1992 Flanders Math Olympiad

Part of Flanders Math Olympiad

Subcontests

(4)

1

a trig approach... (long)

A,B,P

positive reals with

P\le A+B

. (a) Choose reals

\theta_1,\theta_2

A\cos\theta_1 + B\cos\theta_2=P

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)

A=\dfrac{1}{2}xy, B=\dfrac{1}{2}wz

P=\dfrac14 \left(x^2+y^2-z^2-w^2\right)

0<x\le y\le x+z+w

z,w>0

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.

1

sphere and conic

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?

1

belgian gangsters (probability)

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?

1

More cool number theory.

For every positive integer

n

, determine the biggest positive integer

k

2^k |\ 3^n+1