Combinatorics · Counting by groups · Extremal bounding · Score distributions · Integer constraints
Problem 5, 2008
← Prev · 2 / 2 · Next →At a national competition the students worked on \(4\) problems. Each problem was marked with a whole number of points, at least \(0\) and at most \(7\). Altogether \(42\) students competed. Exactly half of the competitors reached at least \(50\%\) of the available points. An award was given for collecting at least \(22\) points, and one sixth of the competitors managed this.
The competitors who received no award scored, all together, three times as many points as all the awarded competitors together. Prove that at least \(6\) competitors scored individually at least \(25\%\) of the available points but less than \(50\%\) of them.
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Slovenian High School Mathematics Competition for Vega Awards (MaSSA), izbirno (selection) round 2008, 1. letnik, category A, problem 5. Organized by DMFA Slovenije (Society of Mathematicians, Physicists and Astronomers of Slovenia). Source