1

Part of 2011 Dutch IMO TST

Problems(2)

x^2 + y^2 + 3^3 = 456\sqrt{x - y} in ZxZ

Source: 2011 Dutch IMO TST1 p1

(x, y)

of integers that satisfy

x^2 + y^2 + 3^3 = 456\sqrt{x - y}

Diophantine equationnumber theory

[(n - 1)^k - (-1)^k] /n is number of ways with k buses, between 2 cities

Source: Dutch IMO TST2 2011 p1

n \ge 2

k \ge1

be positive integers. In a country there are

n

cities and between each pair of cities there is a bus connection in both directions. Let

A

B

be two different cities. Prove that the number of ways in which you can travel from

A

B

by using exactly

k

buses is equal to

\frac{(n - 1)^k - (-1)^k}{n}