Try this beautiful problem from TOMATO Objective no. 13 based on Calendar Problem. This problem is useful for BSc Maths and Stats Entrance Exams. Problem: June 10, 1979, was a SUNDAY. Then May 10, 1972, was a (A) Wednesday; (B) Friday; (C) Sunday; (D) Tuesday; Solution: In a (non-leap) year there are 365 days. $365 […]
Try this beautiful problem from TOMATO Subjective Problem no. 173 based on the Sum of Polynomials. Problem : Sum of polynomials Let [latex] {{P_1},{P_2},...{P_n}}[/latex] be polynomials in [latex] {x}[/latex], each having all integer coefficients, such that [latex] {{P_1}={{P_1}^{2}+{P_2}^{2}+...+{P_n}^{2}}}[/latex]. Assume that [latex] {P_1}[/latex] is not the zero polynomial. Show that [latex] {{P_1}=1}[/latex] and [latex] {{P_2}={P_3}=...={P_n}=0}[/latex] Solution : As [latex] […]
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 […]
Try this beautiful problem from Algebra based on real numbers from PRMO 2017. You may use sequential hints to solve the problem.
Try this beautiful problem from AMC 10A, 2004 based on Mensuration: Cylinder. You may use sequential hints to solve the problem.
Try this beautiful problem from AMC-10A, 2004 based on Triangle. You may use sequential hints to solve the problem.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1994 based on Points of Equilateral triangle.
Try this beautiful problem from the Pre-RMO, 2017 based on ratio and proportion. You may use sequential hints to solve the problem.
Try this beautiful problem from Number Theory based on largest possible value from AMC-10A, 2004. You may use sequential hints to solve the problem.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1994 based on Complex roots and equations.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1994 based on Length and Inequalities.
Try this beautiful problem from the Pre-RMO, 2017 based on Trigonometry & natural numbers. You may use sequential hints to solve the problem.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1994 based on Right angled triangle.