NL-NWO 1965

We consider the sequence $t_1,t_2,t_3,...$ By $P_n$ we mean the product of the first $n$ terms of the sequence. Given that $t_{n+1} = t_n \cdot t_{n+2}$ for each $n$ , and that $P_{40} = P_{80} = 8$ . Calculate $t_1$ and $t_2$ .

Prove that $S_1 = (n + 1)^2 + (n + 2)^2 +...+ (n + 5)^2$ is divisible by $5$ for every $n$ . Prove that for no $n$ : $\sum_{\ell=1}^5 (n+\ell)^2$ is a perfect square. Let $S_2=(n + 6)^2 + (n + 7)^2 + ... + (n + 10)^2$ . Prove that $S_1 \cdot S_2$ is divisible by $150$ .

Given are the points $A$ and $B$ in the plane. If $x$ is a straight line is in that plane, and $x$ does not coincide with the perpendicular bisectror of $AB$ , then denote the number of points $C$ located at $x$ such that $\vartriangle ABC$ is isosceles, as the "weight of the line $x$ ". Prove that the weight of any line $x$ is at most $5$ and determine the set of points $P$ which has a line with weight $1$ , but none with weight $0$ .

We consider a number of points in a plane. Each of these points is connected to at least one of the other points by a line segment, in such a way that a figure arises that does not break up into different parts (that is, from any point along drawn line segments we can reach any other point).. We assign a point the "order" $n$ , when in this point $n$ line segments meet. We characterize the obtained figure by writing down the order of each of its points one after the other. For example, a hexagon is characterized by the combination $\{2,2,2,2,2,2\}$ and a star with six rays by $\{6,1,1,1,1,1,1\}$ . (a) Sketch a figure' belonging to the combination $\{4,3,3,3,3\}$ . (b) Give the combinations of all possible figures, of which the sum of the order numbers is equal to $6$ . (c) Prove that every such combination contains an even number of odd numbers.

The function ƒ. which is defined for all real numbers satisfies: $$f(x+y)+f(x-y)=2f(x)+2f(y)$$ Prove that $f(0) = 0$ , $f(-x) = f(x)$ , $f(2x) = 4 f (x)$ , $$f(x + y + z) = f(x + y) + f(y + z) + f(z + x) -f(x) - f(y) -f(z).$$