[/quote]
,
be an acute triangle with orthocenter 
.
be the point such that the quadrilateral 
is a parallelogram. Let 
be the point on the line 
such that 
bisects 
.
at 
and 
.
.
be a non-cyclic quadrilateral. Let 
and 
be the circumcentre and the circumradius of the triangle 
.
and 
in a similar way. Prove that![\[\frac{1}{O_1A_1^2-r_1^2}+\frac{1}{O_2A_2^2-r_2^2}+\frac{1}{O_3A_3^2-r_3^2}+\frac{1}{O_4A_4^2-r_4^2}=0.\]](http://latex.artofproblemsolving.com/f/f/5/ff575c3012e0b86c5aeeec32aae5268a4dad74d2.png)
Prove that
,
is rational for 
.
is called a rational point in 
-
.
and 
in 
and 
as 
.
,
,
be reals such that 
.Prove that
Where 
the equation 
and integers 
,
satisfying 
for 
(with the convention 
)![\[
\sum_{j=1}^n a_j^{2m-1} \leq \frac{mt}{2} \left( \sum_{j=1}^n a_j^{m-1} \right)^2.
\]](http://latex.artofproblemsolving.com/5/c/d/5cd52e9ce307563bde96f697a75079229d4bfcc2.png)
.
intersects side 
and 
and 
(other than 
be the midpoint of side 
intersects sides 
and 
and 
(other than 
be the midpoint of segment 
,
.
is tangent to the circumcircle of triangle 
.
prove that
and incircle 
.
and 
and 
respectively. Let 
and 
be points outside 
satisfying ![\[\angle BDX = \angle XEA = \angle YDC = \angle AFY = 45^{\circ}.\]](http://latex.artofproblemsolving.com/4/7/0/470b0486fa1342c4b6444ec1e3731c23bbb842fb.png)
Prove that the circumcircles of 
and 
.
such that 
be a fixed integer. The cells of an 
table are filled with the integers from 
to 
with each number appearing exactly once. Let 
of 
touch 
,
denote the line tangent to 
.
such that 
.
of 
.
Y by GeoKing, itslumi
On 
squares, we cover the area with several S-Tetrominos (=Z-Tetrominos) along the square so that in every square, there are two or fewer tiles covering that (tiles can be overlap). Find the maximum possible number of squares covered by at least one tile.
squares, we cover the area with several S-Tetrominos (=Z-Tetrominos) along the square so that in every square, there are two or fewer tiles covering that (tiles can be overlap). Find the maximum possible number of squares covered by at least one tile.This post has been edited 1 time. Last edited by Kei0923, Feb 11, 2023, 11:22 AM
tetronominos such that each but the central square are covered.
is not possible, consider the following coloring:
is the lower left corner, colour all 
fields with both coordinates even red and all 
fields with both coordinates odd blue.
tetrominos but then it is impossible to cover all 
![[asy]
draw((0,0)--(5,0)--(5,5)--(0,5)--(0,0));
draw((1,0)--(1,5));
draw((2,0)--(2,5));
draw((3,0)--(3,5));
draw((4,0)--(4,5));
draw((0,1)--(5,1));
draw((0,2)--(5,2));
draw((0,3)--(5,3));
draw((0,4)--(5,4));
fill((1,1)--(2,1)--(2,2)--(1,2)--cycle, black);
fill((1,3)--(2,3)--(2,4)--(1,4)--cycle, black);
fill((3,3)--(4,3)--(4,4)--(3,4)--cycle, black);
fill((3,1)--(4,1)--(4,2)--(3,2)--cycle, black);
[/asy]](http://latex.artofproblemsolving.com/e/d/5/ed523383a151dc7858c31be9aba9dc63ec0c6c22.png)
![[asy]
draw((0,0)--(5,0)--(5,5)--(0,5)--(0,0));
draw((1,0)--(1,5));
draw((2,0)--(2,5));
draw((3,0)--(3,5));
draw((4,0)--(4,5));
draw((0,1)--(5,1));
draw((0,2)--(5,2));
draw((0,3)--(5,3));
draw((0,4)--(5,4));
fill((0,0)--(1,0)--(1,1)--(0,1)--cycle, blue);
fill((0,2)--(0,3)--(1,3)--(1,2)--cycle, blue);
fill((0,4)--(0,5)--(1,5)--(1,4)--cycle, blue);
fill((2,0)--(3,0)--(3,1)--(2,1)--cycle, blue);
fill((2,2)--(3,2)--(3,3)--(2,3)--cycle, blue);
fill((2,4)--(3,4)--(3,5)--(2,5)--cycle, blue);
fill((4,0)--(5,0)--(5,1)--(4,1)--cycle, blue);
fill((4,2)--(5,2)--(5,3)--(4,3)--cycle, blue);
fill((4,4)--(4,5)--(5,5)--(5,4)--cycle, blue);
[/asy]](http://latex.artofproblemsolving.com/0/2/2/022e835a6bf0ebc5991a2b7de3fc605524d7020d.png)
![[asy]
draw((0,0)--(5,0)--(5,5)--(0,5)--(0,0));
draw((1,0)--(1,1)--(0,1));
draw((0,3)--(1,3)--(1,2)--(2,2)--(2,0));
draw((2,2)--(2,4)--(1,4)--(1,5));
draw((5,0)--(4,0)--(4,1)--(3,1)--(3,5));
draw((4,5)--(4,4)--(5,4));
draw((3,3)--(4,3)--(4,2)--(5,2));
[/asy]](http://latex.artofproblemsolving.com/d/5/d/d5dd0831f1b90f124251bbc4963d66c765187315.png)
![[asy]
draw((0,0)--(5,0)--(5,5)--(0,5)--(0,0));
draw((0,1)--(1,1)--(1,2)--(3,2)--(3,1)--(2,1)--(2,0));
draw((3,2)--(5,2));
draw((5,1)--(4,1)--(4,0));
draw((1,5)--(1,4)--(0,4));
draw((0,3)--(2,3)--(2,4)--(3,4)--(3,5));
draw((2,3)--(4,3)--(4,4)--(5,4));
[/asy]](http://latex.artofproblemsolving.com/9/8/6/9861d75ee61886019065bbbb921043ecfc4f257c.png)
rectangle that has two corners missing
