MathDB
Round-Robin Tournament

Source: 2014 AMC 12A #13

February 4, 2015
AMC

Problem Statement

A league with 1212 teams holds a round-robin tournament, with each team playing every other team once. Games either end with one team victorious or else end in a draw. A team scores 22 points for every game it wins and 11 point for every game it draws. Which of the following is <spanclass=latexbold>not</span><span class='latex-bold'>not</span> a true statement about the list of 1212 scores?
<spanclass=latexbold>(A)</span>There must be an even number of odd scores.<span class='latex-bold'>(A) </span>\text{There must be an even number of odd scores.}
<spanclass=latexbold>(B)</span>There must be an even number of even scores.<span class='latex-bold'>(B) </span>\text{There must be an even number of even scores.}
<spanclass=latexbold>(C)</span>There cannot be two scores of 0.<span class='latex-bold'>(C) </span>\text{There cannot be two scores of 0.}
<spanclass=latexbold>(D)</span>The sum of the scores must be at least 100.<span class='latex-bold'>(D) </span>\text{The sum of the scores must be at least 100.}
<spanclass=latexbold>(E)</span>The highest score must be at least 12.<span class='latex-bold'>(E) </span>\text{The highest score must be at least 12.}