[/quote]
,
,
,
,
.
at 
.
similarly.Prove that 
are collinear.
grid with 
rows and 
be the average of all the numbers in the grid. Prove that if 
the there exist two identical rows (meaning same numbers in the same order).
such that ![$$\left(a^{3}+b^{3}+c^{3}-3abc\right)^{2}-\left[a^{3}+b^{3}+c^{3}+3abc-ab(a+b)-bc(b+c)-ca(c+a)\right]^{2}\ge 2k\cdot(a-b)^{2}(b-c)^{2}(c-a)^{2}$$](http://latex.artofproblemsolving.com/d/2/4/d24b46e65a587fbd977467b6e4e31838a0eb57aa.png)
forall 
such that p is a prime, 
and 
is a divisor of 
.
such that![\[ \frac {2^n + 1}{n^2}
\]](http://latex.artofproblemsolving.com/9/2/2/922444e7669c326efcad5fde3ea00f8c2d9fa5d8.png)
is an integer.
and 
Find the ratio 
,
is a point on arc 
is maximized.
and 
be the altitudes. Let the points 
and 
fulfil the following requirements:
is the circumcentre of triangle 
and 
are equal to the circumradius of triangle 
has the same direction as the oriented segment 
.
has the same direction as 
,
has the same direction as 
.
,
be the intersections of the perpendicular bisectors of 
and 
and 
.
and 
is a divisor of 
.
be a positive integer, 
,
.
,
passing through 
at point 
be a positive integer. Claire writes 
in a row on a blackboard. In a 
William can erase a number 
and replace it with either 
or 
at the same location. His goal is to perform a sequence of moves such that after he is done, the number are strictly increasing from left to right.
independent of 
such that William can always reach his goal in at most 
moves.
independent of 
moves.
and 
,
to be the unique polynomial 
such that we can write 
,
is a polynomial such that either 
or the degree of 
with real coefficients such that 
Y by HamstPan38825, Adventure10, Mango247
There are 
cities, 
airline companies in a country. Between any two cities, there is exactly one -way flight connecting them which is operated by one of the two companies. A female mathematician plans a travel route, so that it starts and ends at the same city, passes through at least two other cities, and each city in the route is visited once. She finds out that wherever she starts and whatever route she chooses, she must take flights of both companies. Find the maximum value of .
cities, 
airline companies in a country. Between any two cities, there is exactly one -way flight connecting them which is operated by one of the two companies. A female mathematician plans a travel route, so that it starts and ends at the same city, passes through at least two other cities, and each city in the route is visited once. She finds out that wherever she starts and whatever route she chooses, she must take flights of both companies. Find the maximum value of .
by the well-known argument, we have a monochrome triangle, so that doesn't work, and for 
a working construction is easy to find (connect vertex pairs {1,2}, {1,3}, {3,4} with red, and the rest blue). The question is, what about 
?
'
edges, so there are at most 
edges. On the other hand 
edges, so 
.
and that's it.
colors we need three acyclic graphs, thus 
,
.
of vertices 
is given by the three paths 
.
,
,
can be partitioned into 
each? or else in some other way into 
,
on 
for a forest 
and 
where 
is the number of edges of 
,
.
-
.
consisting of all the blue edges, and the graph 
edges, we have 
hence done.
edges of some color, say red. The maximal number of edges of a graph on ![\[\left\lceil \frac{\binom n2}{2} \right\rceil \leq n-1 \implies n \leq \boxed{4},\]](http://latex.artofproblemsolving.com/8/4/5/8455657f5f58ff6105675ce451b35d0bc14030f5.png)
which is easily constructable.
.
edges and thus,![\[\frac{n(n-1)}{2} \leq 2(n-1)\]](http://latex.artofproblemsolving.com/3/f/0/3f0ed5612ece892ed7db5d9ab83037660c5224ff.png)
from which we conclude that 
as desired.
.![\[\binom{n}{2} \le 2(n-1) \implies n \le 4,\]](http://latex.artofproblemsolving.com/7/3/6/73644107379b5afa70efb5187a53a84ee12064df.png)
there are three edges coming tru a single vertex in the same color, because of Dirihlet. Now the edges connecting the edges with the same color, should be colored in the other color leading to getting a monochromatic triangle - impossible. If n = 5, after bruteforcing the cases there is either a monochromatic triangle or pentagon, contradiction. For n = 4, the coloring is easy: if 
,
must contain a triangle consisted of only blue or red edges, so any value of 
fails.
.
c
.
Mark the edges 
as red and the rest as blue. This graph works, so the maximum value of 
