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.
be a positive integer. Determine all sequences 
of positive integers such that
for all positive integers 
.
be an integer. Prove that 
is not a perfect cube of an integer.
be an integer. Alice and Bob play the following game: Alice chooses 
and draws a 
table, then he fills the cells of the first row with different numbers from 
. Then, Bob fills on the second row some of the cells (eventually none) with distinct numbers from , and the rest of them with 
. Finally, on each cell of the third row we write the sum of the two cells above. Show that regardless how Alice plays, Bob can guarantee that on the third row he can obtain, in some order, the terms of a non-constant arithmetical progression.
for some positive integer .
, 
an satisfy![\[
\frac{a}{b}=\frac{a^2+n^2}{b^2+n^2}.
\]](http://latex.artofproblemsolving.com/d/6/a/d6aa6a1879870ef5ad758ba8c8650fccfcc5efcc.png)
Prove that 
is an integer.
such that for all 
,![\[f(x+yf(x))+y = xy + f(x+y).\]](http://latex.artofproblemsolving.com/c/f/3/cf3d20a041c27244e90876119b4568b7a3e13c03.png)
+5 w, 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.
of positive real numbers such that
be positive reals. Prove that there is a cyclic permutation 
of 
such that the inequality:![\[
\frac{a}{xa + yb + zc} + \frac{b}{xb + yc + za} + \frac{c}{xc + ya + zb} \geq \frac{3}{x + y + z}
\]](http://latex.artofproblemsolving.com/5/5/3/553527fc40a1b7d29dbdfb4b5e63635308f46c85.png)
and 
.
+3 w
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 the set of positive real numbers.
such that there exists a function 
with the property:![\[
f(y^2f(x) + y + c) = xf(x+y^2)
\]](http://latex.artofproblemsolving.com/e/3/0/e30ee6d9f69cc43422a5c2b9813192a632d359cb.png)
for all 
.
be real numbers such that 
.![\[
3 + (2 - \sqrt{3}) \cdot \frac{(b-c)^2}{b+(\sqrt{3}-1)c} \leq a+b+c
\]](http://latex.artofproblemsolving.com/5/a/8/5a83eaa7470bcca6546599ff4e296e4efec69b16.png)
and determine all the cases when the equality occurs.
has all of its last 
digits equal to 
.
there are three circles 
each of which is tangent to two sides of the triangle and to its incircle . The radii of are 
, and 
. Determine the radius of 
be the triangle and 
be the circle next to the vertex 
where 
Let 
be the triangle’s incenter. Let 
and 
be the projection of 
and 
on 
Then we obviously have the following:
and similarly 
we have:
and summing equation 
over 
we finally get:
and 
and rearranging, we get 
Because 
we have 
and 
Prove that
,
