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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 nn that maximizes the volume of unit nn-ball {(x1,x2,,xn)Rnx12+x22++xn21}? \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 nn-ball of radius RR is Vn(R)=πn2Γ(n2+1)Rn,V_n(R)=\frac{\pi^{\frac n2}}{\Gamma \left(\frac n2+1\right)}R^n, where Γ()\Gamma(\cdot) denotes the gamma function.

The numbers 1,2,3,...,2017,20181,2,3,...,2017,2018 are divided into 2 groups:

a1<a2<<a1009andb1>b2>>b1009.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 a1b1+a2b2++a1009b1009?|a_1-b_1|+|a_2-b_2|+\cdots+|a_{1009}-b_{1009}|?

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

If S(n)S(n) is an odd integer, what is the sum of all possible 1n?\frac1n?

A fair, 66-sided die is rolled 2020 times, and the sequence of the rolls is recorded.

CC 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 6.6. These subsequences don't have to be separate and can overlap each other. For example, the sequence of 2020 rolls 12334222111366141523 12334222111366141523 contains the ten subsequences 123,33,42,222,2211,1113,6,6,141,15123, 33, 42, 222, 2211, 1113, 6, 6, 141, 15 which all add up to 6,6, so C=10C=10 in this case.

The expected value of CC is equal to ab\frac{a}{b} for coprime positive integers aa and b.b.

What is a+b?a+b?

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