[/quote]
,
such that 
divides 
for any positive integers 
and 
.
of real numbers such that 
for all positive integers 
.
denotes the greatest integer less than or equal to 
.
distinct real numbers there exist two pairs 
and 
with 
or 
,![\[ \left| \frac{a-b}{c-d} - 1 \right|< \frac{1}{100000}. \]](http://latex.artofproblemsolving.com/3/b/0/3b03250ffbd03fd657fa00bc2b63a2c17ee75061.png)
is fixed on a plane with largest side 
be an arbitrary diameter of its circumscribed circle, and the point 
lies on the smaller arc 
is on the smaller arc 
.
are feet of perpendiculars dropped from point 
lies on a fixed circle.
such that
,
be an abitary prime number then we can find out that if 
then 
then they say that exist integer 
such that 
i don't know why 
,
such that![\[ x f(x + xy) = x f(x) + f \left( x^2 \right) f(y) \quad \forall x,y \in \mathbb{R}.\]](http://latex.artofproblemsolving.com/b/0/f/b0f1a1d3b9de5d23f1b5a2e9be3b88d8bd575883.png)
be points on the sides 
respectively of a triangle 
and 
meet at 
and 
,
and 
,
,
.
to the area of 
.
with the property that ![\[f(x-f(y))=f(f(x))-f(y)-1\]](http://latex.artofproblemsolving.com/f/2/5/f25cc1e8ae0be1fd02b347fd94be4fab88af1d46.png)
holds for all 
.
to be perfect squares.
be a positive integer and let 
be a finite set of odd prime numbers. Prove that there is at most one way (up to rotation and reflection) to place the elements of 
for some positive integer 
be a fixed triangle, and let 
,
,
be the midpoints of sides 
,
,
,
be a variable point on the circumcircle. Let lines 
,
,
meet the circumcircle again at 
,
,
,
,
,
,
,
,
form a triangle. Prove that the area of this triangle does not depend on 
,
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?
Y by
An acute triangle 
is given on the plane. Let the points 
, 
and 
be on the sides 
, 
and 
, respectively. (, and are different from the vertices 
, 
and 
) Also the points 
, 
and 
are taken such that 
is an equilateral hexagon which the opposite sides are parallel. Suppose that the circumcenters of and 
are coincident. Then determine the least possible value of:
Note: 
area of 
is given on the plane. Let the points 
, 
and 
be on the sides 
, 
and 
, respectively. (, and are different from the vertices 
, 
and 
) Also the points 
, 
and 
are taken such that 
is an equilateral hexagon which the opposite sides are parallel. Suppose that the circumcenters of and 
are coincident. Then determine the least possible value of:
Note: 
area of 
This post has been edited 1 time. Last edited by Ege_Saribass, Apr 27, 2025, 11:10 AM
Reason: missing information
Reason: missing information
be the circumcenter of 
.
.
,
.
.
and 
.
.
hence 
thus 
.
.
.
goes to 
via inversion.
,
,
.
and 
and 
.
.
are linear.![$[E'F']$](http://latex.artofproblemsolving.com/b/8/1/b8142d200cf7f09b6a3c009b17fe198a406538ec.png)
intersects 
,
and 
.
,
and 
.
which is a contradiction.
.
and 
.
.
and 
.
and circumradius of the 
.
and 
,
hence 
.
