To what extent should you prove something in math contests?

Question

Next year, I'll be attending the IMO and although I've done many math contests over the years, most of the questions wanted numerical answers and the ones that required a proof were basic and simple so I'm not really used to formal proof writing; in fact in math class, my teacher always tells me that I do a lot of the steps in my head and although she knows that I know how to prove something, I can't simply do some of the steps in my head.

Example:

Prove for integers numbers $a_1, a_2,..., a_n$, there are integers $x_1, x_2, ..., x_n$ that satisfy $a_1x_1 + a_2x_2 + ... + a_nx_n = d$ where $d = gcd(a_1,a_2,...,a_n)$.

My proof:

Assume this is true for all the natural numbers less than $n$ and now we'd like to prove it for $n$(here, I just assume the reader knows Bezout's Identity).Let $d' = gcd(a_1,a_2,...,a_{n-1})$ so we have $d = gcd(d', a_n)$. There are integers(for example, this is one of the parts she says i do in my head) $y_1, y_2,...,y_{n-1}$ such that $a_1y_1+a_2y_2+...+a_{n-1}y_{n-1} = d'$. We also know there are integers $x,y$ such that $d'x + a_ny = d$(again, this is another one of those instances).

Now, I've been trying to stop myself from doing this but it got me thinking, to what extent should I prove something? What I mean is, what should I assume the person who's reviewing my proof knows? Is it just basic arithmetics? Properties of certain things such as GCD and LCM? Basic properties such as $gcd(a,b) = gcd(a, c) = 1$ iff $gcd(a,bc) = 1$? I know this might be too broad but I'm just asking for proofs in number theory and not geometry, algebra, etc.

P.S: If you think this question doesn't feet the criteria of math.se, what's an appropriate place to post this question?

Thank you so much in advance!

Answer 1

If I were grading this proof, the questions I would have are:

1. It looks like you are proceeding by induction. Have you established the base case?
2. How do you know $d=\text{gcd}(d', a_n)$?

Note, I am not concerned about the existence of $y_1,\dots,y_{n-1}$, since that follows from the induction hypothesis.

3. How do you know such $x$ and $y$ exist? (This is the same as establishing the base case for the induction.)

It's not so much a matter of what the reader knows but rather what you are able to justify. Given that the whole point of the problem is showing that the GCD of a bunch of numbers can be written as an integer combination of those numbers, you should definitely justify that for the case $n=2$, and not just assume it.