1999 French Mathematical Olympiad

Part of French Mathematical Olympiad

Subcontests

(5)

1

collinearity iff relation between triangle centers holds

Prove that the points symmetric to the vertices of a triangle with respect to the opposite side are collinear if and only if the distance from the orthocenter to the circumcenter is twice the circumradius.

1

eating candies in a specific way

On a table are given

1999

red candies and

6661

yellow candies. The candies are indistinguishable due to the same packing. A gourmet applies the following procedure as long as it is possible:
(i) He picks any of the remaining candies, notes its color, eats it and goes to (ii). (ii) He picks any of the remaining candies, and notes its color: if it is the same as the color of the last eaten candy, eats it and goes to (ii); otherwise returns it upon repacking and goes to (i).
Prove that all the candies will be eaten and find the probability that the last eaten candy will be red.

1

maximal ratio of small side to inradius

For which acute-angled triangles is the ratio of the smallest side to the inradius the maximum?

1

diophantine, exponential sum

Find all natural numbers

n

(n+3)^n=\sum_{k=3}^{n+2}k^n.

1

maximal volume of cylinder in a cone

What is the maximum possible volume of a cylinder inscribed in a cone and having the same axis of symmetry as the cone? What is the maximum possible volume of a ball inscribed in the cone with center on the axis of symmetry of the cone? Compare these three volumes.