INMO 2020 Problem 4

Let be an integer and be real numbers satisfying . If then prove that . Hint 1Hint 2Hint 3 The conditions hint at inequalities involving an order, such as the rearrangement and Chebychev inequalities. Also note that for all is an equality case, hence we should try to...
Beautiful problems from Coordinate Geometry

Beautiful problems from Coordinate Geometry

The following problems are collected from a variety of Math Olympiads and mathematics contests like I.S.I. and C.M.I. Entrances. They can be solved using elementary coordinate geometry and a bit of ingenuity. The equation \( x^2 y – 3xy + 2y = 3 \)...
An interesting biquadratic from ISI Entrance 2005

An interesting biquadratic from ISI Entrance 2005

What are we learning ? Competency in Focus: Biquadratic Equations Main idea: It is usually hard to solve biquadratic equations. However, sometimes, combining tools from geometry and strategies like completing the square it is possible to do so.   Look at the knowledge...
cos(sin(x)) function in ISI Entrance

cos(sin(x)) function in ISI Entrance

Get motivated… try this quiz Understand the problem In the range 0f ( 0 leq x leq 2 pi ) , the equation ( cos (sin (x) )  = frac{1}{2} )  has how many solutions?  Tutorial Problems… try these before watching the video. 1. Show geometrically that ( cos ( ...
Geometry of AM GM Inequality

Geometry of AM GM Inequality

Get motivated… try this quiz Understand the problem Arithmetic mean and Geometric mean are useful algebraic tools. They have beautiful geometric interpretation as well. The following video takes a tour into the ideas. Tutorial Problems… try these before...