Question 10
An even number of teams play a round robin: every pair meets exactly once, every team plays exactly once in each round, and there are no draws.
(a) Prove that after round 2 the number of teams with two wins equals the number with two losses.
(b) Find the least number of teams for which, after round 3, the number with three wins may differ from the number with three losses.
(c) Prove that after round 4 the number with four wins equals the number with four losses if and only if the number with exactly three wins equals the number with exactly three losses. (5 marks)
Investigation: prove that for every even $k\ge4$ there is a round robin of $k$ teams in which the number winning $k-1$ matches differs from the number losing $k-1$ matches. (3 bonus marks, separate from the base score.)
(a) Prove that after round 2 the number of teams with two wins equals the number with two losses.
(b) Find the least number of teams for which, after round 3, the number with three wins may differ from the number with three losses.
(c) Prove that after round 4 the number with four wins equals the number with four losses if and only if the number with exactly three wins equals the number with exactly three losses. (5 marks)
Investigation: prove that for every even $k\ge4$ there is a round robin of $k$ teams in which the number winning $k-1$ matches differs from the number losing $k-1$ matches. (3 bonus marks, separate from the base score.)