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, 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.
Try this beautiful problem from Geometry:Area inside the rectangle but outside all three circles.AMC-8, 2014. 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 Squares and Triangles.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2008 based on Percentage. you may use sequential hints.
Try this beautiful problem from the Pre-RMO, 2019 based on Smallest Positive Integer. You may use sequential hints to solve the problem.
Try this beautiful problem from PRMO, 2018 based on Angles in a circle. You may use sequential hints to solve the problem.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2012 based on Circles and triangles.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2012 based on Complex Numbers and Triangles.
Try this beautiful problem from the Pre-RMO, 2019 based on Triangles and Internal bisectors. You may use sequential hints to solve the problem.