First, we consider an example of how one might make up a question.

Here is a nice fact:

(1/2)+(1/3)+(1/6)=1

Let’s have some fun with this fact by making it the answer of a question. Think of it as a mystery equation, and it’s our job to point the problem-solver in the right direction to find it. Hmm…

The sum of these 3 denominators is 11. After a bit of thought we see that {2,3,6} is the only set of 3 positive integers with sum 11, such that the sum of reciprocals is equal to 1. So we will tell the solver that we have x numbers whose sum is f(x) and we’ll pick a function f so that f(3)=11. We’ll also mention that the sum of reciprocals is exactly 1.

Now we need a hint so that the solver can be directed towards x=3.

Consider the more general situation where the x numbers are only required to be positive real numbers instead of positive integers. If the sum of reciprocals is 1, then it is well known that the sum f(x) is minimized if the x numbers are equal to each other, and thus equal to x (think of famous inequalities!). Hence x^2<=f(x).

Let’s pick f(x)=5x-4. Then x^2<=5x-4 and this places x between 1 and 4. We are ready to state the problem. Here’s a version of it, close to the way it appeared on the 2005 William Lowell Putnam Mathematical Competition:

Find all positive integers x, k_1, k_2, …, k_x, such that the sum k_1+k_2+…+k_x=5x-4 and such that (1/k_1)+…+(1/k_x)=1.

Our mystery equation corresponds to the case where x=3. There are a couple of other solutions, corresponding to x=1 and x=4.

To be clear, I did not make up this question. I also have no knowledge of what was going through the mind of the person who did make up this question. I was merely having fun with this beautiful problem!

My point is that this question, while incredibly difficult, may not seem so bad from the point of view of the *problem-maker*. It’s only difficult for the problem-solver. More importantly, I believe that the problem-solver has a much better chance of answering this type of question after having experience as a problem-maker. As much as possible, we want to be proactive rather than reactive.

So I propose that we practice making up good questions.

Good luck on the Putnam!

(A version of this post will be included in an up-coming book I’m working on, about mathematical problem-solving).

(Click here for a pdf version of this post, with better equation formatting).

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