AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
for some positive integer 
.
. Find all functions 
such that for all 
,![\[
f(xf(x+y)+z) = f(z) + f(x)y + f(xf(x)).
\]](http://latex.artofproblemsolving.com/f/9/3/f93b7526ff4338eea89a1c59bc009aa001ffc15d.png)
. Prove that
Let 
. Prove that
Let 
. Prove that
Let 
. Prove that
, define![\[
E(x,y) = x^{-\!y} + y.
\]](http://latex.artofproblemsolving.com/6/1/2/6124224ada5e77a6c03e4cfe24e69435dc996ec4.png)
, prove that![\[
E(x,y) + E(y,x) \geq 4.
\]](http://latex.artofproblemsolving.com/0/6/5/065d58fba8e8799a307655dafdcfdadb2e351eea.png)
, prove that![\[
\frac{2}{1+xy} + x + y < E(x,y) + E(y,x) < 2 + \frac{x}{y} + \frac{y}{x}.
\]](http://latex.artofproblemsolving.com/8/c/9/8c912cad7eeabd17e145a08fc87ecb6b5c5b2abc.png)
be a positive integer. Do there exist distinct -digit positive integers 
and 
, with each of their digits being 
or 
, such that 
divides 
?
denote the set of all positive integers that can be expressed as a sum of three or fewer powers of 
. Let 
be the smallest positive integer that cannot be expressed in the form 
, where 
. Find the remainder when is divided by 
.
, 
be positive integers. Julia and Florian play a game on a 
board. Julia cells. such that Florian has a strategy to always deduce the entire tiling.
be a positive integer. A sequence of integers 
is called -pop if the following holds: for every 
, 
is equal to the number of distinct elements in the set 
. Determine, as a function of , how many -pop sequences there are.
of positive real numbers that satisfy the system:![\[
\begin{aligned}
11bc - 36b - 15c &= abc \\
12ca - 10c - 28a &= abc \\
13ab - 21a - 6b &= abc.
\end{aligned}
\]](http://latex.artofproblemsolving.com/1/7/0/170e1cfaee90f5c37428fa75192ac889261a8f83.png)
be a positive integer. Prove that there are infinitely many primes, each of which divides at least one term of the sequence 
, 
.
and 
be positive integers. is a divisor of 
and is a divisor of 
.
.
. It crossed the segment 
at point 
. Find the angle 
if it is known that the lines 
and 
perpendicular.
and 
Prove that
.
.
.
is iscosceles, so 
.
.
