NL-NWO 1999

Let $f: \mathbb{Z} \rightarrow \{-1,1\}$ be a function such that $$f(mn) =f(m)f(n),\ \forall m,n \in \mathbb{Z}.$$ Show that there exists a positive integer $a$ such that $1 \leq a \leq 12$ and $f(a) = f(a + 1) = 1$ .

A $9 \times 9$ square consists of $81$ unit squares. Some of these unit squares are painted black, and the others are painted white, such that each $2 \times 3$ rectangle and each $3 \times 2$ rectangle contain exactly 2 black unit squares and 4 white unit squares. Determine the number of black unit squares.

Let $ABCD$ be a square and let $\ell$ be a line. Let $M$ be the centre of the square. The diagonals of the square have length 2 and the distance from $M$ to $\ell$ exceeds 1. Let $A',B',C',D'$ be the orthogonal projections of $A,B,C,D$ onto $\ell$ . Suppose that one rotates the square, such that $M$ is invariant. The positions of $A,B,C,D,A',B',C',D'$ change. Prove that the value of $AA'^2 + BB'^2 + CC'^2 + DD'^2$ does not change.

Consider a matrix of size $8 \times 8$ , containing positive integers only. One may repeatedly transform the entries of the matrix according to the following rules:

-Multiply all entries in some row by 2. -Subtract 1 from all entries in some column.

Prove that one can transform the given matrix into the zero matrix.

Let $c$ be a nonnegative integer, and define $a_n = n^2 + c$ (for $n \geq 1)$ . Define $d_n$ as the greatest common divisor of $a_n$ and $a_{n + 1}$ . (a) Suppose that $c = 0$ . Show that $d_n = 1,\ \forall n \geq 1$ . (b) Suppose that $c = 1$ . Show that $d_n \in \{1,5\},\ \forall n \geq 1$ . (c) Show that $d_n \leq 4c + 1,\ \forall n \geq 1$ .