1. Let be an acute angled triangle such that . Let be a point on such that . Let be an internal point of the segment . Prove that if and only if . 2. Show that there are infinitely many positive integers such that . 3. Show that is always composite for all natural

1. For which , does hold? 2. For a polynomial with integral coefficients, i.e. for all with , if where , are coprime integers with then show that: i. ii. 3. Let be a triangle with side-lengths , , corresponding to sides , and respectively and let , and be the lengths of the medians from

1. Given a straight line and points and on the same side of it find the shortest path from to which touches . 2. Given a function such that Show that for all . Thus, infer that for such a function knowing its value at one non-zero point is sufficient for finding