What chess player second round? (Mathematics olympiad Netherlands) Let $A,B$ and $C$ denote chess players in a tournament. The winner of each match plays the next match against the oponent that did not play the current. At the end of the tournament $A$, $B$ and $C$ played $10$, $15$ and $17$ times respectively. Each match only ended up in a win. Question: Which player lost the second match?
UPDATE: So I think I got the answer. Denote $n$ as the amount of matches between $A$ and $B$. Since $A$ plays the same amount of matches against $B$ as the otherway around, we have $15 - n = 17 - (10-n) \implies n = 4$. So $A$ plays a total of 10 matches, while there are a total of 21 matches. This is only possible if $A$ plays all the even matches and he loses that match (else contradiction to amount of matches played).
 A: $A$ must lose the second match.
To see why, we'll first compute how many matches have been played in total. This can be computed using a simple linear system of equations.
The result is $21$. Now, $A$ has played only $10$ matches. The only way this is possible is if $A$ did not play in the first match and then lost every match (otherwise $A$ would have played at least 11 matches).
A: This is disappointingly easy, due to the extremely uneven scores.
$10+15+17=42$ counts every match twice, so there were $21$ matches. The rules are such that every player plays at least once in every pair of consecutive matches. The only way $A$ can have played in less than half of the matches, as is apparently the case, is by playing every second match, and starting (and ending) by not playing. This in turn implies $A$ loses every match (s)he played, the first of which is the second match.
One can also conclude that the opponent of $A$ was $B$ four times and $C$ six times, so before the final match $B$ had won $8$ times and $C$ had won $12$ times (since $A$ was such a loser, every win between $B$ and $C$ was followed by a win against $A$). One cannot know who wont the final match between $B$ and $C$, though of course $C$ won the "tournament".
