It depends on the direction of the applied force. Consult the attached free body diagram.
If the block is held in place by a force acting perpendicular to the wall, then the block will never move. So the applied force must be acting at some angle θ between 0° and 90° (with respect to the positive horizontal axis).
By Newton's second law, we have
• net perpendicular force
∑ F (⟂) = F cos(θ) - N = 0
• net parallel force
∑ F (//) = F sin(θ) - f - mg = ma
However, we're interested in the minimum force required to get the block moving. This minimum force would exactly counter the frictional force keeping the block in place, so that
F sin(θ) - f - mg = 0
The magnitude of friction is 0.5 N (that is, half the normal force's magnitude, not 0.5 Newtons). So
F sin(θ) - 0.5 N - mg = 0 ⇒ F sin(θ) - 0.5 F cos(θ) = mg
⇒ F (sin(θ) - 0.5 cos(θ)) = mg
⇒ F = mg/(sin(θ) - 0.5 cos(θ))
(where m = 10 kg and g = 9.8 m/s²)
