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is described on a circle 
, and is not a trapezius inscribed in a circle. Let the tangency points of and 
be 
respectively. A circle with a center 
, different from is tangent to the segement 
and lines 
. A circle with center 
, different from is tangent to segment 
and lines 
. A circle with center 
, different from is tangent to segment 
and lines . A circle with center 
, different from is tangent to segment 
and lines . Prove that the lines 
are concurrent.
such that 
is a divisor of 
.
, such that 
and 
.
such that 
is prime, 
with 
and ?
. He says to Cecilia: "If you pick any subset of 10 elements from , it will be disjoint with at least one subset from my list." What is the smallest possible number of subsets Miguel's list can have if what he tells Cecilia is true?
be an acute-angled triangle and let 
be its orthocentre. Let 
and 
be the centroids of the triangles 
and 
respectively. If the area of 
is 
units, what is the area of 
?
![$[G_1G_2G_3]$](http://latex.artofproblemsolving.com/0/b/9/0b944f096ecfb1d255604adc2f12ec47b1a926cd.png)
in terms of ![$[MNP]$](http://latex.artofproblemsolving.com/c/c/2/cc26c9ba3b0a503a41448454f50c48a9b8e8b783.png)
where 
are the midpoints of 
.
![$\frac{{[{G_1}{G_2}{G_3}]}}{{[MNP]}} = {\left( {\frac{2}{3}} \right)^2} \Leftrightarrow [{G_1}{G_2}{G_3}] = \frac{4}{9}[MNP] = \frac{4}{9} \cdot \frac{1}{4}[ABC] \Leftrightarrow $](http://latex.artofproblemsolving.com/6/e/9/6e930a45036c5e6cae1177b2e2bbaa5de116a711.png)
![$\boxed{[ABC]=9[G_1G_2G_3]}$](http://latex.artofproblemsolving.com/3/7/5/37529529bd600033aa1211370de1e7696af0e830.png)