Let be a prime number. Prove that there exists an integer with such that neither nor is divisible by .

a) Prove that there doesn't exist sequence such that:

b) Let be an odd prime number. Prove that there exist sequence such that:

Let Prove that

In an acute angled triangle with , let denote the incenter and the midpoint of side . The line through perpendicular to intersects the tangent from to the incircle (different from line ) at a point > Show that is tangent to the circumcircle of triangle .

Proposed by Tejaswi Navilarekallu

Find all with and homeomorphismes of , is an homoemorphisme too.

Find all pairs of prime numbers , such that and are both integers.

Let a,b,c>=0: ab+bc+ca=3 then maximize P=\frac{a^2b+b^2c+c^2a+9}{a+b+c}+\frac{abc}{2}.

Let be real numbers such that Prove that Let be real numbers such that Prove that

Given a triangle ABC. A1, B1 and C1 are the points of contact of the inner circumcircle of the triangle with the sides BC, AC and AB respectively. The point of contact of AA1 with B1C1 and the circumcircle are called L and Q respectively. M is the midpoint of B1C1. The point of intersection of lines BC and B1C1 is called T. P is the foot of the perpendicular drawn to AT from point L. Show that points A1, M, Q and P lie on a circle.

Solve the attached