AoPS Academy
be positive real numbers such that : 
. Prove that : 
candies. A starts first. On a move, if in the box there are 
candies, the player chooses positive integer 
so that 
and 
, and eats candies from the box. The player who eats the last candy wins. Who has winning strategy, in terms of 
.
![$\ [\text{Blog Post 141}] $](http://latex.artofproblemsolving.com/e/2/e/e2e1563c16c5d526310409de610870dc615964d8.png)
be a triangle and 
, 
, 
the midpoints of the sides 
, 
and 
respectively. Prove that if 
is a point in the plane of the triangle such that ![\[ \frac{MA}{MA_1} = \frac{MB}{MB_1} = \frac{MC}{MC_1} = 2 , \]](http://latex.artofproblemsolving.com/1/6/e/16e50b7418889f94c1346f3103d4325c5db73e99.png)
then is the centroid of the triangle.
be a triangle and , , the midpoints of the sides , and respectively. Prove that if is a point in the plane of the triangle such that then is the centroid of the triangle.
be a triangle and , , the midpoints of the sides , and respectively. Prove that if is a point in the plane of the triangle such that then is the centroid of the triangle.
be the centroid of triangle . Let 
be the reflection of 
about . See that 
so 
(Appolonius, as 
). Similarly. define 
. We see that is the point of concurrence of 
. It is easy to see that these three circles cannot be coaxial. So, 
. 
such that 
and 
.
be 
,
. From conditions above we have 
,
as 
,
as 
and 
as 
we obtain 
,
and 
. After suming this 
,
and 
therefore 
~ 
,
.
is paralel to 
are colinear.By the same manner we prove that 
are colinear, hence M is the centroid of the triangle and the problem is solved!