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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 \(P(z) = az^2+ bz+c\), […]
Problems Problem 1 Let \( a, b, c \) be positive real numbers such that $$ \frac{a}{1+a} + \frac{b}{1+b} + \frac{c}{1+c} = 1 $$ Prove that \( abc \leq \frac{1}{8} \)
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 […]
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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 \(\ […]