such that for all 
,![\[f(x+yf(x))+y = xy + f(x+y).\]](http://latex.artofproblemsolving.com/c/f/3/cf3d20a041c27244e90876119b4568b7a3e13c03.png)
be an acute scalene triangle 
be the orthogonal projection of 
on 
,
and 
are the midpoints of 
and 
respectively. Let 
and 
of the circumcircle of triangle 
.
and 
are tangent if and only if 
and 
on 
and 
.
with 
and 
are 
and 
respectively, and the intersections of 
with 
and 
are 
,
and 
are concurrent.
,
,
and circumcentre 
.
is a parallellogram. Suppose that there exists a point 
on 
to 
.
be the midpoint of 
.
,
.
Y by Adventure10
A circle and a point 
higher than the circle lie in the same vertical plane. A particle moves along a straight line under gravity from to a point 
on the circle. Given that the distance travelled from in time 
is equal to 
, where 
is the angle of inclination of the line 
to the horizontal, give a geometrical characterization of the point for which the time taken from to is a minimum.
higher than the circle lie in the same vertical plane. A particle moves along a straight line under gravity from to a point 
on the circle. Given that the distance travelled from in time 
is equal to 
, where 
is the angle of inclination of the line 
to the horizontal, give a geometrical characterization of the point for which the time taken from to is a minimum.
.
and it is minimum when 
is minimum, i.e., when 
is minimum (because the square root is an increasing function). The power p of the point P to the circle (O) is constant for any angle 
and this expression is minimum, when 
is minimum or when 
is maximum (because the reciprocal value is a decreasing function). Let the vertical line through P intersect the horizontal line through R at a point S. The distance PS is equal to 
.
