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2017-06-19 Advanced

         

I have a quadrilateral with perimeter 4 and area 1.

Find the maximum value of the product of its diagonal lengths to 2 decimal places.

What is the minimum value of nn such that there exist integers a1,a2,,ana_1, a_2, \ldots, a_n that satisfy

a15+a25++an5=28?a_1^5+a_2^5+\cdots+a_n^5=28?

Alice and Bob stand on opposite vertices of a regular octahedron. At the beginning of every minute,

  • each chooses an adjacent vertex uniformly at random and moves towards it;
  • each moves at a constant rate of 1 edge per minute;
  • they will stop when they meet up.

The expected value of the number of minutes until they meet up is equal to pq\frac{p}{q}, where pp and qq are coprime positive integers.

Find the value of p+q.p + q.


Note: It is possible that they would meet at a vertex or at the midpoint of an edge.

Evaluate limn010101nx1+x2++xndx1dx2dxn. \lim_{n \to \infty} \displaystyle \int_0^1 \int_0^1 \ldots \int_0^1 \dfrac{n}{x_1+x_2+\dots+x_n}\, dx_1 dx_2 \ldots dx_n.

Clarification: In the answer options, e(2.71828)e \, (\approx 2.71828) is the Euler's number.


Note: This problem is a generalization of my previous problem.

You have recently bought the Random Number Generator 4000 from your local hardware store. This remarkable machine outputs numbers randomly and uniformly in the range [0,1][0, 1]. You decide to create a list of these numbers and find the total sum.

What is the expected number of times the machine should operate to generate a number before the total sum is at least 1?

Answer to 3 decimal places.

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