[/quote]
,
Prove that
be the incentre of acute triangle 
.
of 
,
and 
,
perpendicular to 
meets 
meets 
.
and 
meet again at 
.
and 
meet on the line through 
perpendicular to 
.
,
be positive real numbers, such that 
.![\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]](http://latex.artofproblemsolving.com/b/e/5/be5819a67c3cd78f2dea35fdccf48688c720ce3c.png)
When does the equality hold?
,
![$f : \mathbb{Q}[\sqrt{2}] \to \mathbb{Q}[\sqrt{2}]$](http://latex.artofproblemsolving.com/c/a/1/ca172b912bf5bf7178cbbd5745fbc0a6a8b27503.png)
such that for all ![$x, y \in \mathbb{Q}[\sqrt{2}]$](http://latex.artofproblemsolving.com/9/f/e/9fede0b2fd22836372b8afc488ccd8a14db5d0c2.png)
,
where ![$\mathbb{Q}[\sqrt{2}] = \{ a + b\sqrt{2} \mid a, b \in \mathbb{Q} \}$](http://latex.artofproblemsolving.com/f/a/c/fac8bf68761534a139d570bf66b1aa5dfb59e8db.png)
.
,
is called interesting if the following two conditions hold:
such that 
.
(where 
denotes the 
-
)
.
for which there is no interesting function
such that the following two conditions hold:
and 
satisfying 
,
for all real numbers 
and 
,
.
and 
Find the value of 
that satisfy![\[f(xf(y) + f(x)) = f(x)f(y) + 2f(x) + f(y) - 1,\]](http://latex.artofproblemsolving.com/4/7/f/47f094ce57fe35d339ec3b93152ec3459a22ec48.png)
for every 
,
for every 
and 
,
.
such that for every real number 
,
is a perfect square if and only if 
is a perfect square. Prove that 
is the square of a polynomial with real coefficients.
Prove that
be an infinite sequence of real numbers, for which there exists a real number 
for all 
,![\[\left\lvert a_i-a_j \right\rvert\geq \frac{1}{i+j} \quad \text{for all }i,\ j \text{ with } i \neq j. \]](http://latex.artofproblemsolving.com/a/4/d/a4d87553590a450365b6525ca62a5d0416fe41cc.png)
Prove that 
.
Y by jhu08, PHSH
Let 
be the center of the equilateral triangle 
. Pick two points 
and 
other than 
, , 
on the circle 
so that on this circle , , , , are placed in this order. Extensions of 
and 
intersects respectively with side 
and 
at points 
and 
. Line 
and 
intersects at point 
. Analogously point 
is defined. Let 
and 
meet again at point 
other than .
Prove that
.
Remark.
denotes the circumcircle of triangle 
.
be the center of the equilateral triangle 
. Pick two points 
and 
other than 
, , 
on the circle 
so that on this circle , , , , are placed in this order. Extensions of 
and 
intersects respectively with side 
and 
at points 
and 
. Line 
and 
intersects at point 
. Analogously point 
is defined. Let 
and 
meet again at point 
other than .Prove that
.Remark.
denotes the circumcircle of triangle 
.This post has been edited 2 times. Last edited by Li4, Mar 29, 2020, 3:17 AM
Reason: typo
Reason: typo
lies on 
and 
so 
and therefore 
is the midpoint of arc 
in 
Let 
be the intersection of 
then 
so note that 
is the bisector of 
we get 
and 
hence 
lie on a circle 
with center 
.
.
and 
.
intersect 
again at 
.
.
wrt. 
and 
(Since 
)
.
.
and 
.![[asy]
size(300);
pair A,B,C,O,R,S,O_1,Q_1,P_0,Q;
A=dir(90);
B=dir(210);
C=dir(330);
O=circumcenter(A,B,C);
P_0=0.7*B+0.3*C;
path ooo = circumcircle(B,O,C);
pair P = IP(circumcircle(B,O,C), A--P_0);
dot("$P_1$",P,dir(270));
draw(unitcircle,blue);
dot("$A$", A, dir(A));
dot("$B$", B, dir(B));
dot("$C$", C, dir(350));
dot("$O$", O, dir(320));
pair T = circumcenter(A,O,P);
R = extension(B,P,A,C);
S = extension(C,P,A,B);
Q=foot(T,B,C);
Q_1=extension(A,P,T,Q);
dot("$R$",R,dir(30));
dot("$S$",S,dir(200));
draw(circumcircle(A,O,P),heavycyan);
dot("$O_1$",T,dir(230));
draw(A--B--C--cycle,darkcyan);
draw(C--S,heavymagenta);
draw(B--R,heavymagenta);
dot("$Q_1$",Q_1,dir(180));
draw(A--T--O,red);
draw(T--Q_1,orange);
draw(Q_1--O,orange);
draw(A--P,deepblue);
draw(A--O,fuchsia+dashed);
draw(Q--Q_1,fuchsia+dashed);
pair M_0=dir(270);
draw(arc(M_0,length(C-M_0),20,160),mediumblue);
[/asy]](http://latex.artofproblemsolving.com/9/1/2/912341617ff44de2140945de7a69b419e28d86bd.png)
lie on a circle. Note that 
.
.
.
.
.
.
,
.
,
.
.
at 
.
we have 
with transversal 
we get : 
.
,
bisects 
.
.
of 
,
and summing, we are done. 
.
(then that would analogously imply 
and we would be done).
lie on some circle 
so 
.
.![[asy]
size(200);
pair A=dir(90),B=dir(-150),C=dir(-30),O=(0,0),P1=waypoint(circumcircle(B,O,C),0.3),R=extension(B,P1,A,C),S=extension(C,P1,A,B),Q1=extension(R,S,A,P1),L=extension(A,O,R,S),T=2*foot(A,circumcenter(A,R,S),(0,0))-A;
dot("$A$",A,dir(A));
dot("$B$",B,dir(B));
dot("$C$",C,dir(C));
dot("$O$",O,dir(-70));
dot("$P_1$",P1,dir(-90));
dot("$R$",R,dir(R));
dot("$S$",S,dir(S));
dot("$Q_1$",Q1,dir(160));
dot("$L$",L,dir(70));
dot("$T$",T,dir(T));
draw(CP(circumcenter(B,O,C),B,15,165)^^unitcircle,red);
draw(circumcircle(A,R,S),green);
draw(A--B--C--A^^B--R^^C--S^^O--A--P1^^R--S^^O--T,magenta);
[/asy]](http://latex.artofproblemsolving.com/3/e/1/3e1106b430ae5bcdf73b8cf7e542cd3188981dcd.png)
.
.
.
which implies our claim (as 
)
and 
coincide).
.
is the internal angle bisector of 
.
.
wrt points 
we obtain
This proves our claim. 
is the internal angle bisector of 
of 
then by Claim 3 we obtain 
is the reflection of 
.
This completes the proof of the problem. 
