NL-NWO 1971

Given a trapezoid $ABCD$ , where sides $AB$ and $CD$ are parallel; the points $P$ on $AD$ and $Q$ on $BC$ lie such that the lines $AQ$ and $CP$ are parallel. Prove that lines $PB$ and $DQ$ are parallel.

A sequence of real numbers is called a Fibonacci sequence if $$t_{n+2} = t_{n+1} + t_n$$ for $n= 1,2,3,. .$ . Two Fibonacci sequences are said to be essentially different if the terms of one sequence cannot be obtained by multiplying the terms of the other by a constant. For example, the Fibonacci sequences $1,2,3,5,8,...$ and $1,3,4,7,11,...$ are essentially different, but the sequences $1,2,3,5,8,...$ and $2,4,6,10,16,...$ are not.

(a) Prove that there exist real numbers $p$ and $q$ such that the sequences $1,p,p^2,p^3,...$ and $1,q,q^2,q^3,...$ are essentially different Fibonacci sequences.

(b) Let $a_1,a_2,a_3,...$ and $b_1,b_2,b_3,...$ be essentially different Fibonacci sequences. Prove that for every Fibonacci sequence $t_1,t_2,t_3,...$ , there exists exactly one number $\alpha$ and exactly one number $\beta$ , such that: $$t_n = \alpha a_n + \beta b_n$$ for $n = 1,2,3,...$ (c) $t_1,t_2,t_3,...$ , is the Fibonacci sequence with $t_1 = 1$ and $t_2= 2$ . Express $t_n$ in terms of $n$ .

Prove that $(0,1)$ , $(0, -1)$ , $( -1,1)$ and $(-1,-1)$ are the only integer solutions of $$x^2 + x +1 = y^2.$$

For every positive integer $n$ there exist unambiguously determined non-negative integers $a(n)$ and $b(n)$ such that $$n = 2^{a(n)}(2b(n)+1),$$ For positive integer $k$ we define $S(k)$ by: $$a(1) + a(2) + ... + a(2^k) = S(k)$$ Express $S(k)$ in terms of $k$ .

Someone draws at least three lines on paper. Each cuts the other lines two by two. No three lines pass through one point. He chooses a line and counts the intersection points on either side of the line. The numbers of intersections turn out to be the same. He chooses another line. Now the intersections number on one side appears to be six times as large as that on the other side. What is the minimum number of lines where this is possible?