AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
. Find all functions 
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)
be an acute triangle with 
and orthocenter 
. Let 
and 
be the feet of the altitudes from 
and 
onto sides 
and 
, respectively. Let 
be the midpoint of 
, and 
be the midpoint of 
. Let the perpendicular line through to 
meet at 
and at 
. The line 
meets at 
and at 
. Let 
be the second intersection point (different from ) of the circumcircles of triangles 
and 
. Prove that the points , , , , and lie on the same circle.
, with 
as its incenter and 
as its circumcircle, 
intersects again at 
. Let 
be a point on the arc 
, and 
a point on the segment , such that 
. If 
is the midpoint of 
, prove that the meeting point of the lines 
and 
lies on .
and 
be the incircle and circumcircle of the acute triangle , respectively. Draw a square 
so that all of its sides are tangent to , and 
, 
are both on . Extend 
and 
, intersecting at and 
, respectively. Prove that 
and 
intersects on ., define:
as its circumcentre, as its orthocentre and 
as its nine-point centre.
, 
and 
to be the midpoint of the smaller arcs. as the isogonal conjugate of the Nagel point (i.e. the exsimillicenter of the incircle and circumcircle) as the ninja point of 
as the ninja point of the contact triangle
Points , and are collinear, that is is the Feuerbach point.
Points , and are collinear
be a triangle and let be the contact point of the 
-mixtilinear incircle with the circumcircle, and let 
be the reflection of over . Prove that the nine-point circle of 
is tangent to the incircle.
be an odd prime number. Suppose and are polynomials with integer coefficients such that 
, there is no nonconstant polynomial dividing both and , and![\[
1 + \cfrac{x}{1 + \cfrac{2x}{1 + \cfrac{\ddots}{1 +
(p-1)x}}}=\frac{P(x)}{Q(x)}.
\]](http://latex.artofproblemsolving.com/b/c/a/bca9034e2affa1d80b167451d77434fbcd8bd88a.png)
Show that all coefficients of except for the constant coefficient are divisible by , and all coefficients of are not divisible by .
, 
and 
, a triangle can be formed. Could it happen that from segments of lengths 
a right-angled triangle can be formed?
be the circumcenter of a triangle 
The line through perpendicular to 
meets the lines 
and 
at and 
respectively. Prove that![\[\frac{\overline{CP}}{\overline{CM}} \cdot \frac{\overline{CQ}}{\overline{CM}}\cdot \frac{\overline{AB}}{\overline{PQ}}= 2.\]](http://latex.artofproblemsolving.com/6/6/f/66faf5e7b09391fce74d137b8eeb07c83d3ea242.png)
is the foot of the C-altitude. 
implies that 
is a diameter, perpendicular to , i.e. the inversion 
sends the circle to the line , hence 
, where 
.
and 
Prove that
Prove that
and 
Prove that
