Training Olympiad

Let A_1, A_2, ..., A_n lies on circle (O) such that A_1A_2=A_2A_3=...=A_nA_1. Prove that:

\displaystyle \frac{1}{A_1A_2^2}+\frac{1}{A_1A_3^2}+...+\frac{1}{A_1A_n^2}=\frac{n^2-1}{12R^2}, \forall{n\ge 3}.

Author: VanKhea

Generalization: RagvaloD

ទ្រឹស្ដីបទមេនេឡុស និងអនុវត្ត


Let P, Q, R be collinears points on the sides BC, CA, AB.

Prove that if  \displaystyle \frac{AR}{AB}-\frac{BP}{BC}+\frac{CQ}{CA}=1 then \displaystyle \frac{AB}{AR}-\frac{BC}{BP}+\frac{CA}{CQ}=-2

Author: Van Khea

Polygone identity


Let A_1, A_2, ..., A_k, k\ge 3 lies on circle  (O) such that A_1A_2=A_2A_3=...=A_kA_1 and let P\in (O) such that arc PA_1=m.arc PA_2. Prove that:

\displaystyle \frac{1}{PA_1^2}+\frac{1}{PA_2^2}+...+\frac{1}{PA_k^2}=\frac{k^2}{4R^2sin^2(m\pi)}

Author: Van Khea

Generalization: Tintarn

Geometry for Test


Given a triangle \Delta ABC such that AC=\sqrt{2}R and \angle B=45^0.

Prove that \displaystyle \frac{1}{BA}+\frac{1}{BC}=\frac{\sqrt{2+\sqrt{2}}}{R}

Author: Van Khea