[/quote]
,
![$f : \mathbb{Q}[\sqrt{2}] \to \mathbb{Q}[\sqrt{2}]$](http://latex.artofproblemsolving.com/c/a/1/ca172b912bf5bf7178cbbd5745fbc0a6a8b27503.png)
such that for all ![$x, y \in \mathbb{Q}[\sqrt{2}]$](http://latex.artofproblemsolving.com/9/f/e/9fede0b2fd22836372b8afc488ccd8a14db5d0c2.png)
,
where ![$\mathbb{Q}[\sqrt{2}] = \{ a + b\sqrt{2} \mid a, b \in \mathbb{Q} \}$](http://latex.artofproblemsolving.com/f/a/c/fac8bf68761534a139d570bf66b1aa5dfb59e8db.png)
.
,
is called interesting if the following two conditions hold:
such that 
.
(where 
denotes the 
-
)
.
for which there is no interesting function
such that the following two conditions hold:
and 
satisfying 
,
for all real numbers 
.
and 
,
.
,
,
,
lie on a line in that order, and points 
and 
are located outside the line such that 
,
and 
.
and 
intersect at points 
and 
,
and 
intersect at points 
and 
.
and 
pass through the midpoint of segment 
.
that are relatively prime be called intertwined if among any two divisors of 
,
and among any two divisors of 
is intertwined.
is equal to the product of the first 
distinct prime numbers.
and 
Find the value of 
be positive real numbers, such that 
.![\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]](http://latex.artofproblemsolving.com/b/e/5/be5819a67c3cd78f2dea35fdccf48688c720ce3c.png)
When does the equality hold?
are collinear with 
is the circle with diameter 
,
is the circle with diameter 
.
at 
and 
.
and 
meet on the perpendicular to 
be a strictly increasing sequence of positive integers such that for all positive integers 
![\[d(2025^{a_n}-1)|a_{n+1}.\]](http://latex.artofproblemsolving.com/8/f/d/8fda531266d39c7f11c58d3c1077d2cab3357bea.png)
Show that for any positive real number 
there is a positive integers 
such that 
for all 
.
denotes the number of positive divisors of the positive integer 
Prove that
be a quadrilateral with 
.
.
be the foot of the perpendicular from the point 
.
meets 
at 
meets 
and 
.
Prove that
Y by
What is the least number required to colour the integers 
such that for any set of consecutive integers taken from the given set of integers, there will always be a colour colouring exactly one of them? That is, for all integers 
such that 
, there will be a colour coloring exactly one integer from the set 
.
such that for any set of consecutive integers taken from the given set of integers, there will always be a colour colouring exactly one of them? That is, for all integers 
such that 
, there will be a colour coloring exactly one integer from the set 
.This post has been edited 1 time. Last edited by kitun, Nov 15, 2021, 10:12 AM
