4

Part of 1973 Bundeswettbewerb Mathematik

Problems(2)

|Same gender on the right| = |Different gender on the right|

Source: Bundeswettbewerb Mathematik 1973, round 1, problem 4

n

persons sit around a round table. The number of persons having the same gender than the person at the right of them is the same as the number of those it isn't true for. Show that

4|n

Multiples of 2^n with n digits, all of them being 1 or 2

Source: Bundeswettbewerb Mathematik 1973, round 2, problem 4

Prove: for every positive integer there exists a positive integer having n digits, all of them being

1

2

's only, such that this number is divisible by

2^{n}

. Is this still true in base

4

6