NL-NWO 1995

A kangaroo jumps from lattice poin to lattice point in the coordinate plane. It can make only two kinds of jumps: $(A)$ $1$ to right and $3$ up, and $(B)$ $2$ to the left and $4$ down. $(a)$ The start position of the kangaroo is $(0,0)$ . Show that it can jump to the point $(19,95)$ and determine the number of jumps needed. $(b)$ Show that if the start position is $(1,0)$ , then it cannot reach $(19,95)$ . $(c)$ If the start position is $(0,0)$ , find all points $(m,n)$ with $m,n \ge 0$ which the kangaroo can reach.

For any point $P$ on a segment $AB$ , isosceles and right-angled triangles $AQP$ and $PRB$ are constructed on the same side of $AB$ , with $AP$ and $PB$ as the bases. Determine the locus of the midpoint $M$ of $QR$ when $P$ describes the segment $AB$ .

Let $101$ marbles be numbered from $1$ to $101$ . The marbles are divided over two baskets $A$ and $B$ . The marble numbered $40$ is in basket $A$ . When this marble is removed from basket $A$ and put in $B$ , the averages of the numbers $A$ and $B$ both increase by $\frac{1}{4}$ . How many marbles were there originally in basket $A?$

A number of spheres with radius $1$ are being placed in the form of a square pyramid. First, there is a layer in the form of a square with $n^2$ spheres. On top of that layer comes the next layer with $(n-1)^2$ spheres, and so on. The top layer consists of only one sphere. Compute the height of the pyramid.

An array $(a_1,a_2,...,a_{13})$ of $13$ integers is called $tame$ if for each $1 \le i \le 13$ the following condition holds: If $a_i$ is left out, the remaining twelve integers can be divided into two groups with the same sum of elements. A tame array is called $turbo$ $tame$ if the remaining twelve numbers can always be divided in two groups of six numbers having the same sum. $(a)$ Give an example of a tame array of $13$ integers (not all equal). $(b)$ Prove that in a tame array all numbers are of the same parity. $(c)$ Prove that in a turbo tame array all numbers are equal.