4

Part of 2012 May Olympiad

Problems(2)

6 collinear point, triangle ineq. May Olympiad (Olimpiada de Mayo) 2012 L2 P4

Source:

Six points are given so that there are not three on the same line and that the lengths of the segments determined by these points are all different. We consider all the triangles that they have their vertices at these points. Show that there is a segment that is both the shortest side of one of those triangles and the longest side of another.

combinatorial geometrycombinatoricstriangle inequality

111 blue + 88 white chips, game May Olympiad (Olimpiada de Mayo) 2012 L2 P4

Source:

111

88

white chips. There is a machine that for every

14

blue chips , it gives

11

white pieces and for every

7

white chips, it gives

13

blue pieces. Decide if Pedro can achieve, through successive operations with the machine, increase the total number of chips by

33

, so that the number of blue chips equals

\frac53

of the amount of white chips. If possible, indicate how to do it. If not, indicate why.

combinatoricsgame