[/quote]
,
and 
such that 
and 
is a prime number
and 
.
Let 
.
Prove that
with incentre 
and excentres 
and 
,
and 
respectively. The incircle touches 
and 
at 
and 
respectively. Prove that the circles 
and 
have a common point other than 
such that:
;
.
and 
such that 
.
are positive isn't necessary.
such that 
is a perfect square.
and 
be non-zero integers such that 
Prove that ![\[(a_1^2-1)(a_2^2-1)\cdots(a_n^2-1)\]](http://latex.artofproblemsolving.com/b/f/f/bffd77e507fc4ddc8368c8239c523a4df40448c4.png)
is a perfect square.
be positive real numbers.Prove that![$$\sqrt[3]{ab}+\sqrt[3]{cd}\leq\sqrt[3]{(a+b+c)(c+d+a)}.$$](http://latex.artofproblemsolving.com/b/7/9/b796d0f9a2d9430bf5efb83b901427b13386f478.png)
When equality holds?
be an integer. Ion draws a regular 
safisty: 
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
