Problems of the Week

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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.

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?