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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...

## 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...

## 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...

## 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: Back to...

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

Find all pairs $$(x,y)$$ with $$x,y$$ real, satisfying the equations $$\sin\bigg(\frac{x+y}{2}\bigg)=0~,~\vert x\vert+\vert y\vert=1$$ Learn more Suppose that $$PQ$$ and $$RS$$ are two chords of a circle intersecting at a point $$O$$. It is given that \(PO=3...

## Injection Principle – Combinatorics

The central goal of Combinatorics is to count things. Usually, there is a set of stuff that you would want to count. It could be number of permutations, number of seating arrangements, number of primes from 1 to 1 million and so on. Counting number of elements in a...