MathDB

2

Part of 2013 CentroAmerican

Problems(2)

Passing Coins Around a Round Table

Source: CentroAmerican 2013 Problem 2

8/24/2013
Around a round table the people P1,P2,...,P2013P_1, P_2,..., P_{2013} are seated in a clockwise order. Each person starts with a certain amount of coins (possibly none); there are a total of 1000010000 coins. Starting with P1P_1 and proceeding in clockwise order, each person does the following on their turn:
[*]If they have an even number of coins, they give all of their coins to their neighbor to the left.
[*]If they have an odd number of coins, they give their neighbor to the left an odd number of coins (at least 11 and at most all of their coins) and keep the rest.
Prove that, repeating this procedure, there will necessarily be a point where one person has all of the coins.
combinatorics unsolvedcombinatoricsProcesses
ABC is an Acute Triangle

Source: CentroAmerican 2013 Problem 5

8/24/2013
Let ABCABC be an acute triangle and let Γ\Gamma be its circumcircle. The bisector of ∠A\angle{A} intersects BCBC at DD, Γ\Gamma at KK (different from AA), and the line through BB tangent to Γ\Gamma at XX. Show that KK is the midpoint of AXAX if and only if ADDC=2\frac{AD}{DC}=\sqrt{2}.
trigonometrygeometryangle bisectortrig identitiesLaw of Sinesgeometry unsolved