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Cauchy Schwarz Problem: Let be a polynomial with non-negative coefficients.Prove that if for ,then the same inequality holds for each . Discussion: Cauchy Schwarz's Inequality: Suppose for real numbers (\ a_{i},b_{i}), where (\ i\in{1,2,\dots,n}) we can say that $${\sum_{i=1}^{n}a_{i}^2}{\sum_{i=1}^{n}b_{i}^2}=\sum_{i=1}^{n}{a_{i}b_{i}}^2$$. Titu's Lemma: Let (\ a_{i},b_{i}\in{\mathbb{R}}) and let (\ a_{i},b_{i}>0) for (\ i\in{1,2,\dots,n}) $$\sum_{i=1}^{n}\frac{a_{i}^2}{b_{i}}\ge\frac{{\sum_{i=1}^{n}a_{i}}^2}{\sum_{i=1}^{n}b_{i}}$$ Proof of Cauchy Schwarz's Inequality: We […]

PROBLEM: Given and are two quadratic polynomials with rational coefficients. Suppose and have a common irrational solution. Prove that for all where is a rational number. SOLUTION: Suppose the common irrational root of (\ f(x)) and (\ g(x)) be (\sqrt{a}+b). Then by properties of irrational roots we can say that the other root of both of […]

Problem 1 Find the sum 𝑆=Σ2015𝑘=1(−1)𝑘(𝑘+1)2⋅𝑘 Problem 2 Suppose in $\triangle A B C$, $A B=\sqrt{3}$, $B C=1$, $C A=2$. Suppose there exists a point $P_{0}$ in the plane of $\triangle A B C$ such that $A P_{0}$+$B P_{0}$+$C P_{0} \leq A P+B P+C P$ for all points $P$ in the plane of $\triangle A […]

Problem: Let ( \ k) be a fixed odd positive integer.Find the minimum value of ( \ x^2+y^2),where ( \ x,y) are non-negative integers and ( \ x+y=k). Solution: According to Cauchy Schwarz's inequality, we can write, ( \ (x^2+y^2)\times(1^2+1^2) \ge)(\ (x\times1+y\times1)^2) =>( \ 2(x^2+y^2)\ge)(\ (x+y)^2) =>( \ x^2+y^2\ge) (\frac{k^2}{2}) Therefore,the minimum value of ( \ x^2+y^2) is […]

Hello, this is a discussion page for the college students who are in various prestigious colleges throughout India, and are keen to pursue Mathematics. Abstract Algebra plays a pivotal role in college mathematics, and it mainly focuses on three things GROUPS, RINGS, and FIELDS. Though Field is not in the course of some colleges, eventually […]

Here is a post that contains some questions of algebra and their solution from NBHM 2015 Examination. Go through it and try them. QUESTION [NBHM(January)(2015) 1.1] Solve the following equation, given that it's roots are in arithmetic progression: [x^3-9x^2+28x-30 =0] DISCUSSION: Now every root in the above mentioned equation is in arithmetic progression. So let […]

Let's discuss a Common but deadly question in Group theory. Question: Is it possible to get an infinite group which has elements of finite order? Discussion To pursue this discussion which is basically a very good concept for the students who are new in group theory, they must know first about the QUOTIENT GROUPS. Particularly […]

Problem (Artin, chapter 4, 7.1) Determine the Jordan form of a matrix $$ \left[ \begin{array}{ccc} 1 & 1 & 0 \ 0 & 1 & 0 \ 0 & 1 & 1 \end{array} \right] $$ Discussion According to the Jordan form of a matrix, we first determine the characteristic polynomial of the above matrix. To do […]

1. Prelude Urysohn's Metrization theorem is one of the 'big boys' of second course in Topology. It works as an excellent survey problem that brings scattered ideas theorems and technical terms together. We will examine a standard proof and review the terms and theorems used in the proof. While examining the various bits and pieces, we revisit […]

Presently I am active in learning and teaching mathematics. This series of articles is intended for recording my experiences in this profession. Paper Folding Geometry Today, in Math Olympiad training session, we discussed paper folding geometry. Historically, art and mathematics of paper folding flourished in Japan where it is popularly known as origami. In the last decade […]