be a triangle with 
let 
be the circumcircle of 
and let 
be its radius. Point 
is chosen on 
such taht 
and point 
is the foot of the perpendicular from 
.
mets 
.
is chosen on line 
such that 
and 
lie on a line in that order. Finally, let 
be a point satisfying 
and 
.
,
has two real roots, so that the one is the square of the other.
,
be all positive integers smaller than 
such that 
for all 
is the largest positive integer that divides both 
.
.
such that: ![\[ x = \sqrt{ x - \frac{1}{x} } + \sqrt{ 1 - \frac{1}{x} } \]](http://latex.artofproblemsolving.com/e/c/f/ecf7fb3c2805290f2cb07b7fdd7c73e1c7a55b99.png)
Y by parmenides51, Adventure10, Mango247, and 1 other user
Here you can download PDF-files with some problems from contest of Ukrainian journal "In the World of Mathematics" ("У світі математики" - "U sviti matematyky").
Official web-site: http://www.mechmat.univ.kiev.ua/probability/Journals/mathw.html
One more site: http://olympiads.win.tue.nl/imo/matworld.html
Problems 148-159, 166-171
Vol. 7 (2001), #1-2,4
Official web-site: http://www.mechmat.univ.kiev.ua/probability/Journals/mathw.html
One more site: http://olympiads.win.tue.nl/imo/matworld.html
Problems 148-159, 166-171
Vol. 7 (2001), #1-2,4
This post has been edited 3 times. Last edited by dmitin, Dec 12, 2005, 2:40 PM
these problems are great!
