[/quote]
,
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.
and 
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 
and 
the perpendiculars in 
respectively, where 
,
and 
.
be the circumcircles of triangles 
.
and 
such that 
and 
.
is a perfect square.
.
that satisfies![\[ f(x+yf(x))f(y)=f(x)(1+yf(y+1)) \]](http://latex.artofproblemsolving.com/f/7/a/f7a138fd2dc76f034525491a10f62f1e83de294d.png)
for all 
.
be positive integers such that 
is a multiple of 
for all positive integers 
.
with 
is called historic if 
.
satisfying 
for all pairs of positive reals 
and 
.
denotes the set of all positive reals.
Y by
Let 
be a triangle. Let 
and 
be the intersections of the 
and 
angle bisectors with the opposite sides. Let 
. Let 
where 
is the major 
arc midpiont.
i)Show that points
and are coconic on a conic 
ii) If intersects 
again at 
, not equal to 
or 
, then prove 
and 
concur on 
I will post solution in ~1 week if noone solves.
be a triangle. Let 
and 
be the intersections of the 
and 
angle bisectors with the opposite sides. Let 
. Let 
where 
is the major 
arc midpiont.i)Show that points
and are coconic on a conic 
ii) If
intersects 
again at 
, not equal to 
or 
, then prove 
and 
concur on 
I will post solution in ~1 week if noone solves.
This post has been edited 1 time. Last edited by TestX01, Feb 25, 2025, 11:35 PM
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