MathDB

3

Part of 2006 Hungary-Israel Binational

Problems(2)

Hungary-Israel Binational 2006_3

Source:

10/27/2008
Let \mathcal{H} \equal{} A_1A_2\ldots A_n be a convex n n-gon. For i \equal{} 1, 2, \ldots, n, let Ai′ A'_{i} be the point symmetric to Ai A_i with respect to the midpoint of A_{i \minus{} 1}A_{i \plus{} 1} (where A_{n \plus{} 1} \equal{} A_1). We say that the vertex Ai A_i is good if Ai′ A'_{i} lies inside H \mathcal{H}. Show that at least n \minus{} 3 vertices of H \mathcal{H} are good.
geometry unsolvedgeometry
Hungary-Israel Binational 2006_6

Source:

10/27/2008
A group of 100 100 students numbered 1 1 through 100 100 are playing the following game. The judge writes the numbers 1 1, 2 2, … \ldots, 100 100 on 100 100 cards, places them on the table in an arbitrary order and turns them over. The students 1 1 to 100 100 enter the room one by one, and each of them flips 50 50 of the cards. If among the cards flipped by student j j there is card j j, he gains one point. The flipped cards are then turned over again. The students cannot communicate during the game nor can they see the cards flipped by other students. The group wins the game if each student gains a point. Is there a strategy giving the group more than 1 1 percent of chance to win?
percentsearchcombinatorics unsolvedcombinatorics