,
,
and 
,
is the intersection point of the diagonals 
and 
.
and radius 
intersects 
and 
.
is tangent to the circles with diameters 
and 
.
satisfying the equation![\[
(f(x+y))^2= f(x^2) + f(2xf(y) + y^2), \quad \forall x, y \in \mathbb{R}.
\]](http://latex.artofproblemsolving.com/2/5/e/25eed911f303ed7cc4e03765a2f943ad4641c451.png)
be a triangle inscribed in a circle, where the tangents to the circle at points 
and 
intersect at the point 
.
be a point on the arc 
and 
.
and 
intersect at point 
.
be the reflection of 
and 
intersect at point 
,
.
.
be real numbers such that 
Prove that
Let 
Prove that
If a problem has not been solved within 
days, then others are eligible to post a new probkem.
,
)
satisfied 
Find the minimum of 
and natural number 
are there such that 
and 
?
Y by
Let 
be a positive integer. In a coordinate grid, a path from 
to 
consists of 
consecutive unit steps 
or 
. Prove that the number of paths that divide the square with vertices 
into 2 regions with even areas is 
be a positive integer. In a coordinate grid, a path from 
to 
consists of 
consecutive unit steps 
or 
. Prove that the number of paths that divide the square with vertices 
into 2 regions with even areas is 
This post has been edited 3 times. Last edited by beansenthusiast505, Apr 27, 2024, 1:16 PM
,
paths in total, the claim is equivalent to saying that there are 
more paths which separate the square in two even regions than those with two odd regions.
,
.
.
through 
or through 
.
.
we can find exactly one path with 2 even areas from 
to 
.
