such that ![\[f(xy-1)+f(x)f(y)=2xy-1,\]](http://latex.artofproblemsolving.com/8/8/8/888ca39f2b7f8cec6d6426bee28d40eade40a66e.png)
for any real numbers 
and 
![\[
f : \mathbb{R}^+ \to \mathbb{R}
\]](http://latex.artofproblemsolving.com/e/7/b/e7bfc5b236fb6f00ef328f850b1b14632cbf8416.png)
satisfying the condition that for all 
,![\[
f(x + y^2) \geq f(x) + y.
\]](http://latex.artofproblemsolving.com/3/a/a/3aa20083835682dbd81be692eba65cab19e923e5.png)
of positive integers with 
prime, 
,
,
is a multiple of 
.
kawaii if it satisfies 
is the number of integers not more than 
Y by Adventure10
The vertices of the convex quadrilateral 
lie on the parabola 
. It is known that is cyclic and 
is a diameter of its circumcircle. Let 
and 
be the midpoints of the diagonals of and 
respectively. Find the length of the projection of the segment 
on the axis 
.
lie on the parabola 
. It is known that is cyclic and 
is a diameter of its circumcircle. Let 
and 
be the midpoints of the diagonals of and 
respectively. Find the length of the projection of the segment 
on the axis 
.
are the only two points 
such that 
which gives 
which obviously means that 
are roots of this equation, then 
equals to 
