[/quote]
,
be the set of positive real numbers.
such that there exists a function 
with the property:![\[
f(y^2f(x) + y + c) = xf(x+y^2)
\]](http://latex.artofproblemsolving.com/e/3/0/e30ee6d9f69cc43422a5c2b9813192a632d359cb.png)
for all 
.
and 
then prove 
be non-negative real numbers such that 
Prove that
and 
.
,![\[ a^{b+c} b^{c+a} c^{a+b} \leq 1 . \]](http://latex.artofproblemsolving.com/0/c/e/0ce097b0b507403c959c2992cae1217ecdd7e581.png)
be a finite sequence of positive integers. Prove that there exist nonnegative integers 
and 
such that 
holds for all integers 
be a positive integer. Dominoes are placed on a 
board in such a way that every cell of the board is adjacent to exactly one cell covered by a domino. For each 
or 
.
of positive real numbers that satisfy the system:![\[
\begin{aligned}
11bc - 36b - 15c &= abc \\
12ca - 10c - 28a &= abc \\
13ab - 21a - 6b &= abc.
\end{aligned}
\]](http://latex.artofproblemsolving.com/1/7/0/170e1cfaee90f5c37428fa75192ac889261a8f83.png)
be positive reals. Prove that there is a cyclic permutation 
of 
such that the inequality:![\[
\frac{a}{xa + yb + zc} + \frac{b}{xb + yc + za} + \frac{c}{xc + ya + zb} \geq \frac{3}{x + y + z}
\]](http://latex.artofproblemsolving.com/5/5/3/553527fc40a1b7d29dbdfb4b5e63635308f46c85.png)
and 
.
.
,
,
.
with diameter 
,
at a point 
.
at 
and 
intersect at a point 
on 
.
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of 
is the sum of 
and let 
,
such that 
,
,
.
intersects 
other than 
and 
meet on 
Y by Mango247
Let 
be a triangle with all angles acute, with 
. Let 
be the circle circumscribed to , and 
the midpoint of the arc 
not containing 
. Let 
and 
be points belonging to segments ![$[AB]$](//latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
and ![$[AC]$](//latex.artofproblemsolving.com/0/9/3/0936990e6625d65357ca51006c08c9fe3e04ba0c.png)
respectively, so that 
. Let 
be the point of intersection, other than , between and the circle circumscribed to 
. Let 
and 
be the respective points of intersection, other than , between and the lines 
and 
. Finally, let 
be the point of intersection between lines 
and 
, and let 
be the point of intersection between lines 
and 
. Prove that the midpoint of the segment ![$[BC]$](//latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
belongs to the line 
.
be a triangle with all angles acute, with 
. Let 
be the circle circumscribed to , and 
the midpoint of the arc 
not containing 
. Let 
and 
be points belonging to segments ![$[AB]$](http://latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
and ![$[AC]$](http://latex.artofproblemsolving.com/0/9/3/0936990e6625d65357ca51006c08c9fe3e04ba0c.png)
respectively, so that 
. Let 
be the point of intersection, other than , between and the circle circumscribed to 
. Let 
and 
be the respective points of intersection, other than , between and the lines 
and 
. Finally, let 
be the point of intersection between lines 
and 
, and let 
be the point of intersection between lines 
and 
. Prove that the midpoint of the segment ![$[BC]$](http://latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
belongs to the line 
.This post has been edited 1 time. Last edited by parmenides51, Sep 28, 2020, 1:13 PM
![[asy]
size(10cm);
pair A = dir(110);
pair B = dir(210);
pair C = dir(330);
pair D = dir(270);
draw(unitcircle);
draw(A--B--C--A--cycle);
pair E = 0.6*B+0.4*A;
pair I = incenter(A,B,C);
pair F = 2*foot(E,A,I)-E;
draw(circumcircle(A,E,F));
pair P = intersectionpoints(circumcircle(A,E,F),unitcircle)[1];
pair G = intersectionpoints(P--E+10*(E-P),unitcircle)[1];
pair H = intersectionpoints(P--F+10*(F-P),unitcircle)[1];
pair J = foot(D,A,B);
pair K = foot(D,A,C);
draw(P--G);
draw(P--H);
draw(B--J--D--H);
pair M = intersectionpoints(circumcircle(A,E,F),A--D)[1];
draw(A--D);
dot("$A$",A,dir(A));
dot("$B$",B,dir(B));
dot("$C$",C,dir(C));
dot("$D$",D,dir(D));
dot("$E$",E,dir(E));
dot("$F$",F,dir(F));
dot("$G$",G,dir(G));
dot("$H$",H,dir(H));
dot("$J$",J,dir(J));
dot("$K$",K,dir(K));
dot("$P$",P,dir(P));
dot("$M$",M,dir(M));
[/asy]](http://latex.artofproblemsolving.com/4/4/3/443559a1f0f9de7b2a0c6e20088304502f7c5a0b.png)
.
.
Note that 
is the angle bisector of 
,
.
,
.
.
.
to 
(which in turn implies that 
!
and 
.
