MathDB

2000 Tournament Of Towns

Part of Tournament Of Towns

Subcontests

(7)
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TOT 2000 Spring AJ6 AS6 no of upsets < 3/4 of no games in chess tournament

In a chess tournament , every two participants play each other exactly once. A win is worth one point , a draw is worth half a point and a loss is worth zero points. Looking back at the end of the tournament, a game is called an upset if the total number of points obtained by the winner of that game is less than the total number of points obtained by the loser of that game. (a) Prove that the number of upsets is always strictly less than three-quarters of the total number of games in the tournament. (b) Prove that three-quarters cannot be replaced by a smaller number.
(S Tokarev)
PS. part (a) for Juniors, both parts for Seniors

TOT 2000 Autumn AJ6 min no of participants in spring round tot

In the spring round of the Tournament of Towns this year, 66 problems were posed in the Senior A-Level paper. In a certain country, each problem was solved by exactly 10001000 participants, but no two participants solved all 66 problems between them. What is the smallest possible number of participants from this country in the spring round Senior A-Level paper?
(R Zhenodarov)

TOT 2000 Autumn AS6 black squares of side 1 lie on a white plane

a) Several black squares of side 11 cm are nailed to a white plane with a nail of thickness 0.10 . 1 cm so that they form a black polygon. Can it happen that the perimeter of this polygon is 11 km long? (The nail is not allowed to touch the boundary of any of the squares . ) (b) The same problem as in (a) but with a nail of thickness 00 (a point ) . (c) Several black squares of side 11 cm lie on a white plane so that they form a black polygon (possibly having more than one piece and/ or having holes) . Can it happen that the ratio of its perimeter (in centimetres) to its area (in square centimetres) is more than 100000100000?
(Hungarian Folklore)
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