AoPS Academy
be a triangle with 
and 
. Let 
and 
be the midpoints of 
and 
respectively. The point 
is inside of 
such that 
is equilateral. Let 
and 
. Prove that 
.
. This has to be covered by three types of tiles - red, blue and black. The red tiles are of size 
, the blue tiles are of size and the black tiles are of size 
. Let 
denote the number of ways this can be done. For example, clearly 
because we can have either a red or a blue tile. Also 
since we could have tiled the plot as: two red tiles, two blue tiles, a red tile on the left and a blue tile on the right, a blue tile on the left and a red tile on the right, or a single black tile.
for all 
.
for all 
.![\[ \binom{m}{r} = \begin{cases}
\dfrac{m!}{r!(m-r)!}, &\text{ if $0 \le r \le m$,} \\
0, &\text{ otherwise}
\end{cases}\]](http://latex.artofproblemsolving.com/b/8/d/b8d653661e421db1fce7d93837e7bce31337bb30.png)
for integers 
.
can be written in the form
where 
for some non-negative 
such that
for every 
.
be real numbers such that ![\[ab^2+bc^2+ca^2=6\sqrt{3}+ac^2+cb^2+ba^2.\]](http://latex.artofproblemsolving.com/3/d/0/3d015292c249a69a44d5f923092201d7a271b896.png)
Find the smallest possible value of 
.
points on the plane with diameter 
and two not necessarily distinct points 
not equal to 
or 
satisfying that![\[|AX-XY|+|BY-XY|\leq \frac{2D}{N-2}.\]](http://latex.artofproblemsolving.com/f/3/6/f3649872c33e1db8880b4aedace38d9439f2b4ae.png)
.
,
as a closer intersection of circle centered at 
with line 
to 
.
denote the distance between 
.
be 
point with maximum distance.
,
.
,
Assume that 
is closer to 
.
,
,
Assume that 
is closer to 
We also have, 
Hence, 
as desired. 
![[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
import graph; size(5cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.3) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
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pen ccqqqq = rgb(0.8,0,0);
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dot((0.3232,3.3505),dotstyle);
label("$Q$", (0.4202618100110891,3.6090905573368164), NE * labelscalefactor);
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dot((-6.469313415723834,5.272212358723019),linewidth(4pt) + dotstyle);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */[/asy]](http://latex.artofproblemsolving.com/c/d/2/cd2a7f81fc999d57c1e7ca839aa2dffc8be3e4e2.png)