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.
players, each pair of players plays against each other exactly one match and there are no draws.Show that the players can be arranged in an order 
such that 
defeats 
for all 
.
and 
be the incircle and circumcircle of the acute triangle 
, respectively. Draw a square 
so that all of its sides are tangent to , and 
, 
are both on 
. Extend 
and 
, intersecting at 
and 
, respectively. Prove that 
and 
intersects on .
. Let the perpendicular line from 
to meets 
at points 
, respectively, and the foot from 
to 
is 
. 
intersects line 
at 
, 
intersects line at 
, and lines 
intersect at 
.
is tangent to 
.
be the circumcenter and orthocenter of triangle , respectively. Let 
and 
intersect at point 
. If 
is an arbitrary point on, and and be the intersection of 
with and 
, respectively.
are concyclic.
satisfy the following conditions:
Find the value of 
.
such that![\[(a^2 + b^2 + c^2)^2 = 3(a^3b + b^3c + c^3a) + 1. \]](http://latex.artofproblemsolving.com/9/e/4/9e400fd807a9535ede983b75dcc307d8b75a35b6.png)
be a triangle with incentre 
and circumcircle . Let 
be the midpoints of the arcs 
of not containing 
respectively. Let 
be the point of diametrically opposite to 
. Show that 
and the midpoint 
of 
lie on a line.
is a permutation of 
. Each time, Alice swaps two neighbouring numbers in the sequence. Her goal is to make any three consecutive numbers in the sequence non-monotone. Find the smallest positive integer 
such that Alice can always achieve her goal in at most swaps.
with real coefficients of degree 
such that 
divides 
.
. A convex polygon is said to be -good if it contains lattice points where any three of them are not collinear.-good convex polygon with area at most 
.
so that any -good convex polygon has area at least 
.
satisfying the following:
; are 
and 
;
;
has all of its coefficients being even.
such that:
and 
Prove that
is tangent to 
intersects line 
,
.