MathDB

In a tennis tournament, any two of the

n

participants played a match (the winner of a match gets

1

point, the loser gets

0

). The scores after the tournament were

r_1\le r_2\le\ldots\le r_n

. A match between two players is called wrong if after it the winner has a score less than or equal to that of the loser. Consider the set

I=\{i|r_1\ge i\}

. Prove that the number of wrong matches is not less than

\sum_{i\in I}(r_i-i+1)

, and show that this value is realizable

Problem Statement