,
denote the number of positive divisors of 
such that there are exactly 
positive integers 
not greater than 
that satisfies the following: the equation![\[ \tau(x) = \tau(y) = \tau(z) = \tau(ax + by + cz) = K \]](http://latex.artofproblemsolving.com/a/8/6/a86ab0dd1aacf681d5d6761a4da73face1647f48.png)
holds for some positive integers 
.
such that 
.
has an incenter 
Suppose 
.
be the midpoint of 
.
.
meets 
again at point 
.
and 
be such that 
and 
.
lies on the plane such that 
is a parallelogram, and suppose the angle bisectors of 
and 
concur on 
.
and 
meet 
at 
and 
.
.
,
be two complex numbers such that 
and 
,
.
satisfy the relation 
.
and 
are invertible.
is solvable over 
for every 
.
,
.
be collinear points (in order) and 
a point in plane. Consider the disc 
of center 
,
.![$\mathcal{D}\cap [AC]$](http://latex.artofproblemsolving.com/0/5/6/0569d4130943000380a87b72be211965b5ff02ad.png)
is either the empty set or a segment of length at most 
.![$P(X)\in\mathbb{C}[X]$](http://latex.artofproblemsolving.com/9/9/e/99e8cde28b0b9bc9414425ebc30fa2dc0386fb37.png)
be a polynomial of degree ![\[\sup_{x\in[0,1]}|P(x)|\le(2n+1)^{n+1}\int\limits_{0}^{1}|P(x)|\mathrm{d}x.\]](http://latex.artofproblemsolving.com/9/1/6/91647cd75888c5e9d291280c8ae3f199cc6d1bfd.png)
![$f \colon [0,1] \to \mathbb{R} $](http://latex.artofproblemsolving.com/2/1/8/21859d37ed035abd07d0be112f49d52598fa477e.png)
be a differentiable function such that its derivative is an integrable function on ![$[0,1]$](http://latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
,
.![\[ \int_0^1 (xf'(x))^2 dx \geq 12 \cdot \left( \int_0^1 xf(x) dx\right)^2 \]](http://latex.artofproblemsolving.com/2/3/e/23e33eed39fc1e20486bbb86261d3566e8fd2017.png)
be an odd prime number, and 
be an odd number not divisible by 
be a field with 
elements, and 
be the set of elements of 
,
.
has at least 
.
and 
be a continuous function for which there exists an antiderivative 
,
,
,
.
for all 
be a positive integer, ![$g \in \mathbb{R}[X]$](http://latex.artofproblemsolving.com/0/1/e/01e7ebcebcb9f16935716ee859f7acfaa66c29d4.png)
,
be a polynomial with all of its roots being real, and 
for any 
for all 
.
;
,
matrices which has only 
and 
,
:![\[(f(x)-g(y))(f'(x)-g'(y))(f''(x)-g''(y))=0\]](http://latex.artofproblemsolving.com/0/2/9/029a935e28ca0a9b36e65702689cca91eeb8f614.png)
Y by
Let 
, 
, 
be positive integers such that 
. Alice and Bob play a game on an (initially uncoloured) 
grid as follows:
- First, Alice paints cells green.
- Then, Bob paints other (i.e.uncoloured) cells blue.
Alice wins if she can find a path of non-blue cells starting with the bottom left cell and ending with the top right cell (where a path is a sequence of cells such that any two consecutive ones have a common side), otherwise Bob wins. Determine, in terms of, and , who has a winning strategy.
, 
, 
be positive integers such that 
. Alice and Bob play a game on an (initially uncoloured) 
grid as follows:- First, Alice paints
cells green.- Then, Bob paints
other (i.e.uncoloured) cells blue.Alice wins if she can find a path of non-blue cells starting with the bottom left cell and ending with the top right cell (where a path is a sequence of cells such that any two consecutive ones have a common side), otherwise Bob wins. Determine, in terms of
, and , who has a winning strategy.
