Sine Rule and Incenter – RMO 2009 Geometry

Sine Rule and Incenter – RMO 2009 Geometry

The Problem! Let ABC be a triangle in which AB = AC and let I be its in-centre. Suppose BC = AB + AI. Find ∠BAC. VideoBig Ideas For any triangle ABC, \( \frac{\sin A}{a} = \frac{\sin B } {b} = \frac {\sin C }{c} \). Addendo: If \( \frac{a}{b} = \frac{c}{d} \) then...
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 \( AB^2 + AC^2 – 2\times AB \times AC \times \cos \theta = BC^2 \) ....
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...
Limit is Euler!

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 ____________ Hint 1 - Sum of oddsHint 2 - Break in partialsHint 3 - Something goes to e! Notice...