Q) If is a polynomial with integer coefficients and three distinct integers, then show that it is impossible to have Solution: The solution revolves around an important property of integer coefficient Polynomials, For any 2 integers , is always divisible by . So from this we have divides which implies divides , divides , divides . Once we have this, the remaining… Continue reading
Category: Algebra
Week -9 | Inequalities – 3
Q-1) Define with for . Show that for all positive integers Solution: Using the AM-HM Inequality we have The RHS on simplification becomes . This is known as the famous Nesbitt’s Inequality Q-2) Find the maximum value of with . Solution: Set and . We thus have , and we need the maximum value of . Applying the… Continue reading
RMO – 2007 | An Insight into Pythagorean triplets
Q) Let be three natural numbers such that and . Suppose there exists an integer such that form the sides of a right-angled triangle. Prove that there exist integers such that Before we go to the solution of this problem, let us get into some theory of Pythagorean Triplets. We define a triplet of integers as a Pythagorean triplet, if form… Continue reading
Week -8 | Inequalities – 2
1.1 Convex Functions and Jenson’s Inequality A function is called a convex function if the line segment between any two points on the graph of the function lies above or on the graph. The following curve illustrates this oncept of a convex function and concave function: Fig 1.1 For a convex function , we… Continue reading
Week -8 | Inequalities – 1
1.1 Introduction We have already introduced the concept of inequations in our ‘Theory of Equations’ modules. To recollect: and with both this implies with this implies . We shall introduce ourselves to some advanced inequalities that are used in the modern world. 2.1 The AM-GM-HM Inequality Consider the positive real numbers . The following inequality is called… Continue reading