Weekly Contest #71
(Ended)
Standings
Q1.
Each face of two noncongruent parallelepipeds is a rhombus whose diagonals have lengths 
and 
. The ratio of the volume of the larger of the two polyhedra to the volume of the smaller is 
, where 
and 
are relatively prime positive integers. Find 
. A parallelepiped is a solid with six parallelogram faces such as the one shown below.
![[asy] unitsize(2cm); pair o = (0, 0), u = (1, 0), v = 0.8*dir(40), w = dir(70); draw(o--u--(u+v)); draw(o--v--(u+v), dotted); draw(shift(w)*(o--u--(u+v)--v--cycle)); draw(o--w); draw(u--(u+w)); draw(v--(v+w), dotted); draw((u+v)--(u+v+w)); [/asy]](https://latex.artofproblemsolving.com/b/c/c/bcc8a67600e8a557236d37e2d01ae1737ef052b5.png)
Q2.
Find the largest prime number 
for which there exists a complex number 
satisfying
- the real and imaginary part of are both integers;
and- there exists a triangle whose three side lengths are
the real part of 
and the imaginary part of 
Q3.
There exists a unique positive integer 
for which the sum
![\[U=\sum_{n=1}^{2023}\left\lfloor\dfrac{n^{2}-na}{5}\right\rfloor\]](https://latex.artofproblemsolving.com/6/b/0/6b010978a55642b7f7aabb2df2f7cf45c9db937a.png)
is an integer strictly between 
and 
. For that unique , find 
.
(Note that 
denotes the greatest integer that is less than or equal to 
.)
Q4.
Alice knows that 
red cards and black cards will be revealed to her one at a time in random order. Before each card is revealed, Alice must guess its color. If Alice plays optimally, the expected number of cards she will guess correctly is 
where and are relatively prime positive integers. Find 