# Problems of the Week

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# 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:

$a_1<a_2<\cdots<a_{1009}\qquad \text{and} \qquad b_1>b_2>\cdots>b_{1009}.$

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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