MathDB

1

Part of 2010 JBMO Shortlist

Problems(2)

Strategical Game with 2010 coins in a pile

Source: JBMO Shortlist 2010 Problem C1 (Combinatorics #1)

1/25/2015
<spanclass=′latex−bold′>ProblemC.1</span><span class='latex-bold'>Problem C.1</span> There are two piles of coins, each containing 20102010 pieces. Two players AA and BB play a game taking turns (AA plays first). At each turn, the player on play has to take one or more coins from one pile or exactly one coin from each pile. Whoever takes the last coin is the winner. Which player will win if they both play in the best possible way?
combinatorics unsolvedcombinatorics
In a right ∆ABC prove that A,Q,R are collinear &amp; AP=AC

Source: JBMO Shortlist 2010 Problem G1 (Geometry #1)

1/25/2015
<spanclass=′latex−bold′>ProblemG1</span><span class='latex-bold'>Problem G1</span> Consider a triangle ABCABC with ∠ACB=90∘\angle ACB=90^{\circ}. Let FF be the foot of the altitude from CC. Circle ω\omega touches the line segment FBFB at point PP, the altitude CFCF at point QQ and the circumcircle of ABCABC at point RR. Prove that points A,Q,RA, Q, R are collinear and AP=ACAP = AC.
geometrycircumcirclereflection