 4, 6 or 8. This means in particular, that the divisibility of n by k=2, depends on the last digit of n. This is not the case when speaking of divisibility by 4, but it’s easy to convince, that the divisibility of n by k=4 depends on two last digits of n. Find all such numbers k>1, that the divisibility by k depends on the last digit of the divided number. For each l, find all such numbers k>1, that the divisibility by k depends on l last digits of the divided number.