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

\[a_1^5+a_2^5+\cdots+a_n^5=28?\]

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 \(\frac{p}{q}\), 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.

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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