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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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