MathDB

2

Part of 2000 IberoAmerican

Problems(2)

15th ibmo - venezuela 2000/q2.

Source: Spanish Communities

4/15/2006
Let S1S_1 and S2S_2 be two circumferences, with centers O1O_1 and O2O_2 respectively, and secants on MM and NN. The line tt is the common tangent to S1S_1 and S2S_2 closer to MM. The points AA and BB are the intersection points of tt with S1S_1 and S2S_2, CC is the point such that BCBC is a diameter of S2S_2, and DD the intersection point of the line O1O2O_1O_2 with the perpendicular line to AMAM through BB. Show that MM, DD and CC are collinear.
projective geometrygeometry unsolvedgeometry
15th ibmo - venezuela 2000/q5.

Source: Spanish Communities

4/15/2006
There are a buch of 2000 stones. Two players play alternatively, following the next rules:
(aa)On each turn, the player can take 1, 2, 3, 4 or 5 stones of the bunch. (bb) On each turn, the player has forbidden to take the exact same amount of stones that the other player took just before of him in the last play.
The loser is the player who can't make a valid play. Determine which player has winning strategy and give such strategy.
inductionnumber theory unsolvednumber theory