[/quote]
,
be real numbers such that 
.
and determine all values of 
such that the minimum value is achived.
cities. The airline Starways wants to establish some one-way flights between pairs of cities in such a way that each city has exactly one flight out of it. Find the smallest positive integer 
such that no matter how Starways establishes its flights, the cities can always be partitioned into 
flights.
distinct circles in a plane, with radius 
.
circles, which form a set 
satisfy:
.
,
and 
by the midpoints of 
and 
respectively, and 
and 
be the feet of the altitudes from 
and 
the line being tangent tot he circumcircle of triangle 
the reflection of 
.
by the intersection of 
and 
be the intersection of 
and 
analogously. If 
is the intersection of 
and 
,
.
and 
.
be the point on the line 
.
and 
intersect at point 
,
is the point on segment 
such that 
.
is isosceles.
such that 
such that:
.
such that if any 
,
and circumcircle 
.
be the midpoints of sides 
be the tangency points of 
be the intersections of line 
and line 
,
of 
.
bisects 
.
Y by High, Generic_Username, acegikmoqsuwy2000, jkoj25, bowenying24, HWenslawski, Adventure10, Mango247, Rounak_iitr
Let 
be an equilateral triangle. Extend side 
beyond 
to a point 
so that 
. Similarly, extend side 
beyond 
to a point 
so that 
, and extend side 
beyond 
to a point 
so that 
. What is the ratio of the area of 
to the area of 
?
be an equilateral triangle. Extend side 
beyond 
to a point 
so that 
. Similarly, extend side 
beyond 
to a point 
so that 
, and extend side 
beyond 
to a point 
so that 
. What is the ratio of the area of 
to the area of 
?
This post has been edited 2 times. Last edited by djmathman, Feb 16, 2017, 6:18 PM
.
.
,
,
,
by a simple area comparison. Noting that the coordinates are already homogenized, we apply the area formula to obtain that![$$[A'B'C']=[ABC] \left| \begin{array}{ccc} 4 & 0 & -3 \\ -3 & 4 & 0 \\ 0 & -3 & 4 \end{array} \right| = (64-27)[ABC]=37[ABC]$$](http://latex.artofproblemsolving.com/9/0/0/90099daf2ecb2d8b660500496694534c8f01cdaa.png)
So the answer is 
times the area of the smaller triangle, for an answer of 
.
. Therefore the answer is 
so that 
.![$$\frac{[A'B'C']}{[ABC]}=\begin{vmatrix}4&0&-3\\ -3&4&0\\ 0&-3&4\end{vmatrix}=\frac{37}{1}$$](http://latex.artofproblemsolving.com/c/7/d/c7d3909ef6fc3256d998bc6505baab9d1dfd3d7d.png)
or 
.
the answer would be 
due to
by LoC, or
since ![$\tfrac{[BB'C']}{[ABC]} = \tfrac{\tfrac{1}{2} (BB') (BC') \sin 120^\circ}{\tfrac{1}{2} (AB) (BC) \sin 60^\circ} = \tfrac{BB' \cdot BC'}{AB \cdot BC} = n(n + 1),$](http://latex.artofproblemsolving.com/3/8/8/3885047fe11cea71a6004a9109ff26319c04eb95.png)
and similarly for ![$[CC'A']$](http://latex.artofproblemsolving.com/c/6/b/c6bf0da5f8248ee5e28515e48305dcd8686dc767.png)
and ![$[AA'B'].$](http://latex.artofproblemsolving.com/6/1/a/61a0eb5e61a1b216ed3e2333710ebeccbc994ef1.png)
![$$\frac{[A'B'C']}{[ABC]}=\begin{vmatrix}n+1&0&-n\\ -n&n+1&0\\ 0&-n&n+1\end{vmatrix}=\frac{(n+1)^3-n^3}{1}$$](http://latex.artofproblemsolving.com/8/e/3/8e3e83c52633e257ffd4e9a7d01af132e925c6fd.png)
