Olympic Dreams Coming True - The Solutions
08/23/2004

 

If Hamm finished just .012 points ahead of Kim Dae Eun, then he must have ended the competition with 57.811 + 0.012 = 57.823. If we take out 9.837 points for each of his last two routines, we see that he had 57.823 – 2(9.837) = 38.149 points after his fourth event.

 It’s possible that he could win 8 silver medals (and no gold or bronze) and the requirements would be satisfied (1 combination). If he wins 7 silver medals, then the 8th medal would have to be gold (1 combination). If he wins 6 silver medals, then he could win 2 golds or 1 gold/1 bronze (2 combinations). If he wins 5 silver medals, then he could have 3 golds or 2 golds/1 bronze (2 combinations). No other combination of gold/bronze would satisfy the condition of winning at least as many gold medals as bronze medals. If he wins 4 silver medals, then he could have 4 golds, 3 golds/1 bronze or 2 golds/2 bronzes (3 combinations). If he wins 3 silver medals, then he could have 5 golds, 4 golds/1 bronze or 3 golds/2 bronzes (3 combinations). Two silver medals would lead to 6 golds, 5 golds/1 bronze, 4 golds/ 2 bronzes or 3 golds/3 bronzes (4 combinations). Winning 1 silver medal allows for 7 golds, 6 golds/1 bronze, 5 golds/2 bronzes or 4 golds/3 bronzes (4 combinations). Finally, if Phelps does not win any silver medals, then he could have 8 golds, 7 golds/1 bronze, 6 golds/2 bronzes, 5 golds/3 bronzes or 4 golds/4 bronzes (5 combinations). This is a total of 1 + 1 + 2 + 2 + 3 + 3 + 4 + 4 + 5 = 25 medal combinations that include 8 medals and at least as many gold medals as bronze medals.

 First, we can see that 100(10) = 1000, so there are ten 100-meter lengths in a kilometer. This means that it would take 10(59.26) = 592.6 seconds to swim a kilometer. Since we are assuming that his rate of speed remains constant, we can set up the proportion, "592.6 seconds is to 0.62137 miles as x seconds is to 1 mile." This translates mathematically into 592.6/0.62137 = x/1. Using cross-products, we get 592.6 = 0.62137x. Dividing both sides by 0.62137 yields x = 953.699 seconds. Converting this into minutes, we have 953.699 / 60 = 16 minutes.

 

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December 27, 2004
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