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January 25th, 2010, 12:52 PM   #1
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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!
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January 28th, 2010, 12:39 PM   #2
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January 28th, 2010, 01:07 PM   #3
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Re: A few olympiads problems I can't solve

2. 1633466135458167330677291 * 6802168021680216802168021
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April 4th, 2010, 08:13 AM   #4
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Re: A few olympiads problems I can't solve

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.
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