MathDB

3

Part of 2010 Thailand Mathematical Olympiad

Problems(2)

2555...553 divisible by 2553

Source: 2010 Thailand Mathematical Olympiad day 1 p3

3/12/2021
Show that there are infinitely many positive integers n such that 2555...55⏟n32\underbrace{555...55}_{n}3 is divisible by 25532553.
number theorydivisibledivides
(AB+AC)/BC = DE/ MN + 1 , projections on angle bisectors

Source: Thailand Mathematical Olympiad 2010 day 2 p3

8/17/2020
Let △ABC\vartriangle ABC be a scalene triangle with AB<BC<CAAB < BC < CA. Let DD be the projection of AA onto the angle bisector of ∠ABC\angle ABC, and let EE be the projection of AA onto the angle bisector of ∠ACB\angle ACB. The line DEDE cuts sides ABAB and AC at MM and NN, respectively. Prove that AB+ACBC=DEMN+1\frac{AB+AC}{BC} =\frac{DE}{MN} + 1
geometryangle bisectorprojection