[/quote]
,
and 
Prove that
Where 
and 
is tangent to the circumcircle of triangle 
.
and 
are the midpoints of segments 
and 
respectively, then line 
and line 
intersect 
and 
respectively. Line 
and line 
intersect the circumcircle of triangle 
and 
respectively. Prove that 
and 
.
Equality holds when 
or 
or 
and 
Prove that 
be real numbers such that 
.
Prove that 
are the midpoints of sides 
,
the circumcircle and Euler circle of 
and 
intersect 
at 
,
is the point such that 
is the midpoint of 
.
are defined similarly.
then 
;
,
are created in the same manner. 
is the orthocenter of 
intersect 
at 
resp. Prove that 
.
Equality holds when 
or 
or 
satisfying 
Prove that 
Proposed by Phan Ngoc Chau
Where 
Where 
,
be positive integers such that 
.
is a sequence of integers satisfying 
,
,
,
Prove that there exists 
such that 
such that 
Y by
It is known that the moving point 
is on the curve 
. There is a parabola 
with focus 
. Two tangent lines to
are drawn through a point 
on 
, and the tangent points are 
and 
respectively. The line 
parallel to the line 
is tangent to at point 
. Question: When the line and 
have two intersection points, find the range of 
.\
This may make some of the information in the question useless, because I deleted the first two questions of this big question in order to avoid making the question too long and get straight to the point.\
According to the calculation, I get the analytical expression of the line as 
.\
At first, I thought it only needed to be not parallel to the parabola asymptotes.\
That is, the slope of the straight line
, then 
, so 
,and 
.\
.\
But when I checked the answer, it was wrong.\
It combines the straight line with the curve to get an equation and then uses Vieta's theorem.
\begin{cases}
y=\frac{x_0}{2} x - \frac{x_0^2}{4}
x^2=4y
\end{cases}
The answer is that according to the question, the equation has two negative roots. I can't understand this, and this is exactly where my problem lies.\
Then it gets the following system of equations:
\begin{cases}
4x_0-16 \neq 0
\Delta > 0
x_1+x_2 <0
x_1\cdot x_2>0
\end{cases}
Solve,
.
So we get
.
I hope you can help me figure out why both roots of that equation are negative.
is on the curve 
. There is a parabola 
with focus 
. Two tangent lines to
are drawn through a point 
on 
, and the tangent points are 
and 
respectively. The line 
parallel to the line 
is tangent to at point 
. Question: When the line and 
have two intersection points, find the range of 
.\This may make some of the information in the question useless, because I deleted the first two questions of this big question in order to avoid making the question too long and get straight to the point.\
According to the calculation, I get the analytical expression of the line
as 
.\At first, I thought it only needed to be not parallel to the parabola asymptotes.\
That is, the slope of the straight line
, then 
, so 
,and 
.\
.\But when I checked the answer, it was wrong.\
It combines the straight line
with the curve to get an equation and then uses Vieta's theorem.\begin{cases}
y=\frac{x_0}{2} x - \frac{x_0^2}{4}
x^2=4y
\end{cases}
The answer is that according to the question, the equation has two negative roots. I can't understand this, and this is exactly where my problem lies.\Then it gets the following system of equations:
\begin{cases}
4x_0-16 \neq 0
\Delta > 0
x_1+x_2 <0
x_1\cdot x_2>0
\end{cases}
Solve,
.So we get
.I hope you can help me figure out why both roots of that equation are negative.
