and 
.
Let 
.
Prove that
and 
Prove that
'is a perfect square.
and 
be positive real numbers with 
.
For which 
such that 
for all 
,
.
and 
,
and 
so that 
.
and 
is equal to 
.
over 
.
![\[\lfloor x \rfloor\]](http://latex.artofproblemsolving.com/d/3/f/d3f8facf62413e1f17c89b9c183a8735c2defe8b.png)
denote the greatest integer less than or equal to x . What is the sum of the squares of the real numbers x for which ![\[ x^2 - 20\lfloor x \rfloor + 19 = 0 \]](http://latex.artofproblemsolving.com/f/a/9/fa96f2d6217a16029e5f03499e0a83deffdfa28a.png)
Y by Adventure10, Mango247
Let 
be an acute triangle. On the median 
we take a point 
so that the rays 
and 
cut the sides 
and 
at the points 
and 
respectively. Show that the 
and 
segments are parallel.
be an acute triangle. On the median 
we take a point 
so that the rays 
and 
cut the sides 
and 
at the points 
and 
respectively. Show that the 
and 
segments are parallel.This post has been edited 2 times. Last edited by parmenides51, Jul 23, 2021, 9:18 PM
