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# Problem Garden

Mathematics is not a spectator sport. Neither is Physics, Computer Science or Chemistry. In this portal, we have gathered (and are adding) problems, discussions and challenges that you may try your hands on.

## 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: https://youtu.be/7Zx5n3nuGmo 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 $$\sin\bigg(\frac{x+y}{2}\bigg)=0~,~\vert x\vert+\vert y\vert=1$$ Suppose...

## 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...

## Orthocenter and equal circles

Orthocenter (or the intersection point of altitudes) has an interesting construction. Take three equal circles, and make them pass through one point H. Their other point of intersection creates a triangle ABC. Turns out, H is the orthocenter of ABC. In this process,...

## Geometry of Motion: Open Seminar

Curving the infinity!Imagine squashing the infinite inside small circular disc! Lines bending or sliding to make room for the 'outside territory' inside.  In the upcoming open slate Cheenta Seminar, we tackle this exciting problem from Geometry. Admission is free but...

## Bijections in Combinatorics (TOMATO Obj 168)

Bijection principle is a very useful tool for combinatorics. Here we pick up a problem that appeared in I.S.I.'s B.Stat-B.Math Entrance. Part 1: The problem and the hints https://youtu.be/EoGqTxQy940 Part 2 https://youtu.be/9gPEKehjxr8  Part 3...

## Algebraic Identity (TOMATO Objective 16)

Algebraic Identities can be tricky. Here we handle a simple case of repeated application of (a+b)(a-b).   https://youtu.be/P3EXpj--Rbk

Problems and discussions from various math olympiads including RMO, INMO (India), USAMO, AMC (United States), BMC and more.

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Problems from InPhO, KVPY, NSEP and other contests.

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#### Informative Articles

Selected articles on books, learning methods, and scholarship opportunities.

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#### I.S.I. & C.M.I. Entrance

Problems and discussions from Test of Mathematics at 10+2 Level, previous year I.S.I. & C.M.I. Entrances.

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#### College Mathematics

Problems and discussions from TIFR, M.Math, Subject GRE, IIT JAM and more.

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