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.
is 
than the set of numbers 
if among all inequalities of the form 
the number of true inequalities is at least 
times greater than the number of false ones. Prove that there do not exist three sets 
such that 
is than 
, is than 
, is than .
be an 
real matrix with minimal polynomial 
. Prove that the trace of 
is zero.
we have 
.
.
and 
.
?
be triangle with circumcenter 
and orthocenter 
. 
intersect again at 
respectively. Lines through parallel to 
intersects 
at 
respectively. Point 
such that 
is a parallelogram. Prove that lines 
and 
are concurrent at a point on 
.
+1 w
satisfying
Prove
and 
be positive integers. Suppose that the sum of integers between and , including and , are equal to 
.
, find the pair such that achieves the minimum.
![$C[a,b]$](http://latex.artofproblemsolving.com/e/e/6/ee63b22d81304485423eec80b34bbf21b655d75d.png)
is the set of all continuous functions on the closed interval ![$[a,b]$](http://latex.artofproblemsolving.com/8/e/c/8ecbd1ba3da8f2adef66a63f2ab32c47e63fa734.png)
. For, define![$$d(f, g) = \sup_{x \in [a, b]} |f(x) - g(x)|.$$](http://latex.artofproblemsolving.com/a/4/a/a4a62b37713788c94992853b0eaadcbabbca86d6.png)
(b) Consider the closedness and compactness of the following set in the space :![$$B = \{ f \in C[a, b] : |f(x)| \leq 1, \text{ for all } x \in [a, b] \}.$$](http://latex.artofproblemsolving.com/2/e/d/2ed13593e5bad1171aad457042dcdc9149d9d931.png)
be an acute-angled triangle with 
. Let 
be the circumcircle of . Let 
be the midpoint of the arc 
of containing . The perpendicular from to 
meets 
at and meets again at 
. The line through parallel to meets line 
at 
. Denote the circumcircle of triangle 
by 
. Let meet again at 
. Prove that the line tangent to at 
meets line on the internal angle bisector of 
.
be a triangle with circumcircle . Let 
and 
be two lines through the points 
and 
, respectively, such that 
. The second intersections of and with are and 
, respectively. Assume that and are on the same side of as . Let 
intersect at 
and let 
intersect at 
. If 
, 
and 
are circumcenters of the triangles , 
and 
, respectively, and is the circumcenter of the triangle 
, prove that 
.
be the open unit disk in the complex plane and let 
be a holomorphic function such that 
. Let the Taylor series of 
be 
. Prove that the number of zeroes of (counted with multiplicities) equals 
.
.
be an ordered set.Then given any set X and a function 
we can define an order on by 
. What is your opinion?
,![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]](http://latex.artofproblemsolving.com/7/b/6/7b6e32d32d89ca058a06974e7276f0a5874d90a7.png)
