[/quote]
,
![\[
\text{Find all prime numbers } (p, q) \text{ such that } p^2 + 3pq + q^2 \text{ is a fifth power of an integer.}
\]](http://latex.artofproblemsolving.com/c/f/8/cf855cb087102024a50d41f49c67e7fd1ef22767.png)
and 
are given on side 
of convex quadrilateral 
(with 
)
and 
.
.
be positive real numbers such that 
.
(player) and 
(items), with valuations given by matrix![$
V = \left[\begin{matrix}
v_{1a} & v_{1b} & v_{1c} & v_{1d} \\
v_{2a} & v_{2b} & v_{2c} & v_{2d}
\end{matrix}\right]
$](http://latex.artofproblemsolving.com/d/4/6/d465348852f12739e1fc2a04a9c2e5fdad0dbcb4.png)
![$v_{ij} \in [0, \infty]$](http://latex.artofproblemsolving.com/7/8/8/788bffd362b46a318a2cb0703a9e2ae77089e4e7.png)
.
'
'
is strictly better than another possible bid 
of his if the total valuation derived from 
.![$B_i = [b^i_1, b^i_2, b^i_3, b^i_4]$](http://latex.artofproblemsolving.com/7/b/2/7b233ca3e9056aa374e937b1133cd9c81726f407.png)
such that ![$b^i_j \in [0,1]$](http://latex.artofproblemsolving.com/b/9/6/b96c2a7618767934c8a44d9238c485fed39b8510.png)
and 
.
,
and 
such that given 
has a right angle at 
be the point on line 
such that 
and 
.
is chosen so that 
and 
is the bisector of 
.
and 
is the bisector of 
.
be the midpoint of 
be the point such that 
is a parallelogram. Prove that 
and 
are concurrent.
satisfying 
,
,
)
.
such that![\[
\sum_{i=1}^{n} \frac{k_i}{P'(a_i)} = \sum_{i=1}^{n} k_i.
\]](http://latex.artofproblemsolving.com/4/4/e/44e5eb17b17531e36c362e062471d9bdffc8144c.png)
be an arbitrary interior point of 
,
,
,
intersect 
,
at 
and 
intersect at 
,
intersects 
intersects 
.
is tangent to 
on the board gets replaced by 
,
is the largest prime divisor of 
Y by
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