touch sides 
and 
at 
and 
respectively. Let 
and 
denote the circumcircles of triangle 
and 
respectively.
and 
intersect at 
and 
and 
intersect at 
and 
and 
and 
Prove that 
and 
Show that 
and 
are concurrent.
, such that for all 
:![\[ f(x+f(y))=\dfrac{f(x)}{1+f(xy)}\]](http://latex.artofproblemsolving.com/1/4/1/1418fad9fd00b31c38bfbc5657e2a322d66ef450.png)
such that, for all positive integers 
and 
,
,
.
be a trapezoid inscribed in circle 
,
.
is the symmetric point of 
,
intersects 
.
is midpoint of 
is orthocenter of triangle 
.
take a point 
so that 
.
coincide
Y by ImSh95
A unit square is removed from the corner of an 
grid, where 
. Prove that the remainder can be covered by copies of the figures of 
or 
unit squares depicted in the drawing below.
![[asy]
import geometry;
draw((-1.5,0)--(-3.5,0)--(-3.5,2)--(-2.5,2)--(-2.5,1)--(-1.5,1)--cycle);
draw((-3.5,1)--(-2.5,1)--(-2.5,0));
draw((0.5,0)--(0.5,3)--(1.5,3)--(1.5,1)--(3.5,1)--(3.5,0)--cycle);
draw((1.5,0)--(1.5,1));
draw((2.5,0)--(2.5,1));
draw((0.5,1)--(1.5,1));
draw((0.5,2)--(1.5,2));
[/asy]](//latex.artofproblemsolving.com/5/7/8/578baca60367441fe9b771689f1a2e91c0ccf984.png)
Note: Every square must be covered once and figures must not go over the bounds of the grid.
grid, where 
. Prove that the remainder can be covered by copies of the figures of 
or 
unit squares depicted in the drawing below.![[asy]
import geometry;
draw((-1.5,0)--(-3.5,0)--(-3.5,2)--(-2.5,2)--(-2.5,1)--(-1.5,1)--cycle);
draw((-3.5,1)--(-2.5,1)--(-2.5,0));
draw((0.5,0)--(0.5,3)--(1.5,3)--(1.5,1)--(3.5,1)--(3.5,0)--cycle);
draw((1.5,0)--(1.5,1));
draw((2.5,0)--(2.5,1));
draw((0.5,1)--(1.5,1));
draw((0.5,2)--(1.5,2));
[/asy]](http://latex.artofproblemsolving.com/5/7/8/578baca60367441fe9b771689f1a2e91c0ccf984.png)
Note: Every square must be covered once and figures must not go over the bounds of the grid.
This post has been edited 2 times. Last edited by Rukevwe, May 6, 2022, 2:54 PM
rectangular, so 
3-unit figure can make 
rectangular.
have been proved. Then when 
,
.
exists, 
with a unit removed from a corner can be made as below.
or 
,
does exist. So we only need to prove whe
.
is trivial.
:
:
:
and 
.
only needs one case: the top-right 
can be reduced to 
to 
we split into three cases, also note that one can build a 
then build a big L shape on the bottom-left and use 
it's the same method but start with a middle L shape
then it's the same method but start with the 
