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 the Pre-RMO, 2018 based on Centroids and Area. You may use sequential hints to solve the problem.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2007 based on Sequence and Integers.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2001 based on Patterns and Integers.
Try this beautiful problem from Singapore Mathematics Olympiad, SMO, 2009 based on Centroid of Triangle. You may use sequential hints to solve the problem.
Try this beautiful problem from the Pre-RMO, 2017 based on GCD and Primes. You may use sequential hints to solve the problem.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2011 based on Planes and distance.
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1985 based on GCD and Sequence.
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 Singapore Mathematics Olympiad, SMO, 2012 based on Prime numbers. You may use sequential hints to solve the problem.