[/quote]
,
be a triangle with incentre 
and circumcircle 
.
be the midpoints of the arcs 
of 
respectively. Let 
be the point of 
.
and the midpoint 
of 
lie on a line.
is a permutation of 
.
such that Alice can always achieve her goal in at most 
with real coefficients of degree 
such that 
divides 
.
Prove that
such that:
.
-
.
so that any 
.
satisfying the following:
;
and 
;
;
has all of its coefficients being even.
and 
Prove that
points on the plane with diameter 
and two not necessarily distinct points 
not equal to 
or 
satisfying that![\[|AX-XY|+|BY-XY|\leq \frac{2D}{N-2}.\]](http://latex.artofproblemsolving.com/f/3/6/f3649872c33e1db8880b4aedace38d9439f2b4ae.png)
.
such that for all 
,![\[
f(xf(x+y)+z) = f(z) + f(x)y + f(xf(x)).
\]](http://latex.artofproblemsolving.com/f/9/3/f93b7526ff4338eea89a1c59bc009aa001ffc15d.png)
![$ a,b \in [0 ,1] . $](http://latex.artofproblemsolving.com/9/1/9/91980d5ca5d97fe4b61797e6fbb378211674d9cd.png)
Prove that
such that 
and 
then 
and 
then find the value of 
where 
Y by Adventure10, Mango247
In the triangle 
the point 
is the center of the inscribed circle. On rays 
and 
, points 
and 
respectively are taken for and such that 
Let 
be the midpoint of the segment 
. Prove that the lines 
and 
are perpendicular.
the point 
is the center of the inscribed circle. On rays 
and 
, points 
and 
respectively are taken for and such that 
Let 
be the midpoint of the segment 
. Prove that the lines 
and 
are perpendicular.This post has been edited 1 time. Last edited by parmenides51, Jul 17, 2019, 2:11 PM
Reason: name typo
Reason: name typo
be foot from 
at 
.
and 
.
.
,
be foot from 
.
be the foot from 
and 
be the foot from 
and let the foot from 
.
,
,
.
.
.
.
,
