AoPS Academy
be non-negative real numbers such that 
Prove that
be an acute non-isosceles triangle with circumcircle 
, circumcenter 
and orthocenter 
. We draw a line perpendicular to 
through and a line perpendicular to 
through . Prove that the points of intersection of these lines with sides 
and 
lie on a circle, which is tangent to . be a triangle with incenter 
and intouch triangle 
. Let 
be the foot of the perpendicular from 
onto 
. Assume that 
, 
intersect the sides , in 
respectively. Finally, let the rays 
, 
meet the circumcircle of 
in 
, 
respectively. Prove that the tangent from to the incircle and the line 
meet on the circumcircle of .
be the set of sequences of length 2018 whose terms are in the set 
and sum to 3860. Prove that the cardinality of is at most![\[2^{3860} \cdot \left(\frac{2018}{2048}\right)^{2018}.\]](http://latex.artofproblemsolving.com/d/e/8/de857941803349e275e9c9aa968a201f2988cb8a.png)
where 
denotes the Fibonacci sequence.**
be a sequence such that 
and 
.
and 
)
is a derivable function with the property 
, for every 
, then f is a polynomial function of degree smaller or equal than 1
be a function that admits a primitive 
such that 
if 
such that 
be holomorphic functions on 
, where is an open, connected subset of 
. Suppose that the function 
given by 
takes a maximum value in . Must each function 
be constant on ?
, let 
be the maximum number of points inside or on the sides of a regular 
-agon of side 
such that the distance between any two points is greater than . Prove that 
for 
.
, let be the maximum number of points inside or on the sides of a regular -agon of side such that the distance between any two points is greater than . Prove that for .
. Place points on the vertices of the -agon and another at its centre. Then, dislocate each of the points a distance 
clockwise along the edge. For , we need not worry about the distances to points other than its neighbors or the centre. The former becomes 
, whereas the latter is finitely greater than and does not change appreciably when we do this infinitesimal movement.
