[/quote]
,
be a fixed integer. There are 
beads on a circular necklace. You wish to paint the beads using 
colors, such that among any 
consecutive beads every color appears at least once. Find the largest value of 
for which this task is 
possible.
,
,
,
denote the feet of the altitudes from 
,
,
,
denote the circumcenter of 
.
and 
denote the projections of 
,
.
such that 
and 
.
and 
lies on 
such that 
,
bisects 
.
,
coin from the left and moves all coins in that chain to the left end of the row. For example, if 
and 
,
would be 
with 
such that for every initial ordering, at some moment during the process, the leftmost 
and 
Prove that
Let 
Prove that
Let 
Prove that
prove that
such that for all naturals 
the function satisfies:![\[a + b \mid a^{f(a)} + b^{f(b)} \]](http://latex.artofproblemsolving.com/e/b/b/ebb44041cedd29de516192d983a334d46e7fe7d6.png)
Bob wants to find all possible functions Alice could have. Help Bob and find all functions that Alice could have.
be the medial triangle of an acute-angle triangle 
meet 
at 
.
analogously. Let 
.
and 
.
meet the 
,
and 
similarly. Let 
be the circumcenter of 
be the circle with diameter 
,
at 
.
and 
.
are collinear.
lie on a circle centered at 
are coaxial.
denote the set of all real numbers. Find all functions 
such that
for all real numbers 
,
.
![\[x^2-cx+1 = \dfrac{f(x)}{g(x)},\]](http://latex.artofproblemsolving.com/1/3/3/1339acdaf1999e06e1a4b7dec7d1217389e2afef.png)
where 
and 
are polynomials with nonnegative real coefficients. For each 
,
exist.
such that every collection of 
.
are in that order and 
and 
.
on line 
,
on line 
and 
are equidistant from the circumcenter of 
be six positive real numbers. Prove that![\[a^2+b^2+c^2+\frac{4(ax+by+cz)\sqrt{ab+bc+ca}}{x+y+z} \geqslant 2(ab+bc+ca).\]](http://latex.artofproblemsolving.com/f/8/0/f808f7857c2150960f2a430837286487666797d8.png)
be a cyclic quadrilateral with circumcentre 
.
be the circumcircle of triangle 
and let 
,
denote the second intersections of 
with 
and 
be the circumcircle of triangle 
.
,
.
be the midpoint of 
be the intersection of the tangents from 
.
intersects 
and 
meet at 
,
and 
meet at 
Y by Adventure10, Mango247
Let 
be a quadrilateral, and 
be intersection points of 
and 
respectively. External bisectors of 
and 
intersect at 
, external bisectors of 
and 
intersect at 
and external bisectors of 
and 
intersect at 
. Prove that 
are collinear.
be a quadrilateral, and 
be intersection points of 
and 
respectively. External bisectors of 
and 
intersect at 
, external bisectors of 
and 
intersect at 
and external bisectors of 
and 
intersect at 
. Prove that 
are collinear.
now from desargue theorem we coclude 
are collinear.
