be distinct natural numbers each of which has exactly 
positive integer factors. For instance, 
has exactly 
.
divides any of the 
'
be an integer. Prove that 
can be written as a polynomial with integer coefficients in the variables 
,
and 
.
,
are real numbers such that 
is an integral multiple of 
.
,
for all positive integers 
.
be the orthocenter of an acute-angled triangle 
.
centered at the midpoint of 
and passing through 
and 
.
,
,
and 
.
with real coefficient. It is known that 
and 
.
can't have 
distinct real roots.
Y by Adventure10, Mango247
Let 
be an equilateral triangle. 
is a point on the side 
such that 
, 
is a point on the side 
such that 
is parallel to 
and 
is a point on the side such that 
is parallel to . Find the ratio of areas 
be an equilateral triangle. 
is a point on the side 
such that 
, 
is a point on the side 
such that 
is parallel to 
and 
is a point on the side such that 
is parallel to . Find the ratio of areas 
This post has been edited 1 time. Last edited by parmenides51, Sep 12, 2018, 6:50 PM
Reason: name edit
Reason: name edit
be the sidelength of 
and 
be the height. 
so 
is equilateral. Thus 
The height of 
So ![$$\frac{[MNP]}{[ABC]} = \frac{\frac{b}{7} \cdot \frac{6h}{7}}{bh} = \frac{6}{49}$$](http://latex.artofproblemsolving.com/6/0/d/60d92df7757a9b1855fceb768829b40c15b8e11a.png)
