VWO 1992

It has come to a policeman's ears that 5 gangsters (all of different height) are meeting, one of them is the clan leader, he's the tallest of the 5. He knows the members will leave the building one by one, with a 10-minute break between them, and too bad for him Belgium has not enough policemen to follow all gangsters, so he's on his own to spot the clanleader, and he can only follow one member.

So he decides to let go the first 2 people, and then follow the first one that is taller than those two. What's the chance he actually catches the clan leader like this?

a conic with apotheme 1 slides (varying height and radius, with $r < \frac12$ ) so that the conic's area is $9$ times that of its inscribed sphere. What's the height of that conic?

Let $A,B,P$ positive reals with $P\le A+B$ . (a) Choose reals $\theta_1,\theta_2$ with $A\cos\theta_1 + B\cos\theta_2=P$ and prove that $$A\sin\theta_1 + B\sin\theta_2 \le \sqrt{(A+B-P)(A+B+P)}$$ (b) Prove equality is attained when $\theta_1=\theta_2=\arccos\left(\dfrac{P}{A+B}\right)$ . (c) Take $A=\dfrac{1}{2}xy, B=\dfrac{1}{2}wz$ and $P=\dfrac14 \left(x^2+y^2-z^2-w^2\right)$ with $0<x\le y\le x+z+w$ , $z,w>0$ and $z^2+w^2<x^2+y^2$ . Show that we can translate (a) and (b) into the following theorem: from all quadrilaterals with (ordered) sidelenghts $(x,y,z,w)$ , the cyclical one has the greatest area.