NL-NWO 1973

Given is a triangle $ABC$ , $\angle C = 60^o$ , $R$ the midpoint of side $AB$ . There exist a point $P$ on the line $BC$ and a point $Q$ on the line $AC$ such that the perimeter of the triangle $PQR$ is minimal. a) Prove that and also indicate how the points $P$ and $Q$ can be constructed. b) If $AB = c$ , $AC = b$ , $BC = a$ , then prove that the perimeter of the triangle $PQR$ equals $\frac12\sqrt{3c^2+6ab}$ .

Prove that for every $n \in N$ there exists exactly one sequence of $2n + 1$ consecutive numbers, such that the sum of the squares of the first $n+1$ numbers is equal to the sum of the squares of the last $n$ numbers. Also express the smallest number of that sequence in terms of $n$ .

The angles $A$ and $B$ of base of the isosceles triangle $ABC$ are equal to $40^o$ . Inside $\vartriangle ABC$ , $P$ is such that $\angle PAB = 30^o$ and $\angle PBA = 20^o$ . Calculate, without table, $\angle PCA$ .

We have an infinite sequence of real numbers $x_0,x_1, x_2, ...$ such that $x_{n+1} = \sqrt{x_n -\frac14}$ holds for all natural $n$ and moreover $x_0 \in \frac12$ . (a) Prove that for every natural $n$ holds: $x_n > \frac12$ (b) Prove that $\lim_{n \to \infty} x_n$ exists. Calculate this limit.

An infinite sequence of integers $a_1,a_2,a_3, ...$ is given with $a_1 = 0$ and further holds for every natural number $n$ that $a_{n+1} = a_n - n$ if $a_n \ge n$ and $a_{n+1} = a_n + n$ if $a_n < n$ . (a) Prove that there are infinitely many numbers in the sequence equal to $0$ . (b) Express in terms of $k$ the ordinal number of the $k^e$ number from the sequence, which is equal to $0$ .