In a country, each two cities are connected with a one-way road.
Prove that there is a city from which you can drive to any other whilst travelling along no more than two roads.
a) What is the minimum number of pieces of wire needed in order to weld a cube’s frame?
b) What is the maximum length of a piece of wire that can be cut from this frame? (The length of the edge of the cube is 1 cm).
Prove that in a bipartite planar graph \(E \geq 2F\), if \(E \geq 2\) (\(E\) is the number of edges, \(F\) is the number of regions).
Find the last digit of the number \(1 \times 2 + 2 \times 3 + \dots + 999 \times 1000\).
Is the number 12345678926 square?
Prove there are no integer solutions for the equation \(4^k - 4^l = 10^n\).
There are 100 notes of two types: \(a\) and \(b\) pounds, and \(a \neq b \pmod {101}\). Prove that you can select several bills so that the amount received (in pounds) is divisible by 101.
If a salary is first increased by 20%, and then reduced by 20%, will the salary paid increase or decrease as a result?
In a room there are some chairs with 4 legs and some stools with 3 legs. When each chair and stool has one person sitting on it, then in the room there are a total of 39 legs. How many chairs and stools are there in the room?
Arrange in a row the numbers from 1 to 100 so that any two neighbouring ones differ by at least 50.