[/quote]
,
such that 
.
.
be a cyclic quadrilateral with circumcenter 
,
is not a diameter of its circumcircle. The lines 
and 
intersect at point 
,
lies between 
and 
lies between 
and 
is acute and let 
be its orthocenter. The points 
and 
on the lines 
and 
.
,
.
which satisfy the condition![\[ f(m + n) \geq f(m) + f(f(n)) - 1
\]](http://latex.artofproblemsolving.com/4/b/d/4bddce124cf0da82cb8bb6c6ae2325239354f27c.png)
Find all possible values of 
.
be a positive integer and 
be a positive integer coprime to 
.
,
,
Find, in terms of 
,
such that 
is divisible by 
.
table are initially white. Alice and Bob play a game. First Alice paints 
rows and 
with 
so that 
The point 
satisfies 
the point 
satisfies 
and the point 
on the line 
Let 
and 
be the circumcenters of the triangles 
and 
respectively. Prove that the lines 
and 
are concurrent.
,
as the product of the digits of 
.
be a function satisfying the following conditions:
,
and 
,
;
is surjective.
.
and 
.
and 
respectively. The point 
such that 
is equilateral. Let 
and 
.
.
.
,
.
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 
.
Y by Adventure10
Let 
be positive integers, where 
. It is given that there exist a positive integer 
such that
holds for all integers
. Prove that the above equation is true for all positive integers 
. (Here 
is the sum of digits of 
taken in base 
).
be positive integers, where 
. It is given that there exist a positive integer 
such that
holds for all integers
. Prove that the above equation is true for all positive integers 
. (Here 
is the sum of digits of 
taken in base 
).
.
where 
.
(
is too small to interfere with 
)
.
.
and so 
.
from 
to 
condition and the 
condition are irrelevant and seems very strange that such a problem would appear in a China TST.
for 
and 
.
