Let $n$ and $k$ be integers satisfying $\binom{2k}{2} + n = 60$.  It is known that $n$ days before Evan's 16th birthday, something happened.  Compute $60-n$.

[i]Proposed by Evan Chen[/i]
1. We start with the given equation:
   \[
   \binom{2k}{2} + n = 60
   \]
   Recall that the binomial coefficient $\binom{2k}{2}$ is given by:
   \[
   \binom{2k}{2} = \frac{(2k)(2k-1)}{2}
   \]
   Simplifying this, we get:
   \[
   \binom{2k}{2} = k(2k-1)
   \]

2. Substitute $\binom{2k}{2}$ into the original equation:
   \[
   k(2k-1) + n = 60
   \]

3. We need to find integer values of $k$ and $n$ that satisfy this equation. Let's test some values of $k$:
   - For \( k = 5 \):
     \[
     k(2k-1) = 5(2 \cdot 5 - 1) = 5 \cdot 9 = 45
     \]
     Substituting this into the equation:
     \[
     45 + n = 60 \implies n = 15
     \]

4. Now, we need to compute $60 - n$:
   \[
   60 - n = 60 - 15 = 45
   \]

The final answer is \(\boxed{45}\).