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. Find the smallest value of such that:
is divisible by 
.
?
such that there exists a prime number 
, such that 
is a power of 
. is a number of the form 
where 
is a positive integer.
atoms called usamons. Each usamon either has one electron or zero electrons, and the physicist can't tell the difference. The physicist's only tool is a diode. The physicist may connect the diode from any usamon 
to any other usamon 
. (This connection is directed.) When she does so, if usamon has an electron and usamon does not, then the electron jumps from to . In any other case, nothing happens. In addition, the physicist cannot tell whether an electron jumps during any given step. The physicist's goal is to isolate two usamons that she is sure are currently in the same state. Is there any series of diode usage that makes this possible?
, 
. the incircle 
of meet 
at point 
, 
meet again at 
. 
is a tangent of , and meet the extension line of at 
. 
, 
. the line 
meet at 
, point 
is on the line segment , the line segment 
meet again at 
. Show that 
such that 
.
be positive integers and be prime. We know that 
and 
. Prove that 
is a perfect square.
be a set of 
residues 
. Prove that there exists a set of of residues such that 
contains at least half of all the residues .
such that for all 
the following equality holds ![\[
f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \]](http://latex.artofproblemsolving.com/f/3/a/f3ac860e90d671f9fb592a6bd55c102a3000daec.png)
where 
is greatest integer not greater than 
be an isosceles triangle such that 
. Let be a dot on the 
side.
at point intersects the circumcircle of 
at . Prove that CD
AB
and 
Prove that
be a function such that 
.
.
