1962 IMO Problems

Day II

Problem 4

Solve the equation .

Solution

Problem 5

On the circle there are given three distinct points . Construct

(using only straightedge and compasses) a fourth point on such that

a circle can be inscribed in the quadrilateral thus obtained.

Solution

Problem 6

Consider an isosceles triangle. Let be the radius of its circumscribed

circle and the radius of its inscribed circle. Prove that the

distance between the centers of these two circles is $d=\sqrt{r(r-

rho)}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 7

The tetrahedron has the following property: there exist five

spheres, each tangent to the edges or to their

extensions.

(a) Prove that the tetrahedron is regular.

(b) Prove conversely that for every regular tetrahedron five such spheres

exist.

Solution