NL-NWO 1962

Given a triangle $ABC$ with $\angle C = 90^o$ . a) Construct the circle with center $C$ , so that one of the tangents from $A$ to that circle is parallel to one of the tangents from $B$ to that circle. b) A circle with center $C$ has two parallel tangents passing through A and go respectively. If $AC = b$ and $BC = a$ , express the radius of the circle in terms of $a$ and $b$ .

The $n^{th}$ term of a sequence is $t_n$ . For $n \ge 1$ , $t_n$ is given by the relation: $$t_n= n^3+\frac12 n^2+ \frac13 n + \frac14$$ The $n^{th}$ term of a second sequence $T_n$ , where $T_n$ represents the smallest integer greater than $t_n$ . Calculate: $$(T_1+T_2+...+T_{1014}) -(t_1+t_2+...+t_{1014})$$

Consider the positive integers written in the decimal system with $n$ digits, the start of which is not zero and where there are no two sevens next to each other. The number of these numbers is called $u_n$ . Derive a relation that expresses $u_{n+2}$ in terms of $u_{n+1}$ and $u_n$ .

Write using with the floor function: the last, the second last, and the first digit of the number $n$ written in the decimal system.

There are three kinds of things, which are designated respectively by the words (stripped of all common meaning) notes, staves, and heads. There can be a certain relationship between a note and a head, which is expressed by the saying: they match. Also, a note and a head can match and two different staves can match. Given are the following axioms: (a) If a note and a head each match the same stave, then they match, (b) If two different notes both match with stave B, and also both match with head V, then B and V match, (c) If two staves match, then there is a note that matches both, (d) If a note and a stave are given, then there is a head that matches both. Prove the following theorem, denoting the axiom you apply by its letter. If three staves that differ from each other, each one matches every other, and no note matches any of the three staves, then there is a head that matches all three staves.