(use in a problem for junior) to practice. Can anyone share to me?
be an odd integer, 
.
integers 
such that for any 
,
and 
.
for which satisfy 
for some 
,
is greater than 
.
such that![\[f(x+yf(x))+f(xf(y)-y)=f(x)-f(y)+2xy\]](http://latex.artofproblemsolving.com/e/b/c/ebccae7e765d9fc3c8605e58a7399a70127b39b1.png)
, such that 
Prove that :
equality cases : ?
Y by Adventure10
In quadrilateral 
sides 
and 
are parallel. In each of the four vertices we draw an angular bisector. The angular bisectors of angles 
and 
intersect in point 
, those of angles and 
intersect in point 
, those of angles and 
intersect in point 
, and those of angles and intersect in point S. Suppose that 
is parallel to 
. Prove that 
.
![[asy]
unitsize(1.2 cm);
pair A, B, C, D, P, Q, R, S;
A = (0,0);
D = (3,0);
B = (0.8,1.5);
C = (3.2,1.5);
S = extension(A, incenter(A,B,D), D, incenter(A,C,D));
Q = extension(B, incenter(A,B,C), C, C + incenter(A,B,D) - A);
P = extension(A, S, B, Q);
R = extension(D, S, C, Q);
draw(A--D--C--B--cycle);
draw(B--Q--C);
draw(A--S--D);
dot("$A$", A, SW);
dot("$B$", B, NW);
dot("$C$", C, NE);
dot("$D$", D, SE);
dot("$P$", P, dir(90));
dot("$Q$", Q, dir(270));
dot("$R$", R, dir(90));
dot("$S$", S, dir(90));
[/asy]](//latex.artofproblemsolving.com/a/7/d/a7d4a098c39d06d4bfd497e2c522d54d30b9ff26.png)
Attention: the figure is not drawn to scale.
sides 
and 
are parallel. In each of the four vertices we draw an angular bisector. The angular bisectors of angles 
and 
intersect in point 
, those of angles and 
intersect in point 
, those of angles and 
intersect in point 
, and those of angles and intersect in point S. Suppose that 
is parallel to 
. Prove that 
.![[asy]
unitsize(1.2 cm);
pair A, B, C, D, P, Q, R, S;
A = (0,0);
D = (3,0);
B = (0.8,1.5);
C = (3.2,1.5);
S = extension(A, incenter(A,B,D), D, incenter(A,C,D));
Q = extension(B, incenter(A,B,C), C, C + incenter(A,B,D) - A);
P = extension(A, S, B, Q);
R = extension(D, S, C, Q);
draw(A--D--C--B--cycle);
draw(B--Q--C);
draw(A--S--D);
dot("$A$", A, SW);
dot("$B$", B, NW);
dot("$C$", C, NE);
dot("$D$", D, SE);
dot("$P$", P, dir(90));
dot("$Q$", Q, dir(270));
dot("$R$", R, dir(90));
dot("$S$", S, dir(90));
[/asy]](http://latex.artofproblemsolving.com/a/7/d/a7d4a098c39d06d4bfd497e2c522d54d30b9ff26.png)
Attention: the figure is not drawn to scale.
This post has been edited 2 times. Last edited by nsato, Feb 14, 2023, 12:22 AM
