In a football tournament $20$ teams participated, each pair of teams played exactly one game. For the win the team is awarded $3$ points, for the draw -- $1$ point, for the lose no points awarded. The total number of points of all teams in the tournament is $554$. Prove that there exist $7$ teams each having at least one draw.
Problem
Source: II Caucasus Mathematical Olympiad
Tags: combinatorics
bigant146
21.03.2018 00:53
I need to create this problem twice because we used it in both juniors and seniors test. See https://artofproblemsolving.com/community/c6h1611845_ii_caucasus_mathematical_olympiad_95
gausskarl
21.03.2018 01:15
bigant146 wrote: I need to create this problem twice because we used it in both juniors and seniors test. See https://artofproblemsolving.com/community/c6h1611845_ii_caucasus_mathematical_olympiad_95 No need to do so, you can still have the same problem used twice. See USAMO and USA(j)MO in this forum.
bigant146
21.03.2018 11:31
gausskarl wrote: bigant146 wrote: I need to create this problem twice because we used it in both juniors and seniors test. See https://artofproblemsolving.com/community/c6h1611845_ii_caucasus_mathematical_olympiad_95 No need to do so, you can still have the same problem used twice. See USAMO and USA(j)MO in this forum. I tried to include one problem twice in contest page (https://artofproblemsolving.com/community/c630740_2017_caucasus_mathematical_olympiad) and it told me that "This problem is already in this contest" ... Maybe there is a better solution for this problem but i didn't find it
AyunPa0310
21.03.2018 15:11
There are 190 matches, so there are 16 matches ended with a draw. If there are only 6 teams having at least one draw, the number pof draw matches will be less then 15 (there are 15 matches between these 6 teams). Contradition!