[/quote]
,
,
is the midpoint of 
the point of intersection of the tangents at points 
and 
of the circumscribed circle of 
is the midpoint of the segment 
.
meets the circumcircle 
.
.
:
be reals such that 
and 
Prove that
Where 
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of 
is the sum of 
,![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]](http://latex.artofproblemsolving.com/7/b/6/7b6e32d32d89ca058a06974e7276f0a5874d90a7.png)
such that 
that satisfy the equation
.
.
and 
,
and 
and 
,
is the Euler line of triangle 
.
,
,
,
are odd primes and![\[2^ap^b=(p+2)^c-1.\]](http://latex.artofproblemsolving.com/5/8/5/58518a8ebed14836b72af743ff100eae2efba5e2.png)
we have 
.
.
Y by
Points 
and 
are marked on the sides 
of an acute-angled triangle ABC so that 
is a bisector and 
is an altitude. Let 
be the feet of the perpendiculars from to 
and 
respectively. Prove that 
.
I. Gorodnin
and 
are marked on the sides 
of an acute-angled triangle ABC so that 
is a bisector and 
is an altitude. Let 
be the feet of the perpendiculars from to 
and 
respectively. Prove that 
.I. Gorodnin
This post has been edited 1 time. Last edited by parmenides51, Jun 14, 2020, 9:06 AM
so 
is cyclic.
and 
.
we have 
.
.
