## Equality in a triangle

Prove that in any triangle it holds that:

## Planes and sphere

Consider two parallel planes that have a distance and intersect a sphere of radius . (both planes intersect the sphere.) Prove that the area of the surface of the sphere enclosed by both planes is only dependant by and not by the position of the two planes with respect to the sphere.

**Solution**

WLOG we assume that the two planes which intersect the sphere of radius are perpedicular to -axis with equations and respectively.

(In the scheme the points and are points of intersection of a meridian cycle with the planes and respectively and the points and are the projections of the points and respectively, on -axis.)

The area in question is the area of the surface by revolution of the meridian’s arc which lies between the planes, rotating it about the -axis. The arc is the graph of the function , and the area in question is

## Factorization

Let be a right triangle at . Factor .

**Solution**

The following equations hold:

(1)

(2)

(3)

We have successively:

## Factorial series

Prove that

**Solution**

We have successively:

## The composition is a metric

Let be a metric , be a strictly increasing function and concave on such that . Prove that is a metric.