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.
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 
of people, we define 
to be the least number of tables needed such that each person from the group sits in one of them with no two friends sitting on the same table, and 
to be the least number of tables needed such that each person from the group sits in one of them with no two enemies sitting on the same table.
such that for each group of 
people we have![\[ f(n) \leqslant f(a(G)) + f(b(G)) \quad \text{and} \quad f(2) = 1.\]](http://latex.artofproblemsolving.com/0/3/d/03dfc700f9284152b39e32031c92c33283d65da6.png)
Find all possible values of 
.
is the set of all positive integers.
such that,
for all non-negative integers 
and 
.
is the set of all non-negative integers and 
is the set of all real numbers. cells is a connected figure consisting of equal-sized cells[1].
cells. It is said to be primitive it its cells cannot be partitioned into two or more dinosaurs. Find with proof the maximum number of cells in a primitive dinosaur. cells, one with 
cells is obtained by adjoining a new cell by making it adjacent to one or more existing cells.
such that 
and 
are coprime if and only if 
?
are positive reals. Consider the following polynomial for each 
in 
:
where indices are taken modulo 
, i.e., 
for any 
in 
. Show that it is impossible that each of these polynomials has all its roots real.
, 
, 
, and 
be the orthocenters of triangles 
, 
, 
, and 
, respectively. Let be the centroid of triangle . is the foot of onto the plane .
be the foot of onto the plane . Notice that 
is perpendicular to plane 
. Thus 
, which implies 
. Similarly, 
. Thus 
, as required. and . Notice the centroids of triangles , , and all coincide at some point 
. Since is the foot of onto the plane , it follows that 
. By the claim, we also get that 
. Thus all five points lie on the sphere with diameter 
, concluding the proof.
be a function such that 
.
.
is given. It is known that triangles 
,
,
,
are all acute. Prove that the orthocenters of these triangles, together with the centroid of triangle 