= Solution
The <condenser lens> should image the point source onto the small projection <lens>. Then every illuminated LCD point directs its central <light ray> into the projection <lens>, rather than wasting light elsewhere. In the paraxial idealization this gives approximately uniform useful illumination.
Put the LCD at $z=0$, the source at $z=-100\,\mathrm{mm}$, the projection <lens> at $z=u=29.5567\,\mathrm{mm}$, and the <condenser lens> at $z=-h$. The cone from the <condenser lens>'s rim, of radius $15\,\mathrm{mm}$, to the projection <lens> has radius
$$
r_{\rm LCD}=15\frac{u}{u+h}\,\mathrm{mm}
$$
at the LCD. Maximum approximately uniform illumination without unnecessary overfilling makes this equal the half-diagonal $7.5\,\mathrm{mm}$. Thus $h=u$: place the <condenser lens> $29.56\,\mathrm{mm}$ before the LCD, or $70.44\,\mathrm{mm}$ after the source. Its object and image distances are $100-h$ and $u+h$, so
$$
\boxed{f_c=\frac{(100-h)(u+h)}{100+u}=32.14\,\mathrm{mm}.}
$$
The illuminated disk just contains the rectangular LCD. Exact spatially uniform <irradiance> is not guaranteed by a single spherical <lens> and a point source; the result uses paraxial <geometrical optics>, negligible <optical aberration>, and an approximately uniform incident cone.
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