$$\underset{\mathrm{0}} {\overset{{x}} {\int}}{f}\left({t}\right){dt}+\underset{{x}} {\overset{\mathrm{2}{x}} {\int}}{tf}\left({t}\right){dt}=\underset{\mathrm{0}} {\overset{\mathrm{2}{x}} {\int}}\left(\mathrm{1}−{t}\right){f}\left({t}\right){dt} \\$$$${f}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)=? \\$$
$$\mathrm{differentiate}\:\mathrm{w}.\mathrm{r}.\mathrm{t}\:{x} \\$$$${f}\left({x}\right)+\mathrm{2}{xf}\left(\mathrm{2}{x}\right)\frac{{d}}{{dx}}\mathrm{2}{x}−{xf}\left({x}\right)=\left(\mathrm{1}−\mathrm{2}{x}\right){f}\left(\mathrm{2}{x}\right)\frac{{d}}{{dx}}\mathrm{2}{x} \\$$$${f}\left({x}\right)+\mathrm{4}{xf}\left(\mathrm{2}{x}\right)−{xf}\left({x}\right)=\mathrm{2}\left(\mathrm{1}−\mathrm{2}{x}\right){f}\left(\mathrm{2}{x}\right) \\$$$${x}=\frac{\mathrm{1}}{\mathrm{2}},\:{f}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)+\mathrm{2}{f}\left(\mathrm{1}\right)−\frac{\mathrm{1}}{\mathrm{2}}{f}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)=\mathrm{0} \\$$$${f}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)=−\mathrm{4}{f}\left(\mathrm{1}\right) \\$$