Typically, a "Theorem", "Lemma", "Inequality" or "Postulate" can be used directly. A common/obvious formula can also be used directly.
For example: Fermat's Little Theorem, Euler's Lemma, Hölder's Inequality or Bertrand's Postulate or the fact that ${n\choose k}=\frac{n!}{n! (n-k)!}$. These can be used directly. Other things like the Law of Sines are considered to be "Theorems", so they can be used directly as well.
However, others like "The Tangent Trick" (I have actually never heard of this), "Euler Line", and "Euler's Product Formula". The first 2 are not considered "Formal Theorems" and the last one is not a common/obvious formula, so they must be proved beforehand.
Another case is the case of "Trivialising the Question". Suppose a question says:
Prove that for all nonnegative reals $a_1, a_2, a_3, \cdots, a_n$, $\frac{1}{n}(a_1+a_2+a_3+\cdots+a_n)\geq (a_1 a_2 a_3 \cdots a_n)^{\frac{1}{n}}$.
This is clearly just the AM-GM Inequality. In this case, you can't just say "oh cos AM-GM, so Q.E.D." (this trivialises the question). If the question is asking you to prove the AM-GM Inequality (or whatever theorem), you have to prove it!
To conclude, a "Theorem", "Lemma", "Inequality" or "Postulate" or a common/obvious formula can be used directly. Those not considered as "Formal Theorems" or not a common/obvious formula must be proved. Hope this helps!
Edit 1: This is not specific to Inequality, but I just wanted to add more details. I am not very sure about the various properties of stuff in Geometry. What I do is that for the most basic properties, I quote it directly, and add a brief explanation. For the more advanced properties, I usually just prove them.
For example, I will write
Let $G$ be the centroid of $\Delta ABC$, and $D$ be the midpoint of $BC$. Then, $AG=2GD$ (This can be easily proven by the ratios between $[\Delta AGB]$, $[\Delta BGC]$ and $[\Delta CGA]$).
But for properties like
Let $H$ be the orthocenter of $\Delta ABC$. Then, the reflection of $H$ about any side of the triangle lies on the circumcircle.
I will prove these beforehand.
Edit 2:
As @CalvinLin has pointed out, you don't need to prove certain facts considered basic/"well-known" For example, at the IMO, you can quote that "squares are congruent to 0 or 1 mod 4" without having to prove it". For well known properties without a unique name (like reflection of orthocenter lies on circumcircle), then you should prove it.
