[/quote]
,
such that for all any 
:
and 
satisfies the following recursion formulas: 
,
.
distinct real numbers, there exist two numbers 
and 
such that![\[
a^2 + b^2 < \sqrt{3}(1 + a^2b^2).
\]](http://latex.artofproblemsolving.com/6/4/4/6447a411df1c43e799555b07de3dc5d7eae0f4fb.png)
Find the smallest possible value of 
be the midpoint of 
on triangle 
.
lies on segment 
such that 
and 
on line 
such that 
.
.
be a convex quadrilateral. The diagonals 
and 
intersect at 
,
.
and 
be the midpoints of sides 
,
,
such that![\[
EG + 3FH \geq k \cdot d + (1 - k) \cdot s,
\]](http://latex.artofproblemsolving.com/b/e/f/bef24b0d482f95d2e6ef22b3797a432c69d98af1.png)
where 
is the semiperimeter of 
is the sum of the lengths of its diagonals (i.e., 
)
such that the equation
and 
passes through the center of the other one. Triangle 
touch 
.
and 
be positive real numbers such that 
.
When equality holds?
such that: 
và 
.
is the circumcircle of 
wwith diameter 
.
and 
intersect at 
. The line perpendicular to 
at point 
intersects the extension of 
.
at length 
.
, find the measure of 
.
.
set of numbers:
,
i
Y by kiyoras_2001
Problem. Let 
be the nonnegative integers in 
where 
. Define 
. Prove that 
.
Bonus problem (Open). Can we prove a tighter result than the one above? That is, is there a function
such that 
where 
, and the statement is still true when 
?
Or, is there a function such that 
where 
, and the statement is still true when ?.
be the nonnegative integers in 
where 
. Define 
. Prove that 
.Bonus problem (Open). Can we prove a tighter result than the one above? That is, is there a function
such that 
where 
, and the statement is still true when 
?Or, is there a function
such that 
where 
, and the statement is still true when ?.This post has been edited 10 times. Last edited by CBMaster, Apr 27, 2025, 5:43 AM
Reason: .
Reason: .
