Geometry is perhaps the most important topic in mathematics as far as Math Olympiad and I.S.I. Entrance goes. The following list of results may work as an elementary set of tools for handling some geometry problems. ‘Learning’ them won’t do any good....

Advanced Mathematics Seminar 2 hours An Open seminar for Pre-RMO and I.S.I. Entrance 2019 aspirants. We will work on topics from Number Theory, Geometry and Algebra. Registration is free. There are only 25 seats available. Date: 29th June, Friday, 6 PM Online...

If a function is 'nice' (!), then for every secant line, there is a parallel tangent line! This is the intuition behind the Mean Value Theorem from Differentiable Calculus. The fifth problem from I.S.I. B.Stat and B.Math Entrance 2018, has a clever application of this...

The Problem Let \(f:(0,\infty)\to\mathbb{R}\) be a continuous function such that for all \(x\in(0,\infty)\), $$f(2x)=f(x)$$Show that the function \(g\) defined by the equation $$g(x)=\int_{x}^{2x} f(t)\frac{dt}{t}~~\text{for}~x>0$$is a constant function. Key Ideas One...

The Problem Let \(f:\mathbb{R}\to\mathbb{R}\) be a continuous function such that for all \(x\in\mathbb{R}\) and for all \(t\geq 0\), $$f(x)=f(e^tx)$$Show that \(f\) is a constant function. Key Ideas Set \( \frac{x_2}{x_1} = t \) for all \( x_1, x_2 > 0 \). Do the same...

The Problem Suppose that \(PQ\) and \(RS\) are two chords of a circle intersecting at a point \(O\). It is given that \(PO=3 \text{cm}\) and \(SO=4 \text{cm}\). Moreover, the area of the triangle \(POR\) is \(7 \text{cm}^2\). Find the area of the triangle \(QOS\). Key...