Romanian Mathematical Olympiad

Question

Michael is a good chess player. He took part in a competition where all the best chess players of the city were invited. The competition had two stages. After the first stage, looking at the partial results, Michael found that the number of players ranked higher than him was a half of the number of players ranked lower than him. In the second stage Michael played better; He managed to surpass four of the players that were ranked higher than him, after the first stage, but he was also surpassed by two ranked lower than him. So, at the end of the competition, the final results showed that the number of players that were ranked higher than him was a quarter of the number of players that were ranked lower. What place did Michael get at the end of the competition? (Justify your answer!)

Accepted Answer

If we denote by x the number of players ranked higher than Michael after the first stage, it turns out that the number of players ranked lower than him will be 2x. Because in the second stage Michael surpassed 4 players that were before him and he was surpassed by 2 players that were after him, he "advanced" 2 places, so, at the end, the number of players that were before him is x - 2 and the number of the players after him is 2x + 2. As the number of players that were placed before him at the end of the competition is a quarter of the number of the players that were placed after him, then 4(x - 2) = 2x + 2, from where x = 5. Michael finished the competition in the fourth place.