Hypothesis testing: do we reject if the p-value is the exact same as the significance level α? Do we reject the null hypothesis if the p-value is the exact same as our level of significance α?
For example, with α = 0.05, we observe p = 0.05.
Should we reject? Or, do we reject only if p is strictly smaller than α?
 A: The common thresholds of 0.1, 0.05, and 0.01 that p-values are evaluated against are intended to be heuristics rather than steadfast rules. The smaller the p-value the better, the less likely it is for the null hypothesis to be observed in the data. Therefore, these thresholds are not meant to represent strict "cutoffs" where decisions are based only on whether a p-value passes that specific cutoff. For more details on interpretation of p-values, please see Wasserstein & Lazar (2016).
Your best recourse would be to indicate that your statistical model has a pretty low p-value and although it doesn't entirely cross the threshold of 0.05, there is generally enough evidence to reject the null hypothesis.
A: A good answer from Anavir. In practice, the value of $\alpha$ one uses is quite arbitrary.
To address your issue more directly, though, the answer is it doesn't matter!
Why? For simplicity's sake, we're going to assume we're working with simple hypotheses, with continuous distributions specified under the null and alternative hypotheses. When we "fix $\alpha$" we really ensure that $Pr(\text{rejecting } H_0 | H_0 \text{ is true}) \leq \alpha$.
For continuous real-valued random variable $X$ and $x \in \mathbb{R}$, as I'm sure you know, $Pr(X = x) = 0$.  Also, notice that the $p$-value, which we'll denote as $P$ is a continuous random variable in and of itself! (In fact, under the null in this case, its a uniform random variable on $[0,1]$, but that's besides the point). The $p$-value which we observed, which we will denote as $p$ is a realization of $P$.
If $Pr(P \leq \alpha) = \alpha$, then
$$Pr(P \leq \alpha) = Pr(P = p) + Pr(P < \alpha) = Pr(P < \alpha) = \alpha$$.
Indeed, rejecting when your p-value is less than or equal to $\alpha$, or strictly less than $\alpha$, makes no difference. We still satisfy the constraints we set out for ourselves.
