MathDB

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Part of 2017 CentroAmerican

Problems(2)

Centroamerican Math Olympiad - 2017 P1

Source: OMCC 2017

6/23/2017
The figure below shows a hexagonal net formed by many congruent equilateral triangles. Taking turns, Gabriel and Arnaldo play a game as follows. On his turn, the player colors in a segment, including the endpoints, following these three rules:
i) The endpoints must coincide with vertices of the marked equilateral triangles.
ii) The segment must be made up of one or more of the sides of the triangles.
iii) The segment cannot contain any point (endpoints included) of a previously colored segment.
Gabriel plays first, and the player that cannot make a legal move loses. Find a winning strategy and describe it.
OMCCCENTROOMCC 2017Game Theory
Centroamerican Math Olympiad 2017 - P4

Source: OMCC 2017

6/23/2017
ABCABC is a right-angled triangle, with ∠ABC=90∘\angle ABC = 90^{\circ}. B′B' is the reflection of BB over ACAC. MM is the midpoint of ACAC. We choose DD on BM→\overrightarrow{BM}, such that BD=ACBD = AC. Prove that B′CB'C is the angle bisector of ∠MB′D\angle MB'D.
NOTE: An important condition not mentioned in the original problem is AB<BCAB<BC. Otherwise, ∠MB′D\angle MB'D is not defined or B′CB'C is the external bisector.
OMCCOMCC 2017CENTROgeometrygeometric transformationreflectionangle bisector