MathDB

3

Part of 2009 Iran MO (2nd Round)

Problems(2)

Geometric Inequality on Angles - Iran NMO 2009 - Problem 3

Source:

9/20/2010
Let ABC ABC be a triangle and the point D D is on the segment BC BC such that AD AD is the interior bisector of ∠A \angle A . We stretch AD AD such that it meets the circumcircle of ΔABC \Delta ABC at M M . We draw a line from D D such that it meets the lines MB,MC MB,MC at P,Q P,Q , respectively (M M is not between B,P B,P and also is not between C,Q C,Q ). Prove that ∠PAQ≥∠BAC \angle PAQ\geq\angle BAC .
inequalitiesgeometrycircumcircletrigonometryangle bisectorgeometry proposedGeometric Inequalities
Cards - Iran NMO 2009 - Problem 6

Source:

9/20/2010
1111 people are sitting around a circle table, orderly (means that the distance between two adjacent persons is equal to others) and 1111 cards with numbers 11 to 1111 are given to them. Some may have no card and some may have more than 11 card. In each round, one [and only one] can give one of his cards with number i i to his adjacent person if after and before the round, the locations of the cards with numbers i−1,i,i+1 i-1,i,i+1 don’t make an acute-angled triangle. (Card with number 00 means the card with number 1111 and card with number 1212 means the card with number 11!) Suppose that the cards are given to the persons regularly clockwise. (Mean that the number of the cards in the clockwise direction is increasing.) Prove that the cards can’t be gathered at one person.
topologygeometrycircumcirclecalculusintegrationinvariantcombinatorics proposed