be positive real numbers such that : 
. Prove that : 
with![\[2x^3zf(z)+yf(y) \ge 3yz^2f(x)\]](http://latex.artofproblemsolving.com/a/b/f/abf10d7e679f6f04db556424a0a0eda26f371fd4.png)
for all 
.
grid as follows. Two players in turn write either an S or an O in an empty square. The first player who produces three consecutive boxes that spell SOS wins. If all boxes are filled without producing SOS then the game is a draw. Prove that the second player has a winning strategy.
be the circumcenter and orthocenter of a scalene triangle 
,
be the intersection point of the lines 
and 
.
meets the side 
.
and 
be the second intersection points of the circumcircles of 
and 
with the circumcircle of 
,
and 
Y by Adventure10, Mango247
Two half-lines 
and 
, with the common endpoint 
, make an acute angle 
. Let 
on and 
on be points such that 
, and let be the line through parallel to . Let 
be the circle with centre and radius 
. We construct a sequence of half-lines 
, all lying inside the angle , in the following manner:
(i)
is given arbitrarily;
(ii) for every natural number
, the circle intercepts on 
a segment that is of the same length as the segment cut on 
by and 
.
Prove that the angle determined by the lines and has a limit as tends to infinity and find that limit.
and 
, with the common endpoint 
, make an acute angle 
. Let 
on and 
on be points such that 
, and let be the line through parallel to . Let 
be the circle with centre and radius 
. We construct a sequence of half-lines 
, all lying inside the angle , in the following manner:(i)
is given arbitrarily;(ii) for every natural number
, the circle intercepts on 
a segment that is of the same length as the segment cut on 
by and 
.Prove that the angle determined by the lines
and has a limit as tends to infinity and find that limit.
