Answer
$N=0.082\ mol$
Work Step by Step
Initial pH:
$pH=pKa+\log([A^-]/[HA])$
$[A^-]=[HA]\rightarrow pH=pKa=4.76$
Case of adding strong acid: $pH'=pH-1=3.76$
$3.76=4.76+\log([A^-]/[HA])$
$0.1=[A^-]/[HA]$
$[A^-]=(0.10\ M\times 1L-N)/V_f$
$[HA]=(0.10\ M\times 1L+N)/V_f$
$0.1=(0.1-N)/(0.1+N)$
$0.01+0.1N=0.1-N$
$0.09=1.1N$
$N=0.082\ mol$