A cylindrical jar has radius $r$. To what height $h$ should it be filled with a liquid so that the force exerted by the liquid on the sides of the jar equals the force exerted on the bottom?

Let a cylinder with radius r be filled with a homogeneous liquid of density $\rho$ up to a height h. Pressure at the bottom of the cylinder, $p_1=\rho gh$ Pressure at the top of the liquid surface, $p_2=0$ Average pressure on the sides of the cylinder,
$p=\frac{p_1+p_2}{2}=\rho gh/2$
Force on the sides of the vessel = average pressure x area $=\frac{\rho gh}{2}\times 2\pi rh$
$=\pi r\rho g{h}^2$
Force on the bottom of the vessel $=p_1\pi r^2$ $=\rho gh\pi r^2$
Two forces will be equal if
$$\begin{gathered} \pi r\rho g{h}^2=\rho gh\pi r^2 \\ h=r \end{gathered}$$