AoPS Academy
such that
is not a polynomial .
for all 
, 
.
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.
be a positive real number. Find all functions 
such that 
and![\[
f(af(x) - f(y) + y) \leq x + f(y) - y, \quad \forall x, y \in \mathbb{R}^+.
\]](http://latex.artofproblemsolving.com/e/2/9/e296b71ad8b8bd977e2bb2a3630a5bc020b8e03a.png)
. 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)
we construct the equilateral triangles 
Prove that the midpoints of the four segments 
and the midpoints of the eight segments 
are the 12 vertices of a regular dodecagon.
, 
, 
and 
.
, 
, 
and 
.
, with 
, and the result follows.
where is the midpoint of . and that the angles at are correct.
is equilateral where 
is the midpoint of 
and 
is the midpoint of 
.
and 
, we see that 
and 
. On the other hand, in the square 
we clearly have 
. But by construction, the triangles 
are all congruent whence 
and hence 
as desired.![[asy]
import olympiad;
size(6cm);
defaultpen(fontsize(10pt));
defaultpen(linewidth(0.4));
dotfactor *= 1.5;
pair A = dir(225), B = dir(315), C = dir(45), D=dir(135), K=extension(A,dir(75),B,dir(105)), L=extension(B,dir(165), C,dir(195)), M=-K, N=-L, E=(K+L)/2, F=(A+K)/2, O=0;
draw(A--B--C--D--A);
draw(B--K--A, dotted);
draw(B--L--C, dotted);
draw(K--L, dashed);
draw(E--F--O--E, dashed);
draw(A--L, dashed);
draw(C--K, dashed);
dot("$A$", A, dir(225));
dot("$B$", B, dir(315));
dot("$C$", C, dir(45));
dot("$D$", D, dir(135));
dot("$K$", K, dir(90));
dot("$L$", L, dir(180));
dot("$M$", M, dir(270));
dot("$N$", N, dir(0));
dot("$O$", O, dir(0));
dot("$E$", E, dir(135));
dot("$F$", F, dir(315));
dot((M+L)/2);
dot((M+N)/2);
dot((N+K)/2);
dot((B+K)/2);
dot((B+L)/2);
dot((C+L)/2);
dot((C+M)/2);
dot((D+M)/2);
dot((D+N)/2);
dot((A+N)/2);
[/asy]](http://latex.artofproblemsolving.com/4/3/b/43bcbb2df29f0d306536032cbadd91c1884688e9.png)
