[/quote]
,
denote the set of positive integers
such that for all 
it holds that 
satisfying 
holds for all real numbers 
.
be positive real numbers such that 
Prove that:
and 
be two sequences defined by:![\[
a_{n+1} = \frac{1 + a_n + a_n b_n}{b_n} \quad \text{and} \quad b_{n+1} = \frac{1 + b_n + a_n b_n}{a_n}
\]](http://latex.artofproblemsolving.com/a/a/0/aa0a5c5acedc990eb345fb50a8395db05307a4e7.png)
for all 
,
and 
.![\[
a_{2024} < 5.
\]](http://latex.artofproblemsolving.com/f/c/a/fca6e6630f33429b5a7ea867d639a2e29ff881f2.png)
and write the numbers 
down on a board. A move consists in erasing any two numbers 
and 
,
and 
on the board, and then removing repetitions (e.g., if the board contained the numbers 
,
and 
,
)
with real coefficients of degree 
)![\[
Q(x) \;=\; P(x) \;+\; a\,P'(x) \;+\; a^2\,P''(x) \;+\; \dots \;+\; a^n\,P^{(n)}(x)
\]](http://latex.artofproblemsolving.com/6/5/e/65e5e4eefcadd5d0019b4929b3472cdf203c13aa.png)
has no real roots.
and 
,
,
the pair 
if for any elements 
we have 
.
be the greatest possible number of elements of a set that escapes the pair 
.
with incircle 
,
be the incenter and 
be the point where 
.
be the point on 
with 
and 
be the orthocenter of 
,
on 
also satisfies 
.
,
,
satisfying 
the following inequality holds:
of the triangle 
.
the orthogonal projection of 
on 
.
and 
are such that 
is a paralellogram, find the locus of point 
when 
.
Y by
For a sequence 
of integers, a pair 
with 
is called interesting if there exists a pair 
of integers with 
such that 
For each 
, find the largest possible number of interesting pairs in a sequence of length 
.
of integers, a pair 
with 
is called interesting if there exists a pair 
of integers with 
such that 
For each 
, find the largest possible number of interesting pairs in a sequence of length 
.This post has been edited 2 times. Last edited by Quidditch, Apr 14, 2024, 10:18 AM
.
and 
for 
and observe that every pair 
is interesting so long as 
,
.
such pairs, so we have the construction.
uninteresting pairs given any sequence 
.
.
.
we see that unless 
one of the pairs 
or 
is uninteresting, since one of 
and 
exceeds 
.
pairs. This finishes, since now we have eliminated 
,
,
,
of 
is interesting. So 
at least 
have one uninteresting pair. So there are at least 
interesting pairs. This can be achieved when 
and 
,
we’ll only need to find pairs that include 
.
,
.
and 
would be boring unless 
,
in range 
must be boring as well, so there are at least 
,
and 
for all 
.
.
,
and 
add to 
,
,
.
.
denote the answer. For 
,
.
and see that there are 2 interesting differences, which gives 
.
.
and in this case 
,
.
for all 
for some numbers 
.
is not interesting. Consider pairs 
.
.
pairs. So, we get at most 
pairs for any 
.
for every 
.
,
are all smaller powers of 2, we have that 
are all interesting. Now, for any 
,
or 
.
,
.
This is achieved by 
for 
.
,
so the 
pairs 
for 
are interesting. Furthermore, the 
for 
are interesting, since 
Also, 
is interesting because 
so we have a total of 
interesting pairs so far, and the other pairs are of the form 
,
so the double of 
because the largest such difference is 
.
for which 
,
.
so since 
Therefore, the double of 
is strictly greater than 
.
,
uninteresting pairs, so at most 
interesting pairs.
with 
.
for 
and 
.
is 
thus any of the 
is not interesting and one of 
and 
is not interesting unless 
since one will be more than half of 
,
or 
is not interesting unless 
is in the midpoint of 
and 
(there is at most one such 
.
and 
work so we eliminate 
is not good
and 
are both good
so if both pairs are good then we have 
and 
;
so we must have equality meaning 
but clearly at most one 
pairs so we are done
.
.
.
,
and 
.
;
is greater than 
;
,
interesting, while it was obviously not counted in 
cannot be interesting. Note for any 
if 
,
cannot be interesting. We need to ensure to find at least 
.
and equality will hold for at most one 
Actually, I find it quite tricky for P4 comparing with a difficulty of day 1..
.
we get the following figure:![[asy]
draw( (0,0)--(8,0), blue ); real e = 0.2; draw( (0,-e)--(0,e), blue ); draw( (1,-e)--(1,e), blue ); draw( (2,-e)--(2,e), blue ); draw( (4,-e)--(4,e), blue ); draw( (8,-e)--(8,e), blue ); label("$0$", (0,e), dir(90), blue); label("$1$", (1,e), dir(90), blue); label("$2$", (2,e), dir(90), blue); label("$4$", (4,e), dir(90), blue); label("$8$", (8,e), dir(90), blue); label("$a_1$", (0,-e), dir(-90), black); label("$a_2$", (1,-e), dir(-90), black); label("$a_3$", (2,-e), dir(-90), black); label("$a_4$", (4,-e), dir(-90), black); label("$a_5$", (8,-e), dir(-90), black);
[/asy]](http://latex.artofproblemsolving.com/0/b/9/0b9c8f9a91dd6a634d7f557b73c29f2acce18a03.png)
are interesting except for the cases where 
and 
.
and 
,
.
and 
.
and 
,
.
.
,
is never interesting.
,
is never interesting.
pairs, one for each index 
.
for every 
,
,
and
which is wrong.
then, 
.
such that both pairs 
are interesting. Let 
then if both of the above are interesting we have 
and 
so adding these together 
so both equalities must hold and 
and 
choose the pair 
and for 
choose 
.
where we choose 
for 
are interesting, and so is the pair 
.
as 
.
for any 
is not interesting due to size constraints and similarly if 
,
is not interesting. this eliminates 
.
.
