Let D, E, F be points on the sides BC, CA, AB respectively of a triangle ABC such that BD = CE = AF and ∠BDF = ∠CED = ∠AFE. Prove that ABC is equilateral.

Call a natural number n faithful, if there exist natural numbers a < b < c such that a divides b, b divides c and n = a + b + c. (i) Show that all but a finite number of natural numbers are faithful. (ii) Find the sum of all natural numbers which are not faithful.

Consider two polynomials and with integer coefficients such that is a prime, and . Suppose there exists a rational number r such that P(r) = Q(r) = 0. Prove that r is an integer.

Suppose five of the nine vertices of a regular nine-sided polygon are arbitrarily chosen. Show that one can select four among these five such that they are the vertices of a trapezium.

Let ABCD be a quadrilateral inscribed in a circle . Let E, F, G, H be the midpoints of the arcs AB, BC, CD, DA of the circle . Suppose AC.BD = EG .FH. Prove that AC, BD, EG, FH are concurrent.

Find all functions such that , for all x, y ∈ R, where R denotes the set of all real numbers.