Problems

Age
Difficulty
Found: 1980

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 \(pqr = 7(p + q + r)\).