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 January 25th, 2010, 12:52 PM #1 Member   Joined: Dec 2009 Posts: 34 Thanks: 0 A few olympiads problems I can't solve 1. Prove that there is no real function such as for every real x,y : $\frac{f(x)+f(y)}{2} - f( \frac{x+y}{2} ) >= |x-y|$ 2. Find 2 positive numbers x,y with the same number of digits such as the product xy is a 50 digits number at least that all of them are 1 (a number of the form 111111111) ... 3. Calculate the square root of the number: 11111...1222...25 where the digit 1 appears 2008 times and the digit 2 appears 2009 times. Good Luck! January 28th, 2010, 12:39 PM #2 Global Moderator   Joined: Dec 2006 Posts: 20,931 Thanks: 2205 3.� 333...5 January 28th, 2010, 01:07 PM #3 Senior Member   Joined: Feb 2009 From: Adelaide, Australia Posts: 1,519 Thanks: 3 Re: A few olympiads problems I can't solve 2. 1633466135458167330677291 * 6802168021680216802168021 April 4th, 2010, 08:13 AM #4 Newbie   Joined: Mar 2010 Posts: 7 Thanks: 0 Re: A few olympiads problems I can't solve 1. let g(x,y) = f(x) + f(y) - f(x+y) let h(x,y) = 2*|x-y| restate the problem as: prove that there does not exist an f, such that for each point in the x,y plane g >= h Note that along the x and y axes, h is unbounded. Note that along the x and y axes, g(x, 0) = f(x) + f(0) - f(x) = f(0) g(0, y) = f(0) + f(y) - f(0) = f(0) If g>=h at every point in the x,y plane, it is greater than every point along the x and y axes as well. this implies f(0) >= 2*|x|, for every x (or f(0) >= 2*|y|, for every y) There is no real number f(0) with this property, therefore no f exists. I'm very keen on the explanation to the other two answers. I don't know how you would go about these without a calculator. Tags olympiads, problems, solve Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Math Events 4 April 16th, 2011 06:20 AM rashi101 Math Books 1 March 30th, 2010 04:42 PM amero Calculus 3 August 14th, 2008 10:54 AM johnny Computer Science 3 November 5th, 2007 08:29 PM verynice2000 Elementary Math 2 December 31st, 1969 04:00 PM

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