[/quote]
,
and 
Prove that
.
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)
such that for all 
,![\[xf(x+f(y))=(y-x)f(f(x)).\]](http://latex.artofproblemsolving.com/e/4/a/e4a3bbb8b91d2aa62d699c24df342fa59be71915.png)
Prove that
be an acute triangle with 
and orthocenter 
.
and 
be the feet of the altitudes from 
and 
onto sides 
and 
,
be the midpoint of 
be the midpoint of 
.
meet 
and 
.
meets 
and 
.
and 
.
and 
Prove that
as its incenter and 
as its circumcircle, 
intersects 
.
be a point on the arc 
,
a point on the segment 
.
is the midpoint of 
,
and 
lies on 
and 
be the incircle and circumcircle of the acute triangle 
so that all of its sides are tangent to 
,
are both on 
and 
,
and 
intersects on 
as its nine-point centre.
,
and 
to be the midpoint of the smaller arcs.
Points 
Points 
-
be the reflection of 
is tangent to the incircle.
be an odd prime number. Suppose 
,![\[
1 + \cfrac{x}{1 + \cfrac{2x}{1 + \cfrac{\ddots}{1 +
(p-1)x}}}=\frac{P(x)}{Q(x)}.
\]](http://latex.artofproblemsolving.com/b/c/a/bca9034e2affa1d80b167451d77434fbcd8bd88a.png)
Show that all coefficients of 
,
and 
,
a right-angled triangle can be formed?
Y by
Given a parallelogram 
with center at the point 
. The circle 
intersects the sides 
and 
at points 
and 
, respectively. The circles 
and 
intersect for the second time at the point 
. Prove that the orthocenter of the triangle 
lies on the perpendicular drawn from the point on the line .
(M. Plotnikov)
with center at the point 
. The circle 
intersects the sides 
and 
at points 
and 
, respectively. The circles 
and 
intersect for the second time at the point 
. Prove that the orthocenter of the triangle 
lies on the perpendicular drawn from the point on the line .(M. Plotnikov)
