This problem is from the Test of Mathematics, TOMATO Subjective Problem no. 172 based on the Round Robin tournament. Problem : Suppose there are [latex] {k}[/latex] teams playing a round robin tournament; that is, each team plays against all the other teams and no game ends in a draw.Suppose the [latex] {i^{th}}[/latex] team loses [latex] […]
This is a Test of Mathematics Solution Subjective 87 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance. Also visit: I.S.I. & C.M.I. Entrance Course of Cheenta Problem: Let \(P(z) = az^2+ bz+c\), […]
This is a beautiful problem based on Some Direct Inequalities from Test of Mathematics Subjective Problem no. 80. Problem: Some Direct Inequalities If \(a,b,c\) are positive numbers, then show that \(\frac{b^2+c^2}{b+c}+\frac{c^2+a^2}{c+a}+\frac{a^2+b^2}{a+b}\geq a+b+c\) Solution: This problem can be solved using a direct application of the Titu's Lemma but we will instead prove the lemma first using […]
This is a beautiful problem based on Solving Equations from Test of Mathematics Subjective Problem no. 20. Problem : Solving equations If \(\ a,b,c,d\) satisfy the equations $$a+7b+3c+5d=0,$$ $$8a+7b+6c+2d=-16,$$ $$2a+6b+4c+8d=16,$$ $$5a+3b+7c+d=-16,$$ then \(\ (a+d)(b+c)\) equals \(\ (A)16 \quad (B)-16\quad (C)0 \quad\) (D)none of the foregoing numbers Solution: $$a+7b+3c+5d=0\dots(1),$$ $$8a+7b+6c+2d=-16\dots(2),$$ $$2a+6b+4c+8d=16\dots(3),$$ $$5a+3b+7c+d=-16\dots(4),$$ \(\ (1)-(3)\), and \(\ […]
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: 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 […]
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 […]
Try this problem from TOMATO Problem 7 based on the Parity of the terms of a sequence. Problem: Parity of the terms of a sequence If \( a_0 = 1 , a_1 = 1 \) and \( a_n = a_{n - 1} a_{n - 2} + 1 \) for \( n > 1 \), then: […]
Try this beautiful Geometry problem from PRMO, 2019, problem-23, based on finding the maximum area. You may use sequential hints to solve the problem.
Try this beautiful problem from Geometry based on Rectangle Pattern from AMC-10A, 2016, Problem 10. You may use sequential hints to solve the problem.
Try this beautiful problem from Geometry: Ratio of area of Circles from AMC-10A, 2009, Problem 21. You may use sequential hints to solve the problem.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1988, Question 11, based on Complex plane.
Try this beautiful problem from algebra, based on Quadratic equation from AMC-10A, 2005. You may use sequential hints to solve the problem.
Try this beautiful problem based on Probability in game from AMC-10A, 2005. You may use sequential hints to solve the problem.
Try this beautiful problem from the Pre-RMO, 2019 based on Covex Cyclic Quadrilateral. You may use sequential hints to solve the problem.
Try this beautiful problem from the Pre-RMO, 2017 based on Sides of Quadrilateral. You may use sequential hints to solve the problem.
Try this beautiful problem from Geometry: Area of the inner square AMC-10A, 2005, Problem-8. You may use sequential hints to solve the problem.
Try this beautiful problem from Geometry: Ratios of the areas of Triangle and Quadrilateral from AMC-10A. You may use sequential hints to solve the problem.