MathDB

3

Part of 2008 South East Mathematical Olympiad

Problems(2)

four points lie on a circle

Source: China south east mathematical olympiad 2008 day1 problem 3

7/14/2013
In △ABC\triangle ABC, side BC>ABBC>AB. Point DD lies on side ACAC such that ∠ABD=∠CBD\angle ABD=\angle CBD. Points Q,PQ,P lie on line BDBD such that AQ⊥BDAQ\bot BD and CP⊥BDCP\bot BD. M,EM,E are the midpoints of side ACAC and BCBC respectively. Circle OO is the circumcircle of △PQM\triangle PQM intersecting side ACAC at HH. Prove that O,H,E,MO,H,E,M lie on a circle.
geometrycircumcircleperpendicular bisectorgeometry unsolved
pirates and treasure chests

Source: China south east mathematical Olympiad 2008 day2 problem 7

7/15/2013
Captain Jack and his pirate men plundered six chests of treasure (A1,A2,A3,A4,A5,A6)(A_1,A_2,A_3,A_4,A_5,A_6). Every chest AiA_i contains aia_i coins of gold, and all aia_is are pairwise different (i=1,2,⋯ ,6)(i=1,2,\cdots ,6). They place all chests according to a layout (see the attachment) and start to alternately take out one chest a time between the captain and a pirate who serves as the delegate of the captain’s men. A rule must be complied with during the game: only those chests that are not adjacent to other two or more chests are allowed to be taken out. The captain will win the game if the coins of gold he obtains are not less than those of his men in the end. Let the captain be granted to take chest firstly, is there a certain strategy for him to secure his victory?
combinatorics unsolvedcombinatoricsGame TheoryTrees