[/quote]
,
find all 
satisfying 
:![\[g(f(x)-y)=f(g(y))+x.\]](http://latex.artofproblemsolving.com/3/d/7/3d7dd5cd43f2936d6e743474744dcd883633bcca.png)
be a positive real number. The scorpion tries to catch the flea on a 
chessboard. The length of the side of each small square of the chessboard is 
.
vertexes of the chessboard and, in each turn, can jump from the vertex where it is to one of the adjacent vertexes. The scorpion moves on the boundary line of the chessboard, and, in each turn, it can walk along any path of length less than 
,
positive composite integers that form an arithmetic progression and are pairwise coprime.
an alphabet with 
(as a function of 
satisfying the following conditions:
,
.
,
.
such that 
and 
.![$P, Q \in \mathbb{R}[x]$](http://latex.artofproblemsolving.com/c/3/0/c301578bade5b93056d77454921cba0a9812212c.png)
be relatively prime nonconstant polynomials. Show that there can be at most three real numbers 
such that 
is the square of a polynomial.
rectangle can be tiled using 
ways to file a fixed 
rectangle 
with 
pieces. (symmetric constructions are supposed to be different.)
be non-negative real numbers such that 
.![\[
6xyz \leq x(1 - x) + y(1 - y) + z(1 - z),
\]](http://latex.artofproblemsolving.com/f/5/b/f5b208ea6439994cb40f6b7f0806c216d9868e86.png)
and determine when equality holds.
of positive integers such that 
is multiple of 
and 
is multiple of 
.
are nonnegative and 
.![\[ \frac{1}{a^2 -4a+9} + \frac {1}{b^2 -4b+9} + \frac{1}{c^2 -4c+9} \]](http://latex.artofproblemsolving.com/4/9/0/49034beb1150fbdf1c5af8f739dfeca9e153fa40.png)
Y by
Quick question because I’m new here, I keep searching keywords in the community search functions to check if they’ve been posted before, and every time it ended up not finding the original, leading to a repost. This has happened so many times I’ve decided not to post anymore until I can somehow find a way to efficiently search for it.
I accidentally reposted a Putnam 2006, even though I searched for the keywords in the question, but since I didn’t know it was a Putnam problem (I had seen it somewhere else), I didn’t manage to find the problem so unknowingly reposted it.
I accidentally reposted a Putnam 2006, even though I searched for the keywords in the question, but since I didn’t know it was a Putnam problem (I had seen it somewhere else), I didn’t manage to find the problem so unknowingly reposted it.
