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JeeJee
Canada5652 Posts
Consider that SaviOr was not always the bonjwa. At some point he was a "noob". Let's say he had a winning percentage of less than 80%. Then something clicked and he started kicking butt, and his overall winning % went above 80%. Show that at some point in between, his win% must've been exactly 80 Bonus: Aside from 80%, for what other numbers does this property hold true? GL!  | ||
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paper
13197 Posts
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paisano
Romania4 Posts
+ Show Spoiler + Never thought of actually solving the two inequalities. StRyKeR's solution is the right one. | ||
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StRyKeR
United States1756 Posts
On February 15 2010 05:54 paisano wrote: No, it would be the intermediate-value theorem for continuous functions; one question though, are you sure winning % isn't discreet; assuming Savior can only play a finite number of games, won't the fraction w/p (w - won; p- played) be rational, and thus the function is undefined for the irrationals? If I'm wrong, I'd appreciate some help on this, as it seems you can't really apply the theorem here. Win % is discrete, which is why the property doesn't hold for every fraction. 80% happens to be one of them. You cannot apply the intermediate value theorem. I believe this was a Putnam problem from years back. + Show Spoiler + Let w be the number of wins and T be the total number of games played right before Savior's win rate was strictly below 80%. That means w/T < 4/5 and (w+1)/(T+1) >= 4/5 First equation gives 5w < 4T, or 5w <= 4T - 1, since w and T are integers. Second gives 5w + 5 >= 4T + 4, or 5w + 1 >= 4T, or 5w >= 4T - 1. Combined with the first equation, we see that we MUST have 5w = 4T - 1. In other words, w = 4T/5 - 1/5 and (w + 1)/(T + 1) = (4T/5 + 4/5)/(T + 1) = 4/5 = 80%, as desired. Basically, the first game that gives Savior a win rate of at least 80% is the game that puts him at EXACTLY 80%. More generally, the trick works for any fractions p/q such that q - p = 1. In this case, q = 5 and p = 4. | ||
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JeeJee
Canada5652 Posts
I think I have to stop with number puzzles . . . TL is too good at them. Unfortunately without them the fortnight is gonna turn into a week >.< lol edit: yes this was a putnam problem from 2004 | ||
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