Given a triangle ABC, let P and Q be the points on the segments AB and AC, respectively such that AP = AQ. Let S and R be distinct points on segment BC such that S lies between B and R, ∠BPS = ∠PRS, and ∠CQR = ∠QSR. Prove that P, Q, R and S […]
Given a triangle ABC, let P and Q be the points on the segments AB and AC, respectively such that AP = AQ. Let S and R be distinct points on segment BC such that S lies between B and R, ∠BPS = ∠PRS, and ∠CQR = ∠QSR. Prove that P, Q, R and S […]
Problem 1: Define a sequence as: Prove that this sequence has a finite limit as Also determine the limit. Problem 2: Let and be two sequences of numbers, and let be an integer greater than Define Prove that if the quadratic expressions do not have any real roots, then all the remaining polynomials also don’t […]
The best way to learn mathematics is to DO mathematics. In fact we can add something more to that. The best way to get inspired about mathematics is to 'experience' beautiful mathematics. In 2012 we are transforming our learning (and teaching) methods. Till today the basic style of our program comprised of: Inside Classroom a […]
Compute I = $latex (\int_e^{e^4}\sqrt{\log(x)}dx)$ if it is given that $latex (\int _1^2 e^{t^2} dt = \alpha )$ I = $latex ([x \sqrt{\log(x)}]_e^{e^4} - \int_e^{e^4} x \frac{1}{2 \sqrt{log(x)}} \frac {1}{x} dx )$ = $latex ([e^4 \sqrt {\log_e e^4} - e \sqrt {\log _e e}] - \frac{1}{2} \int_e^{e^4}\frac{1}{\sqrt{log(x)}} dx )$ = $latex (2 e^4 - e […]
P148. Show that there is no real constant c > 0 such that $latex (\cos\sqrt{x+c}=\cos\sqrt{x})$ for all real numbers $latex (x\ge 0)$.Solution: If the given equation holds for some constant c>0 then, f(x) = $latex (\cos\sqrt{x}-\cos\sqrt{x+c}=0)$ for all $latex (x\ge 0)$$latex (\Rightarrow 2\sin\frac{\sqrt{x+c}+\sqrt{x}}{2}\sin\frac{\sqrt{x+c}-\sqrt{x}}{2}=0)$Putting x=0, we note$latex (\Rightarrow\sin^2\frac{\sqrt{c}}{2}=0)$As $latex (c\not=0)$$latex (\sqrt{c}=2n\pi)$$latex (\Rightarrow c=4n^2\pi^2)$We put n=1 and […]
P164. Show that the area of the bounded region enclosed between the curves $latex (y^3=x^2)$ and $latex (y=2-x^2)$, is $latex (2\frac{2}{15})$. Solution: Note that $latex (y=x^{\frac{2}{3}})$ is an even function (green line). P165. Find the area of the region in the xy plane, bounded by the graphs of $latex (y=x^2)$, x+y = 2 and $latex […]
That is a good start. And a demanding one. All good starts are demanded by birthright. They ask you to do more in the subsequent days. This article is mainly targeted at class 12 pass-outs who are targeting I.S.I. 2012 (or those 12th graders who are able to devote some serious time to mathematics). Target […]
Indian Statistical Institute (I.S.I.), Chennai Mathematical Institute (C.M.I.) and Institute of Mathematics and Application (I.M.A.) can be regarded as three Indian institutions that provided world class mathematics course at undergraduate level. The B.Stat Course at I.S.I. is also world famous. The courses at C.M.I. and I.M.A. have computer science as second major. Each of these […]
Try this beautiful problem from Singapore Mathematics Olympiad, SMO, 2012 based on Prime numbers. You may use sequential hints to solve the problem.
Try this beautiful problem from Singapore Mathematics Problem, SMO, 2012 based on Series. You may use sequential hints to solve the problem.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2008 based on Logic and speed.
Try this beautiful problem from the American Invitational Mathematics Examination, AIME, 1984 based on Function and symmetry. You may use sequential hints.
Try this beautiful problem from Geometry based on Triangle from PRMO 2018. You may use sequential hints to solve the problem.
Try this beautiful problem from Algebra based on LCM from AMC-8, 2016. You may use sequential hints to solve the problem.
Try this beautiful problem from Algebra aboutRatio of Lcm & gcf from AMC-8, 2013. You may use sequential hints to solve the problem.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2008 based on Equations and Integers.
Try this beautiful problem from Singapore Mathematics Olympiad, SMO, 2009 based on Problem on Series. You may use sequential hints to solve the problem.
Try this beautiful problem from Geometry: The area of the region, AMC-8, 2017. You may use sequential hints to solve the problem.
This is a subjective problem from TOMATO based on Graphing integer value function. Problem: Graphing integer value function Let [x] denote the largest integer (positive, negative or zero) less than or equal to x. Let $y= f(x) = [x] + \sqrt{x - [x]} ,s=2 $ be defined for all real numbers x. (i) Sketch on […]
This is a Test of Mathematics Solution Subjective 116 (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. Problem If A, B, C are the angles of a triangle, then show that $ […]
This is a Test of Mathematics Solution Subjective 115 (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. Problem If $\displaystyle { \frac{\sin^4 x }{a} + \frac{\cos^4 x }{b} = \frac{1}{a+b} }$ , […]
This is a Test of Mathematics Solution Subjective 113 (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. Problem: Find the vertices of the two right angles triangles, each having area 18 and […]
This is a Test of Mathematics Solution Subjective 110 (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 ABCD be a […]
This is a Test of Mathematics Solution Subjective 107 (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: If a, b and c […]
This is a Test of Mathematics Solution Subjective 90 (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: : Draw the region of […]
This is a Test of Mathematics Solution Subjective 88 (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: A pair of complex numbers […]
This is a Test of Mathematics Solution Subjective 84 (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. Problem Show that there is exactly one value of \(x\) that satisfies the equation: \(2 […]
This is a Test of Mathematics Solution Subjective 83 (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. Problem If a and b are positive real numbers such that a + b = […]