There’s a common error I keep seeing on OTIS applications, so I’m going to document the error here in the hopes that I can pre-emptively dispel it. To illustrate it more clearly, here is a problem I made up for which the bogus solution also gets the wrong numerical answer:

**Problem**: Suppose for positive real numbers , , . Find the minimum possible value of .

The wrong solution I keep seeing goes like so:

**Nonsense solution**: By AM-GM, the minimum value of is . Equality occurs if , which means . Since , this gives , so the minimum possible value is .

The issue is that the first line does not make sense. It’s worse than just “false” or “wrong”: it’s a type-error, meaning it cannot even be formulated into a statement which could then be regarded as either true or false.

What do I mean by “type-error”? In mathematics, coherent statements are usually either true or false. Examples of false statements include or (from the Indiana Pi bill and *1984*, respectively). However, it’s possible to write statements that are not merely false, but not even “grammatically correct”, such as

To call these equations false is misleading. If your friend asked you whether , you would say “no”. But if your friend asked whether equals the identity matrix, the answer is a different kind of “no”; in the words Tom Leinster (section 3.3), the best response is “your question makes no sense”.

In this case, one seeks a minimum of a function in three variables , , satisfying some constraint. So this minimum should be an absolute constant, hence independent of , , .

In other words, if and are nonconstant functions, then

- the statement “ with equality when ” does make sense; but
- “ is the minimum of with equality when ” is a type-error. The minimum value of a function is not supposed to depend on the inputs.

And of course, the *actual* minimum value to this example problem is . Or rather, although , it can take any value as close to as you want, say by taking , .

By the way, food for thought: what’s the maximum possible value of ?

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## Published by Evan Chen (陳誼廷)

I am a math olympiad coach and a PhD student at MIT.
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