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## 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. Hint 1 - Triangular Inequality If a, b, c are sides of a triangle, triangular inequality...

## Real Surds – Problem 2 Pre RMO 2017

ProblemHint 1Hint 2Hint 3Final Answer 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 +...

## 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...