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# Cheenta Reading Room

Outstanding problems, discussion and more

## Integers in a Triangle – AMC 10A

There is an intuitive definition of perpendicularity. It does not involve angle. Instead, it involves the notion of distance. Consider a point P and a line L not passing through it. If you wish to walk from P to L, the path of shortest distance is the perpendicular...

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## Kite in a Circle – Pre RMO 2017, Problem 13

Cyclic Quadrilaterals are often important objects in a Geometry problem. Recognizing them can lead to a path to the solution. A case in point is this problem from Pre RMO 2017. How to recognize cyclic quadrilaterals?  Opposite angles add up to $$\pi$$ Two angles...

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## Tools in Geometry for Pre RMO, RMO and I.S.I. Entrance

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. One should 'find'...

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## PreRMO and I.S.I. Entrance Open Seminar

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 Students...

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## I.S.I. 2018 Problem 5 – a clever use of Mean Value Theorem

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...

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## Leibniz Rule, ISI 2018 Problem 4

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...

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## Functional Equation – ISI 2018 Problem 3

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...

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## Power of a Point – ISI 2018 Problem 2

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...

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## Solutions of equation – I.S.I. 2018 Problem 1

Find all pairs $$(x,y)$$ with $$x,y$$ real, satisfying the equations $$\sin\bigg(\frac{x+y}{2}\bigg)=0~,~\vert x\vert+\vert y\vert=1$$ Discussion: https://youtu.be/7Zx5n3nuGmo Back to...

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## I.S.I. B.Stat, B.Math Entrance 2018 Subjective Paper

Find all pairs $$(x,y)$$ with $$x,y$$ real, satisfying the equations $$\sin\bigg(\frac{x+y}{2}\bigg)=0~,~\vert x\vert+\vert y\vert=1$$ Suppose...

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