MathDB
Chess tournament

Source: Turkish NMO 1st Round 2013 - Problem 36

April 20, 2013
combinatorics proposedcombinatorics

Problem Statement

A chess club consists of at least 1010 and at most 5050 members, where GG of them are female, and BB of them are male with G>BG>B. In a chess tournament, each member plays with any other member exactly one time. At each game, the winner gains 11, the loser gains 00 and both player gains 1/21/2 point when a tie occurs. At the tournament, it is observed that each member gained exactly half of his/her points from the games played against male members. How many different values can BB take?
<spanclass=latexbold>(A)</span> 5<spanclass=latexbold>(B)</span> 4<spanclass=latexbold>(C)</span> 3<spanclass=latexbold>(D)</span> 2<spanclass=latexbold>(E)</span> 1 <span class='latex-bold'>(A)</span>\ 5 \qquad<span class='latex-bold'>(B)</span>\ 4 \qquad<span class='latex-bold'>(C)</span>\ 3 \qquad<span class='latex-bold'>(D)</span>\ 2 \qquad<span class='latex-bold'>(E)</span>\ 1