Pythagoras Extended! – RMO 2008 Problem 6

Pythagoras Extended! – RMO 2008 Problem 6

The Problem  Find the number of integer-sided isosceles obtuse-angled triangles with perimeter 2008. Video LectureKey ideas Cosine Rule: If ABC is any triangle, \( \angle BAC  = \theta \) then \( AB^2 + AC^2 – 2\times AB \times AC \times \cos \theta = BC^2 \) ....
A rejoinder to the ‘Discovery’

A rejoinder to the ‘Discovery’

Nehru writes, ‘very little original work on mathematics was done in India after the twelfth century till we reach the modern age. ‘Discovery of India’ was written over five months when Nehru was imprisoned in the Ahmednagar Fort. It was first...
Cyclic Pentagon – RMO 2008 Problem 1

Cyclic Pentagon – RMO 2008 Problem 1

Problem Let ABC be an acute-angled triangle, let D, F be the mid-points of BC, AB respectively. Let the perpendicular from F to AC and the perpendicular at B to BC meet in N. Prove that ND is equal to circum-radius of ABC. VIdeo DiscussionTheorems and tools The...
Set of Nilpotent Matrices – TIFR 2017

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 \( A^n = 0 \) )....