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.
we consider a 
rectangular frame, consisting of the 
boundary squares of a rectangle.
for which Renate has a winning strategy.
be distinct real numbers such that 
Find the value of 
be distinct real numbers such that 
Find the value of 
be distinct real numbers such that 
Find the value of 
be a semicircle with diameter 
and midpoint 
. Let 
be a point on different from 
and 
.
touches in a point 
, the segment 
in a point 
, and additionally the segment 
. The circle 
touches in a point 
and additionally the segments 
and .
and 
are perpendicular.
we consider the last digit different from zero in the decimal expansion of 
. The infinite sequence of these digits starts with 
. Determine all digits which occur at least once in this sequence, and show that each of those digits occurs in fact infinitely often.
, then jumps in some order on each of the numbers 
exactly once and finally returns with his last jump to . Can the total distance he travelled with these 
jumps be a) 
, b) 
?
![$Q\in \{-1,1,0\}[x]$](http://latex.artofproblemsolving.com/b/4/0/b404c8d6415aaf3d4442cc5381f582c1f8f8b996.png)
with 
.![$\exists Q \in \mathbb Z[x]$](http://latex.artofproblemsolving.com/5/6/2/562f1db4bbf5e402e3d16c1979dede295e2f5eae.png)
palindrom with 
?
such that 
.
we denote the set of positive real numbers.
be a non-zero real number. Find all functions 
such that![\[
xf(x+y)=(x+\alpha y)f(x)+xf(y)
\]](http://latex.artofproblemsolving.com/e/c/5/ec5a49e0735e3b13fbef38905c0d7405f44d09b2.png)
for all 
.
![$P \in \mathbb R[x,y]$](http://latex.artofproblemsolving.com/8/a/0/8a0576299e976c51607959fb68c7c742b2a49c6a.png)
with ![$P \not\in \mathbb R[x] \cup \mathbb R[y]$](http://latex.artofproblemsolving.com/a/f/7/af71ba8c643d5bd8a2928fa6714ccd91713bd641.png)
and 
homeomorphismes of 
, 
is an homoemorphisme too.
and 
. Prove that
Let and 
. Prove that
Let and 
. Prove that
Let and 
. Prove that
and 
Prove that 
be real numbers such that 
. Find the minimum of 
and 
Prove that 
has real roots 
where 
.
has real roots 
where 
.
(s non zero), then 
, and for a positive integer 
, prove that
.
be positive reals such that 
. Find the maximum value of 
Given 
non-zero reals 
such that their sum is 
. Prove that there exists two numbers 
such that 
by using the condition