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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 \(n\) such that there exist integers \(a_1, a_2, \ldots, a_n\) that satisfy


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 \(\frac{p}{q}\), where \(p\) and \(q\) are coprime positive integers.

Find the value of \(p + q.\)

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

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

Clarification: In the answer options, \(e \, (\approx 2.71828)\) 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 \([0, 1]\). 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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