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.
such that 
and 
people at a round table. Some of them are knights who always tell the truth, and the rest are liars who always tell lies. Each of those sitting said the phrase: “among the 
people sitting clockwise from where I sit there are as many knights as among the people seated counterclockwise from where I sit”. For what 
could this happen?
such that the inequality 
holds for all positive integers 
.
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 
?
