[/quote]
,
for which
.![\[\sqrt{a+\frac{9}{b+2c}}+\sqrt{b+\frac{9}{c+2a}}+\sqrt{c+\frac{9}{a+2b}}+\frac{2(ab+bc+ca)}{9}\ge\frac{20}{3}\]](http://latex.artofproblemsolving.com/b/3/2/b32e0bcba967ca847e878ac186c3168cd6c5fc66.png)
is a prime number
denote the set of positive integers. Find all functions 
such that for positive integers 
and 
![\[f(a^2 + b^2) = f(a)f(b) \text{ and } f(a^2) = f(a)^2.\]](http://latex.artofproblemsolving.com/c/1/f/c1f8ffe04cfcc45b498f3931a4796d5c56dc04d0.png)
such that there exists a sequence of positive real numbers 
,
,![\[
a_1 + \cdots + a_{n+1} < \alpha \cdot a_n.
\]](http://latex.artofproblemsolving.com/9/b/7/9b73deb46032844b324288b02470ff0623ac0ff1.png)
Proposed by Pavle Martinović
,
be the point which minimizes the sum of squares of distances from the sides of the triangle. Let 
the projections of 
is the barycenter of 
.
be the set of all positive real numbers. Find all strictly monotone (increasing or decreasing) functions 
such that there exists a two-variable polynomial 
with real coefficients satisfying
for all 
.
.
.
be the midpoint of the arc 
not containing 
and define 
similarly. Show that the orthocenter of 
is the incenter 
of 
Y by
Is there a rough difficulty comparison between IMO shortlist questions and USAMO questions? For example,
SL 1, 2, 3 -> USAMO P1
SL 4, 5, 6 -> USAMO P2
SL 7, 8, 9 -> USAMO P3
(This is just my guess; probably not correct)
Also feel free to compare it with other competitions(like the jmo) as well!
SL 1, 2, 3 -> USAMO P1
SL 4, 5, 6 -> USAMO P2
SL 7, 8, 9 -> USAMO P3
(This is just my guess; probably not correct)
Also feel free to compare it with other competitions(like the jmo) as well!
