[/quote]
,
satisfying: 
,
and the function 
defined by![\[f(n)=\left\lfloor \frac{an+b}{cn+d}\right\rfloor\text{ for all } n\in\mathbb{Z}_{\ge 0}.\]](http://latex.artofproblemsolving.com/8/f/b/8fb372df4b44c3d7c0690b310b12c684134fee74.png)
Prove that the following are equivalent:
is surjective;
,
and 
.
and 
be on sides 
and 
respectively in 
such that 
.
be the circumcenter of 
and let 
.
be the n-th prime number in increasing order for 
.
satisfying 
for all 
be an acute triangle. Let 
be the midpoint of side 
,
be the feet of the altitudes from 
and 
,
and 
intersect at a point 
,
is perpendicular to line 
such that ![\[f(f(x)) + xf(xy) = x + f(y)\]](http://latex.artofproblemsolving.com/a/6/f/a6f84dd53eec985b90bad3df6580eeaf29d73195.png)
for all positive real numbers 
and 
.
,
with center 
and radius 
intersects the circle 
and 
.
.
such that for any real value of 
we have:
and 
are points, such that 
,
= 
and 
and 
passes through the fix point.
Consider an 
-
in the plane. An eligible set is a non-empty set of the form 
where 
determine the smallest possible number of eligible subsets 
such that :
Y by PikaPika999
An 
-tromino is a figure made up of three squares, obtained by removing one square from a 
board.
We have a
board consisting of 
unit segments. A configuration of several -trominoes is optimal if the -trominoes do not overlap, each one covers exactly three squares of the board, and moreover, no unit segment of the board belongs to two -trominoes. Below is an optimal configuration of 5 -trominoes:
Determine the largest possible value of
for which there exists an optimal configuration of L-trominoes on the board.
-tromino is a figure made up of three squares, obtained by removing one square from a 
board.We have a
board consisting of 
unit segments. A configuration of several -trominoes is optimal if the -trominoes do not overlap, each one covers exactly three squares of the board, and moreover, no unit segment of the board belongs to two -trominoes. Below is an optimal configuration of 5 -trominoes:
Determine the largest possible value of
for which there exists an optimal configuration of L-trominoes on the board.This post has been edited 2 times. Last edited by EmersonSoriano, Apr 5, 2025, 6:47 PM
Reason: change
Reason: change
