2

Part of 1972 Bundeswettbewerb Mathematik

Problems(2)

Rolling beer mats: number of rotations

Source: Bundeswettbewerb Mathematik 1972, round 1, problem 2

In a plane, there are

n \geq 3

circular beer mats

B_{1}, B_{2}, ..., B_{n}

B_{k}

B_{k+1}

k=1,2,...,n

B_{n+1}=B_{1}

. The beer mats are placed such that another beer mat

B

of equal size touches all of them in the given order if rolling along the outside of the chain of beer mats. How many rotations

B

makes untill it returns to it's starting position¿

geometrygeometric transformationrotation

79 integers => 13 divides one of their digit's sum's

Source: Bundeswettbewerb Mathematik 1972, round 2, problem 2

79

consecutive positive integers, one can find at least one whose sum of digits is divisible by

13

. Show that this isn't true for

78

consecutive integers.

number theory proposednumber theory