[/quote]
,
be an acute triangle with orthocenter 
and circumcenter 
.
and 
be points on the line segments 
and 
respectively such that 
is a parallelogram. Prove that 
.
that satisfy 
for all 
and 
satisfies the following recursion formulas: 
,
.
and 
be points on segment 
denote the circle passing through 
.
denote the circle passing through 
at 
.
)
.
the feet of the altitudes lying on 
respectively. One of the intersection points of the line 
and the circumcircle is 
The lines 
and 
meet at point 
Prove that 
-
and 
.
cities, 
airline companies in a country. Between any two cities, there is exactly one 
and 
verifying 
. 
be integer numbers. Prove that:
is divisible by 
and circumscribed about the incircle 
,
.
,
,
,
,
,
,
,
(distinct from 
intersects 
(distinct from 
be the midpoint of the arc 
of 
at 
and 
,
at 
.
.
but doesn't tell it to anyone. Bernaldo thens pick an arbitrary positive integer 
.
be a set of 2025 positive real numbers. For a subset 
,
as the median of 
when all elements of 
.![\[
P(A) = \sum_{\substack{T \subseteq A \\ |T| \text{ odd}}} M_T, \quad Q(A) = \sum_{\substack{T \subseteq A \\ |T| \text{ even}}} M_T.
\]](http://latex.artofproblemsolving.com/1/d/f/1df2ea8487289111939a53dcc49576fbe6fdef37.png)
Find the smallest real number 
such that for any set ![\[
P(A) - Q(A) \leq C \cdot \max(A),
\]](http://latex.artofproblemsolving.com/c/d/e/cdea343d11c722a423e857aeb01202863405773e.png)
where 
denotes the largest element in 
in 
colors, there exists an arithmetic progression of size 
.
from this?
Y by
From P(4,5), the chord of contact to the conic 3x² + 4y² = 1 is AB, we are to find the midpoint of this chord.
I used T(4,5)=0 to get eqn. of AB, then assuming (h,k) to be the midpoint, T(h,k)=S1(h,k) should give the equation of AB again. But comparing both equations to get h,k does not give me the correct answer.
What am I doing wrong?
I used T(4,5)=0 to get eqn. of AB, then assuming (h,k) to be the midpoint, T(h,k)=S1(h,k) should give the equation of AB again. But comparing both equations to get h,k does not give me the correct answer.
What am I doing wrong?
.
.
and the midpoint 
and 
.
? 
?
is wrong, because the midpoint 
and 
,
are points of the ellipse.
