Definition. For and , Then , we define say that is residue of nth-degree modulo m , If there exist integer solutions for – Otherwise, It is non-residue of nth-degree modulo m. For we call it as quadratic, cubic and bi-quadratic residues respectively. Theorem : There are quadratic residues in . Proof : Consider […]
The Area Way Incases , where a relation with areas is possible, It is quite efficient to use it to solve the problem. Example – If are the altitudes of a triangle from A,B,C to the opposite sides and r is the inradius. Prove that – 1. 2. Proof Let I be the incentre. Note […]
With the on-going trend, Algebraic Inequalities are almost erasing the existence of Geometrical Inequalities . Nevertheless, They are equally beautiful (if not more) , although they seem a lot harder to me. Yes, they can mostly be proved by brute-force trigononmetry or complex (whatever it is , they are equally good solutions) , but I […]
Problem Prove that – P.S. It is straightforward holder, but what about a solution by Cauchy or AM-GM ? Solution By, AM-GM we have , Write up analogous inequalities, and add to get – which re-arranges to the desired question 🙂
LockHartsLament The article talks for itself.. Beautifully written.. I wish one day i could put forward my thought in such an organized and calm manner despite the frustrations and anger and emotions..
We define is is the exponent of p in the expansion of a. Thus by definition we directly have the following facts – 1. 2. 3. Equality when 4. 5. We also have the legendre’s formula – 6. This function gets to be seriously useful many more plus it is beautiful !! . Setting such […]
Problem Prove that for positive reals with sum 1 , Proof Homogenize the inequality to – Since the inequality is cyclic, let Substitute – and let , The inequality by the above substitution is transformed to – By AM-GM inequality , therefore the inequality is true with equality which gives us the equality cases –