MathDB
Turkey NMO 2008 1st Round - P16 (Combinatorics)

Source:

August 25, 2012
inequalities

Problem Statement

A class of 5050 students took an exam with 44 questions. At least 11 of any 4040 students gave exactly 33, at least 22 of any 4040 gave exactly 22, and at least 33 of any 4040 gave exactly 11 correct answers. At least 44 of any 4040 students gave exactly 44 wrong answers. What is the least number of students who gave an odd number of correct answers?
<spanclass=latexbold>(A)</span> 18<spanclass=latexbold>(B)</span> 24<spanclass=latexbold>(C)</span> 26<spanclass=latexbold>(D)</span> 28<spanclass=latexbold>(E)</span> None of the above <span class='latex-bold'>(A)</span>\ 18 \qquad<span class='latex-bold'>(B)</span>\ 24 \qquad<span class='latex-bold'>(C)</span>\ 26 \qquad<span class='latex-bold'>(D)</span>\ 28 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the above}