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Lemma
Inequalities Difficulty 4

TMUA practice: Inequalities, problem 2

A TMUA-calibre inequalities problem at difficulty 4 of 5, with the full worked solution.

The question
In a season a particular team played 60 games, each ending in a win or a loss. The team never lost three games consecutively and never won five games consecutively. If NN is the number of games the team won, then NN satisfies
A 24≤N≤5024 \le N \le 50
B 20≤N≤4820 \le N \le 48
C 12≤N≤4012 \le N \le 40
D 18≤N≤4218 \le N \le 42

The correct answer is highlighted. B

Worked solution

Split the 60 games into consecutive blocks.

Upper bound. Partition the season into 12 blocks of 5 consecutive games. In each block the team won at most 4 (five wins in a block would be five consecutive wins). Hence N≤12×4=48N \le 12 \times 4 = 48.

Lower bound. Partition the season into 20 blocks of 3 consecutive games. In each block the team lost at most 2 (three losses would be three consecutive losses). Hence the total number of losses is at most 20×2=4020 \times 2 = 40, so N=60−(losses)≥60−40=20N = 60 - (\text{losses}) \ge 60 - 40 = 20.

Therefore 20≤N≤4820 \le N \le 48.

Answer: B.