Archive for the ‘Problems from IMO’ Category

Problems from IMO 1960


1-Find all the three-digit numbers for which one obtains, when dividing the number by 11, the sum of the squares of the digits of the initial number.

2-For which real numbers x does the following inequality hold :

3-A right-angled triangle ABC is given for which the hypotenuse BC has length a and is divided into n equal segments, where n is odd. Let α be the angle with which the point A sees the segment containing themiddle of the hypotenuse.Prove that:

where h is the height of the triangle.



Prove the following equality:

Geometric inequality


Denote by a,b, c the lengths of the sides of a triangle. Prove that: