[/quote]
,
be a right-angled triangle where 
=
be an altitude of that triangle and points 
and 
be the midpoints of 
,
is the circumcenter of the triangle 
,
and 
,
be a point 
)
be the anticomplement of 
be the 
-
is parallel to the real asymptote of the circular pivotal isocubic with pivot 
and 
.
such that 
.
is a power of a prime number, but for arbitrary 
be a positive integer. Define a sequence by setting 
and, for each 
,
be the unique integer in the range 
for which 
is divisible by 
.
the obtained sequence is 
.
eventually becomes constant.
lie on sides 
,
.
meet at orthocenter 
.
such that 
and 
.
and 
intersect at point 
.
.
.
.
over 
intersects lines 
and 
and 
and 
.
lies on 
.
such that 
prove that the triangle must have a right angle.
such that for any real numbers 
and 
,![\[
f\left(x^2+2y f(x)\right) + (f(y))^2 \leq (f(x+y))^2
\]](http://latex.artofproblemsolving.com/8/4/3/8437449cd3f8f0da24d17a60b55dcb563e6b5655.png)
defined by 
and![\[
u_{n+1} = \frac{1}{2}u_n^2 - 4 \quad \text{for all } n \in \mathbb{N}.
\]](http://latex.artofproblemsolving.com/9/9/4/994aa754cc1288ce4f28a95a0276e64282fb5f66.png)
a) Prove that there exist infinitely many positive integers 
such that 
.![\[
\lim_{n \to \infty} \frac{2u_{n+1}}{u_0u_1\cdots u_n}.
\]](http://latex.artofproblemsolving.com/f/f/1/ff174e7431cbfbc17c650d109651241286756a1a.png)
is a cyclic quadrilateral. Let 
and 
.
and 
the orthocenters of triangles 
and 
,
,
be a triangle. A circle passing through 
and 
intersects the sides 
and 
again at 
and 
respectively. Prove that 
,
and 
are concurrent, where 
and 
are the orthocentres of triangles 
respectively.
.
and 
=
.
is the foot of the altitude from 
in triangle 
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
