Problem: The number of terms in the expression of $latex [(a+3b)^2 (a-3b)^2]^2 $ A) 4; B) 5; C) 6; D) 7; Solution: $latex [(a+3b)^2 (a-3b)^2]^2 $ $latex = [\{(a+3b)(a-3b)\}^2]^2 $ $latex = \{ (a^2 -9b^2)^2\}^2 = (a^2 - 9b^2)^4 $ By Binomial Theorem, the given expression contains 5 terms (since $latex (x +y)^n $ has […]
Try this beautiful problem from TOMATO Objective no. 27 based on Closure of a set of even numbers. Problem: Closure of a set of even numbers S is the set whose elements are zero and all even integers, positive and negative. Consider the 5 operations- [1] addition; [2] subtraction; [3] multiplication; [4] division; and […]
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 […]
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1990 based on combinatorics in Tournament.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1990 based on Interior Angle.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1990 based on Smallest positive Integer.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1990 based on Proper divisors.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1990 based on Algebraic value.
Try this beautiful problem from Probability based on dice from AMC-10A, 2011. You may use sequential hints to solve the problem
Try this beautiful problem from Geometry: Area of Region in a Circle from AMC-10A, 2011, Problem -18. You may use sequential hints to solve the problem.
Try this beautiful problem from Algebra based smallest positive value from PRMO 2019. You may use sequential hints to solve the problem.
Try this beautiful problem from combinatorics based on Regular Polygon from PRMO 2019. You may use sequential hints to solve the problem.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1990 based on Positive solution.