# 2018-08-13 Advanced

In the 17th century, French mathematician Pierre de Fermat challenged his colleagues with a variation of this question:

26 is an integer that is one more than a perfect square and one less than a perfect cube. Find another integer that has this property.

Does another integer like this exist?

What is the value of the dimension $n$ that maximizes the volume of unit $n$-ball $\left\{ \left( x_1,x_2,\cdots,x_n \right) \in\mathbb{R}^n\, \mid \, x_1^2+x_2^2+\cdots+x_n^2\le 1 \right\}?$

Hint: The volume of $n$-ball of radius $R$ is $V_n(R)=\frac{\pi^{\frac n2}}{\Gamma \left(\frac n2+1\right)}R^n,$ where $\Gamma(\cdot)$ denotes the gamma function.

The numbers $1,2,3,...,2017,2018$ are divided into 2 groups:

$a_1b_2>\cdots>b_{1009}.$

What is the sum of all possible values of $|a_1-b_1|+|a_2-b_2|+\cdots+|a_{1009}-b_{1009}|?$

If $n$ is a positive integer, let $S(n)$ be the sum of all the positive divisors of $n$.

If $S(n)$ is an odd integer, what is the sum of all possible $\frac1n?$

A fair, $6$-sided die is rolled $20$ times, and the sequence of the rolls is recorded.

$C$ is the number of times in the 20-number sequence that a subsequence (of any length from one to six) of rolls adds up to $6.$ These subsequences don't have to be separate and can overlap each other. For example, the sequence of $20$ rolls $12334222111366141523$ contains the ten subsequences $123, 33, 42, 222, 2211, 1113, 6, 6, 141, 15$ which all add up to $6,$ so $C=10$ in this case.

The expected value of $C$ is equal to $\frac{a}{b}$ for coprime positive integers $a$ and $b.$

What is $a+b?$

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