## Archive for the ‘Problems from IMO’ Category

### Problems from IMO 1960

15/01/2012

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.

### Equality

13/01/2012

Prove the following equality:





### Geometric inequality

13/01/2012

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