[/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 
be a finite set of cells in a square grid. On each cell of 
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
Three circles were drawn on the plane, any two of which intersect at two points. Then all the intersection points were marked, and the circles themselves were erased.
1. Let
points be marked. Can they be the vertices of a regular hexagon?
2. Let
points be marked, and they lie at the vertices of a rhombus with angles 
and 
. The radius of the smaller circle is 
. What are the radii of the other two?
3. Let points be marked. Can different sets of circles correspond to them?
4. Suppose there were not three, but
circles, and 
points were marked. Prove that the original circles can be recovered uniquely.
1. Let
points be marked. Can they be the vertices of a regular hexagon?2. Let
points be marked, and they lie at the vertices of a rhombus with angles 
and 
. The radius of the smaller circle is 
. What are the radii of the other two?3. Let
points be marked. Can different sets of circles correspond to them?4. Suppose there were not three, but
circles, and 
points were marked. Prove that the original circles can be recovered uniquely.This post has been edited 1 time. Last edited by parmenides51, Feb 28, 2021, 12:47 PM
