When I am working on my research, or on MO/math.SE questions, I often find myself thinking in a way which reminds me of the feel of solving Olympiad problems. If I then solve the problem, I try to find a very special case of my problem which is still challenging, and can be stated and solved using only material on the Olympiad curriculum. I then e-mail Kiran Kedlaya and say "Hey Kiran, do you think this would be a good Olympiad problem?" If he thinks so, he proposes it to the USAMO committe.
I wrote Problem 2 on the 2010 USAMO in this way; it is Theorem 3.2 of this paper specialized to the case that $W_0$ is the group $S_n$. The fact that the "total number of moves" referred to in the theorem is at most $\binom{n}{3}$ is computed in Section 5.2.
I think I send about one problem a year; but most of them get rejected.
UPDATE 2014 Problem B4 of the 2014 Putnam was mine. Let $F(x,y,z) = \sum F_{ijk} x^i y^j z^k$ be a homogenous polynomial of degree $n$ with positive real coefficients. We say that $F$ is hyperbolic with respect to the positive orthant if, for all $(u_1,v_1,w_1)$ and $(u_2,v_2,w_2) \in \mathbb{R}_{> 0}^3$, the polynomial $f(t) = F(tu_1+u_2,tv_1+v_2,tw_1+w_2)$ has $n$ negative real roots.
In this paper, I show that there are constants $V_1$ and $V_2$ (dependent on $n$) so that,
(1) if $F$ is hyperbolic with respect to the positive orthant, then $F_{i(j+1)(k+1)} F_{(i+1)j(k+1)} > V_1 F_{i(j+1)(k+1)} F_{(i+2)jk}$ and the same for all permutations of the indices
(2) if $F_{i(j+1)(k+1)} F_{(i+1)j(k+1)} > V_2 F_{i(j+1)(k+1)} F_{(i+2)jk}$ and the same for all permutations of the indices, then $F$ is hyperbolic with respect to the positive orthant.
The proof is nonconstructive; I also (Theorem 20) give an explicit value of $V_1$. I was thinking about whether I could give a concrete value for $V_2$. The problem was too hard, so I thought instead about homogenous polynomials in two variables, which is the same as inhomogenous polynomials in one variable. At this point, I was basically looking for a converse to Newton's inequality: I wanted a constant $C$ so that, if $a_k^2 > C a_{k-1} a_{k+1}$, then all the roots of $\sum_{k=0}^n a_k z^k$ are real. The result in one variable wasn't worth publishing, but I figured I could make a nice problem by choosing a particular polynomial and asking people to prove the roots were real.
UPDATE 2020 Problem 6 of the 2020 USOMO was mine. It is the key computation from this MO answer, specialized to the case that the matrix $X$ has rank one, so $X_{ij} = x_i y_j$. The rank one case turned out not to be easier than the general case, which is why it doesn't show up in that MO thread, but it is one of the things I thought about when working on that answer, and I noticed at the time that it looked like a strengthening of the rearrangement inequality.
Given that you mention that you have "no prior experience [in] writing proofs", you may find Velleman's How to Prove It: A Structured Approach to be a helpful preparation for Spivak's wonderful book. And it is a wonderful book, so please keep at it, or come back to it after Velleman. Note that I haven't read Velleman, but it seems to get universal praise for "preparing students to make the transition from solving problems to proving theorems".
I'd recommend that you do not obsess about finishing all, or even most, of the problems on first reading. I think it's worthwhile to give it a first read without worrying that you are getting every ounce out of it. If you feel like you're getting most of it, keep going. Do plan to give it a second read, though.
Also, there's some good discussion on how to read mathematics books in how-to-read-a-book-in-mathematics.
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As far as advanced math goes, Spivak's problems are rather easy. I recommend you not let this discourage you though. Instead you should spend a very long time solving the problems until eventually you have brought yourself up to that level. This is the way to succeed in advanced math.