This post contains RMO 2018 solutions, problems, and discussions. RMO 2018, Problem 1: Let \(ABC\) be a triangle with integer sides in which \(AB < AC\). Let the tangent to the circumcircle of triangle \(ABC\) at \(A\) intersect the line \(BC\) at \(D\). Suppose \(AD\) is also an integer. Prove that gcd\((AB,AC) >1\). RMO 2018, […]
A convex polygon \( \Gamma \) is such that the distance between any two vertices of \( \Gamma \) does not exceed 1. Prove that the distance between any two points on the boundary of \( \Gamma \) does not exceed 1. If X and Y are two distinct points inside \( \Gamma \), prove that […]
The Problem Let ABC be a triangle in which AB = AC and let I be its in-centre. Suppose BC = AB + AI. Find ∠BAC. Big 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 each of […]
Recently, French mathematician Cedric Villani's team came up with '21 measures for the teaching of Mathematics'. I read through the report, with great curiosity. I happily noted that Cheenta's Thousand Flowers program has already implemented some of his recommendations.
Pythagoras theorem can be extended! What happens if the triangle is obtuse-angled (instead of right-angled?) We explore the idea by using a problem from Math Olympiad.
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. Theorems and tools The discussion uses the following Theorems: Midpoint Theorem: The line […]
Problem In a quadrilateral $ABCD$. It is given that $AB=AD=13$, $BC=CD=20$, $BD=24$. If $r$ is the radius of the circle inscribable in the quadrilateral, then what is the integer close to $r$? Hint 1: First, notice that the quadrilateral is a kite. Diagonals of a kite bisect each other (Prove this!) If $X$ is the point […]
A book is published in three volumes, the pages being numbered from 1 onwards. The page numbers are continued from the first volume to the third. The number of pages in the second volume is 50 more than in the first volume, and the number pages in the third volume is one and a half times that in […]
Problem 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)). Hint 1 This problem will use the following elementary algebraic identity: $(x+y)^3=x^3+y^3+3 x^2 y+3 x y^2$ Can you identify what is x and what is y? Hint 2 background_video_pause_outside_viewport="on" tab_text_shadow_style="none" body_text_shadow_style="none"] Set $x=\sqrt{a}, y=\sqrt{b}$. Then the […]