Find all functions who's self-convolution is the same as their square.
To make it explicit, find all $f:\mathbb{R} \rightarrow \mathbb{R}$ such that: $$ \int_{-\infty}^{\infty} f(t - \tau)f(\tau) d\tau = (f(t))^2 $$
Or the same problem on a limited domain $f:(-a,a) \rightarrow \mathbb{R}$ $$ \int_{-a}^{a} f(t - \tau)f(\tau) d\tau = (f(t))^2 $$
Edit: Anne Bauval has asked for a motivation for this problem. If you consider any non-linear ODE in 1D with derivatives beyond the first, there are no simple methods of solution. For example:
$$x''' + (x'')^2 + x^2 x' = 0$$
The fourier transform of these types of problems become polynomials of multiplication and convolution for example: $$(i\omega)^3\hat{x}(\omega) + [(i\omega)^2\hat{x}(\omega)] * [(i\omega)^2\hat{x}(\omega)] + [(i\omega)\hat{x}(\omega)] * \hat{x}(\omega) * \hat{x}(\omega) = 0$$
The above example is the simplest polynomial using multiplication and convolution. So, solving this problem would be a first step towards solving more complex convolution and multiplication polynomials. At some point, this method could allow you to solve previously unsolvable non-linear ODEs.
Bruno B.: Pointed out a related, but not identical question math.stackexchange.com/q/3336615/1104384