Problems

Age
Difficulty
Found: 1942

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).

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.

Recall that a natural number \(x\) is called prime if \(x\) has no divisors except \(1\) and itself. Solve the equation with prime numbers \(pqr = 7(p + q + r)\).