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# Problem Garden

Mathematics is not a spectator sport. In this portal, we have gathered (and are adding) problems, discussions and challenges that you may try your hands on.

## Set of Nilpotent Matrices – TIFR 2017

State True or False: The set of nilpotent matrices of $$M_3 (\mathbb{R} )$$ spans $$M_3 (\mathbb{R} )$$ considered as an $$\mathbb {R}$$ - vector space ( a matrix A is said to be nilpotent if there exists $$n \in \mathbb{N}$$  such that $$A^n = 0$$ )....

## Sequence of isosceles triangles – I.S.I. Entrance 2018 Problem 6

Let, $$a \geq b \geq c > 0$$ be real numbers such that for all natural number n, there exist triangles of side lengths $$a^n,b^n,c^n$$  Prove that the triangles are isosceles. If a, b, c are sides of a triangle, triangular inequality assures that difference of two...

## Bases, Exponents and Role reversals (I.S.I. Entrance 2018 Problem 7 Discussion)

Let $$a, b, c$$ are natural numbers such that $$a^{2}+b^{2}=c^{2}$$ and $$c-b=1$$. Prove that (i) a is odd. (ii) b is divisible by 4 (iii) $$a^{b}+b^{a}$$ is divisible by c Notice that $$a^2 = c^2 - b^2 = (c+b)(c-b)$$ But c - b = 1. Hence $$a^2 = c + b$$. But c...

## Limit is Euler!

Let $$\{a_n\}_{n\ge 1}$$ be a sequence of real numbers such that $$a_n = \frac{1 + 2 + ... + (2n-1)}{n!} , n \ge 1$$ . Then $$\sum_{n \ge 1 } a_n$$ converges to ____________ Notice that $$1 + 3 + 5 + ... + (2n-1) = n^2$$. A quick way to remember this is sum of...

## Real Surds – Problem 2 Pre RMO 2017

Suppose $$a,b$$ are positive real numbers such that $$a\sqrt{a}+b\sqrt{b}=183$$. $$a\sqrt{b}+b\sqrt{a}=182$$. Find $$\frac{9}{5}(a+b)$$. This problem will use the following elementary algebraic identity: $$(x+y)^3 = x^3 + y^3 + 3x^2y + 3xy^2$$ Can you...

## 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...

## 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'...

## 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...

## 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...

## 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...

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

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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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