NL-NWO 1964

Given a triangle $ABC$ , $\angle C= 60^o$ . Construct a point $P$ on the side $AC$ and a point $Q$ on side $BC$ such that $ABQP$ is a trapezoid whose diagonals make an angle of $60^o$ with each other.

Given is a flat plane $V$ containing a rectangular coordinate system $xOy$ . We consider quartets of numbers $(p,q,r,s)$ ; $p\le 0$ , $q \le 0$ , $r \le 0$ , $s \le 0$ . On every quartet we add a point $S$ from $V$ in a way that is in the accompanying figure is displayed. In this figure $OP = p$ , $PQ = q$ , $QR = r$ , $RS = s$ , $\angle OPQ = \angle PQR = \angle QRS = 135^o$ . (a) What is the set of the points of $V$ , which are added to these quartets ? (b) Which of these points has been added to only one quartet? How many quartets have the other points been added? (c) What is the set of points added to the quartets for which $p + q = 1$ and $r = s = 0$ ? (d) What is the set of points added to the quartets for which $p + 1 =$ and $r + s = 1$ ?

$O$ $P$ $Q$ $R$ $S$ $X$ $Y$

Solve $(n + 1)(n +10) = q^2$ , for certain $q$ and maximum $n$ .

The function $ƒ$ is defined at $[0,1]$ , and $f\{f(x)\} = ƒ(x)$ . $\exists _{c\in [0,1]} \left[f(c) =\frac12 \right]$ Determine $f\left(\frac12 \right).$ $\forall _{t\in [0,1]}\exists _{s\in [0,1]}[f(s) = t]$ . Determine $f$ .

Prove that the function $g$ , with $g(x) = x$ , $0 \le x \le k$ , $g(x) = k$ , $k \le x \le 1$ satisfies the relation $g\{g(x)\} = g(x)$ .

Consider a sequence of non-negative integers g $_1,g_2,g_3,...$ each consisting of three digits (numbers smaller than $100$ are also written with three digits; the number $27$ , for example, is written as $027$ ). Each number consists of the preceding by taking the product of the three digits that make up the preceding. The resulting sequence is of course dependent on the choice of $g_1$ (e.g. $g_1 = 359$ leads to $g_2= 135$ , $g_3= 015$ , $g_4 = 000$ ).Prove that independent of the choice of $g_1$ : (a) $g_{n+1}\le g_n$ (b) $g_{10}= 000$ .