AoPS Academy
table are initially white. Alice and Bob play a game. First Alice paints 
of the fields in red. Then Bob chooses 
rows and columns from the table and paints all fields in them in black. Alice wins if there is at least one red field left. Find the least value of such that Alice can win the game no matter how Bob plays.
be an interior point of the acute triangle 
with 
so that 
The point 
on the segment 
satisfies 
the point 
on the segment 
satisfies 
and the point 
on the line satisfies 
Let 
and 
be the circumcenters of the triangles 
and 
respectively. Prove that the lines 
and 
are concurrent.
, define 
as the product of the digits of (in decimal). Find all positive integers such that 
.
be a function satisfying the following conditions:
, if 
and 
, then 
;
is surjective.
. 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 
.
be a triangle with orthocenter 
and circumcircle 
. Let be the reflection of 
in 
and let the reflection of in 
. Let 
be the midpoint of segment 
. Show that the tangent to in is perpendicular to 
.
be the midpoint of 
Reflect over to 
and Reflecting over sends to . Hence 
is a parallelogram.
is the nine-point circle of 
then showing 
be midpoint of 
. Take as the origin. 
. From 
. From 
. Take altitudes from and to and respectively and get equations of lines and their intersection which is . So, 
. We need to prove 
is parallel to . Which means their slopes should be equal.
.
,
,
;
,
,
.Then 
and 
.Note that the displacement vectors can be written as 
and 
.Then,by EFFT,if we have that ![$$a^2[c^2\frac{S_{AC}-S^2}{S^2}-b^2\frac{S_{AB}-S^2}{S^2}]+b^2[(b^2-c^2)\frac{S_{AB}-S^2}{S^2}+c^2\frac{S^2+S_{BC}}{S^2}]+c^2[(b^2-c^2)\frac{S_{AC}-S^2}{S^2}-b^2\frac{S^2+S_{BC}}{S^2}]=0,$$](http://latex.artofproblemsolving.com/c/b/0/cb0c8dd0d07f62dd2103e507d98a1d32b1e19c67.png)
we are done. This can be shown by some algebraic manipulations.
on unit circle, then 
.
and 
be the circumcentre and centroid of 
maps 
so 
.