A. 413. Let O be the point of intersection of the diagonals AC, BD of a convex quadrilateral ABCD. Let G_{1} and G_{2} be the centroids of triangles OAB and OCD, respectively. Let H_{1} and H_{2} be the orthocenters of triangles OBC and ODA, respectively. Prove that G_{1}G_{2} is perpendicular to H_{1}H_{2}.
(Vietnamese competitoon problem)
(5 points)
B. 3955. In a theatre performance of Hamlet, Acts 2 to 5 are played in a part of the auditorium chosen at random by means of a draw. The stalls are divided into four sectors. The part of the audience with seats in the sector selected will move to another sector, taking also their chairs with them. Provided that each of the four sectors are large enough (that is, if one chooses a sector, then there will always be enough space for one's chair there) and assuming that each sector is equally likely to be chosen, what is the proportion between the probabilities that one needs to move twice during the play or only once?
(4 points)
K. 103. Andrew, Bill, Charlie and Dennis are playing cards. In each game, only three of the four boys are playing. They get points from one another, depending on the outcome of the game: either one player gets points from the other two or two players get points from the third one. (If there are two losers or two winners then they do not necessarily lose or gain the same number of points.) At the beginning, everyone had 100 points. The boys have played four games, each time leaving out a different person. The points that each participant had during the card play are tabulated on the diagram. Which player was left out from the respective games?
Andrew 
Bill 
Charlie 
Dennis 
100 
100 
100 
100 
110 
110 
90 
80 
125 
105 
80 
110 
85 
85 
100 
130 

(6 points)
This problem is for grade 9 students only.
K. 105. A book contains 700 psalms, numbered from 1 to 700. The congregation sings one of them every Sunday. The number of the chosen psalm is displayed as a 3digit number on a special counter, fabricated from three small wooden cubes. (One or twodigit numbers are displayed with leading zeros, e.g. 3 appears as 003 and 28 as 028.) The is one digit on faces of the cubes, respectively, and 6 can also be interpreted as a 9. a) How many cubes are needed to display every possible number? How the digits are arranged on the faces? b) How many more psalms can be included in the book without adding further cubes?
(6 points)
This problem is for grade 9 students only.
K. 107. If two digits of the fourdigit number 1234 are cancelled in every possible way and the remaining digits are read as a twodigit number, the numbers 12, 13, 14, 23, 24, 34 are obtained. Their sum is 120. Find a fourdigit number for which this sum is a) 540; b) 220.
(6 points)
This problem is for grade 9 students only.
K. 108. A square patch of grass, 100 m on a side is surrounded by a paved road along its boundary. Most people want to get from corner A of the square to the midpoint F of an opposite side. They walk along the path ABF, but some of them have started a path through the grassy area. They walk along the side AB for a while, then they are heading towards F along a straight line through the grass, as shown in the diagram. Thus they walk 25% less compared to the path along the border. How far is the turning point K of the path from the point A?
(6 points)
This problem is for grade 9 students only.