# Are functional-equation problems on topic?

Please vote up if you think this is on-topic, and vote-down if it isn’t. My functional equation questions always attract close votes, and some people say that functional equations are math textbook style. It is very frustrating to have people argue over, and today, someone argued with me about if functional equations are on topic.

• Can't vote because this is like asking whether math problems are on topic. It depends on the sort of math problem. Jun 7 '20 at 5:10
• Having answered the question whose comment thread prompted this post, I enjoyed solving it and do not feel it was trivial/textbook, contrary to one of the comments thereto. But I agree with what Rand says below that the blanket statement "functional equations are on topic" needs to be tempered with care that the topic does not become repetitive or mundane. Jun 7 '20 at 13:47
• I like math a lot and enjoy solving math problems (have been active on MSE some 4 years now), but funtional equations are for me pure math. I see no reason they would fit better on PSE than MSE.
– Jens
Jun 8 '20 at 0:07
• @Jens You're creating a false dichotomy: being able to ask about something on MSE doesn't mean we shouldn't be able to ask about it here too. Discussion of our site scope should be done without reference to other sites and their scopes. Jun 8 '20 at 13:00

## They are a subset of mathematics and should be treated as such.

In other words, the answer to "are functional equation problems on-topic" is the same as the answer to "are maths problems on-topic": namely, it depends. We have an established policy on which maths puzzles are on-topic and which are not:

So, what makes something a math puzzle rather than math problem? I think there's a few features.

• Clever or elegant solution, often an "aha" moment
• Unexpected problem statement.
• Unexpected or counterintuitive result.

In contrast, math problems tend to be "textbook". And by that I don't mean that they have to come from textbooks (or that textbooks can't contain math puzzles), but that they use standard, staightforward methods than anyone familiar with the subject is expected to know. They can be difficult, but their goal is to test comprehension of the material, not ingenuity. This doesn't apply to problems from math olympiads like the Putnam exam, which are designed to have clever solution.

## Most functional equation puzzles are on-topic.

Of course there are examples of trivial functional-equation problems which could be closed: solving something like $$\frac{d}{dx}f(x)=f(x)$$ (yes, differential equations are functional equations too) is a pretty standard problem and not really a puzzle.

But Olympiad-style functional equations, like most/all of the ones we've seen here so far, are likely to be counted as maths puzzles, like most other Olympiad-style maths questions: thinking outside the box, not using standard straightforward methods, is rather the point.

## Functional equations should not, in general, be on topic

As a former math Olympian myself, I say this for several reasons.

1. Not all Olympiad questions should be on topic.

While numerous Olympiad problems are purely "elementary" in that any laymen has enough knowledge to solve it, some are way too technical.

The year I qualified for USAMO, the following FE was present:

(USAMO 2018 #2) Find all functions $$f:(0,\infty) \rightarrow (0,\infty)$$ such that

$$f\left(x+\frac{1}{y}\right)+f\left(y+\frac{1}{z}\right) + f\left(z+\frac{1}{x}\right) = 1$$for all $$x,y,z >0$$ with $$xyz =1.$$

I did not solve it, despite my efforts. In fact, I couldn't, because there was some technical mathematical theory that is required to resolve this functional equation.

To drive the point home, would the following question really be on topic for this site?

Let the side $$AC$$ of triangle $$ABC$$ touch the incircle and the corresponding excircle at points $$K$$ and $$L$$ respectively. Let $$P$$ be the projection of the incenter onto the perpendicular bisector of $$AC$$. It is known that the tangents to the circumcircle of triangle $$BKL$$ at $$K$$ and $$L$$ meet on the circumcircle of $$ABC$$. Prove that the lines $$AB$$ and $$BC$$ touch the circumcircle of triangle $$PKL$$.

Just because a mathematical problem is "clever" does NOT mean it is on topic. Yes, there are many clever math problems that are as much math as they are puzzle. However, we must draw a line.

1. If functional equations are on topic, what else is?

The FE that the OP contributed was, fortunately, elementary. It could be resolved using standard techniques. But there are many, MANY FE's that are completely inaccessible to the vast, vast majority of the PSE userbase.

Remember the USAMO FE I presented? If you think that's on topic, I hope you'll enjoy reading the solutions here.

If we allow OP's FE to be on topic, will that USAMO FE be on topic? Will ALL FE's be on topic? Will ALL Olmypiad questions be on topic? The slope is slippery. There is a place for Olympiad questions on Art of Problem Solving, there is no need for them here. Allowing such Olympiad questions hurts our identity. Trust me, you do not want all math Olympians flocking over here posting their FE's, Olympiad geometry configurations, monstrous classical inequalities, and so forth.

Could we determine whether an Olympiad question is "on topic" based on the technicality of its resolution? Possibly. Though I get the feeling that such antics are silly.

I may be being a bit extreme in my opinion here, but my point stands that we should be very careful around letting some standard Olympiad questions to be on topic.

• This is a well articulated and reasoned counterpoint. Jun 7 '20 at 23:21