[/quote]
,
be the midpoint of side 
in 
,
is such that the quadrilateral 
is cyclic and the circumcircle of 
is tangent to the line 
.
is the second common point of the line 
and the circumcircle of 
.
be an isosceles triangle with 
,
be its incenter. Incircle touches sides 
at 
,
to 
, respectively. Rays 
intersect 
at 
,
touches incircle at 
.
from the positive integers to the positive integers is called Canadian if it satisfies 
for all pairs of positive integers 
and 
.
such that 
for all Canadian functions 
.
on 
respectively. Circumcircle 
of triangle 
intersects 
.
intersects 
at 
cuts 
respectively. Construct parallelogram 
.
are collinear.
-
and 
such that 
.
there exists a positive integer 
have at least one block of 
in their decimal representation.
be real numbers such that 
.
such that for every 
is satisfied? Explain the answer.
,
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?
,
,
denotes the digit sum of 
,such that 
P
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 
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 
