Let ”ABC” be a triangle, ”I” its in-centre; be the reﬂections of I in BC, CA, AB respectively. Suppose the circum-circle of triangle passes through A. Prove that are concyclic, where is the in-centre of triangle .

Find all triples such that , where is a prime and are natural numbers.

Let ”A” be a set of real numbers such that ”A” has at least four elements. Suppose ”A” has the property that is a rational number for all distinct numbers ”a, b, c” in ”A”. Prove that there exists a positive integer ”M” such that is a rational number for every ”a” in ”A”.

All the points with integer coordinates in the xy-plane are coloured using three colours, red, blue and green, each colour being used at least once. It is known that the point (0, 0) is coloured red and the point (0, 1) is coloured blue. Prove that there exist three points with integer coordinates of distinct colours which form the vertices of a right-angled triangle.

Let ”ABC” be a triangle; be three equal, disjoint circles inside ”ABC” such that touches ”AB” and ”AC”; touches ”AB” and ”BC”; and “” touches ”BC” and ”CA”. Let be a circle touching circles externally. Prove that the line joining the circumcenter ”O” and the in-centre ”I” of triangle ABC passes through the centre of .

Let be a given polynomial with integer coefficients. Prove that there exist two polynomials and , again with integer coefficients, such that