[/quote]
,
every next number (from the fifth number) is the unit number of the sum of the four numbers preceding it. Is there any cases that we get the numbers 
and 
in this series?
with two altitudes 
and 
.
be the point on the line 
.
and 
intersect at point 
,
is the point on segment 
such that 
.
is isosceles.
such that 
is unbounded and![\[2f(m)f(n)-f(n-m)-1\]](http://latex.artofproblemsolving.com/0/f/7/0f77b5addbf8f5e603eb1b6923cdac795cd132e9.png)
is a perfect square for all integer 
and 
be the circumcircle and the circumcentre of an acute-angled triangle 
.
intersects 
.
be the circle with diameter 
.
and 
intersect 
and 
respectively. The point 
is chosen on the line 
so that 
.
.
,
be the midpoint of 
.
on 
is such that 
is an altitude of the triangle 
be the reflection of 
,
are 
and 
be the point such that the circumcentre of triangle 
is the midpoint of the segment 
.
.
and 
.
,
and 
,
.
.
a grid of 9*9, we choose a little square in 
be positive reals, s.t 
,
Y by
Let 
be the set of all lines in the plane and 
be a set of 
points in the plane. For a line 
let 
be the number of distinct points among the orthogonal projections of the points from on 
. Find the maximum possible number of distinct values of (this quantity is computed for any line ) as varies.
be the set of all lines in the plane and 
be a set of 
points in the plane. For a line 
let 
be the number of distinct points among the orthogonal projections of the points from on 
. Find the maximum possible number of distinct values of (this quantity is computed for any line ) as varies.This post has been edited 1 time. Last edited by Assassino9931, Apr 27, 2025, 10:18 PM
.![[asy]
draw((-1,0)--(5,0),green);
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draw((-1,3)--(5,3),green);
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dot((4,1));
dot((0,2));
dot((1,2));
dot((2,2));
dot((3,2));
dot((4,2));
dot((1,3));
dot((2,3));
dot((3,3));
[/asy]](http://latex.artofproblemsolving.com/7/1/a/71a6b1182ee72c3b10c93bfd57855c3ecf945fe5.png)
![[asy]
draw((0,-1)--(0,4),green);
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draw((4,-1)--(4,4),green);
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dot((1,0));
dot((2,0));
dot((3,0));
dot((0,1));
dot((1,1));
dot((2,1));
dot((3,1));
dot((4,1));
dot((0,2));
dot((1,2));
dot((2,2));
dot((3,2));
dot((4,2));
dot((1,3));
dot((2,3));
dot((3,3));
[/asy]](http://latex.artofproblemsolving.com/b/0/0/b00767e3d62432eb03c82ad9f2932593bc00ebe7.png)
![[asy]
draw((-1,1)--(2,4),green);
draw((-1,0)--(3,4),green);
draw((-1,-1)--(4,4),green);
draw((0,-1)--(4,3),green);
draw((1,-1)--(5,3),green);
draw((2,-1)--(6,3),green);
dot((0,0));
dot((1,0));
dot((2,0));
dot((3,0));
dot((0,1));
dot((1,1));
dot((2,1));
dot((3,1));
dot((4,1));
dot((0,2));
dot((1,2));
dot((2,2));
dot((3,2));
dot((4,2));
dot((1,3));
dot((2,3));
dot((3,3));
[/asy]](http://latex.artofproblemsolving.com/7/6/a/76a5a455bbcfec1c8dc6064634b7b589ff8af6be.png)
![[asy]
draw((-1,1)--(1,-1),green);
draw((-1,2)--(2,-1),green);
draw((-1,3)--(3,-1),green);
draw((0,3)--(4,-1),green);
draw((0,4)--(4,0),green);
draw((1,4)--(5,0),green);
draw((2,4)--(5,1),green);
dot((0,0));
dot((1,0));
dot((2,0));
dot((3,0));
dot((0,1));
dot((1,1));
dot((2,1));
dot((3,1));
dot((4,1));
dot((0,2));
dot((1,2));
dot((2,2));
dot((3,2));
dot((4,2));
dot((1,3));
dot((2,3));
dot((3,3));
[/asy]](http://latex.artofproblemsolving.com/d/8/0/d80bb006bd717b1d974d151737935aa20bc97c85.png)
![[asy]
draw((1,-1)--(5,1),green);
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draw((-1,-0.5)--(5,2.5),green);
draw((-1,0)--(4,2.5),green);
draw((-1,0.5)--(4,3),green);
draw((-1,1)--(4,3.5),green);
draw((-1,1.5)--(3,3.5),green);
draw((0,2.5)--(2,3.5),green);
dot((0,0));
dot((1,0));
dot((2,0));
dot((3,0));
dot((0,1));
dot((1,1));
dot((2,1));
dot((3,1));
dot((4,1));
dot((0,2));
dot((1,2));
dot((2,2));
dot((3,2));
dot((4,2));
dot((1,3));
dot((2,3));
dot((3,3));
[/asy]](http://latex.artofproblemsolving.com/e/b/e/ebeb5a7ecc83b65fa3bfd0efc8630690677658f8.png)
to 
is possible. Then either 
to 
is possible. If 
is possible then all the points lie inside some 
affine box, which is less than 
is the next smallest, which means that there cannot be more than 
affine box that contains all the points and transform it into a regular array. Then note that the best possible slopes are 
and 
and if we want 
to be possible, the points must be in the form of the construction, and then that leaves no room for 
to be possible.
