A question on frictional force?

A 35 kg crate is at rest on the floor. A man attempts to push it across the floor by

applying a 100 N force horizontally. (a) Take the coefficient of static friction

between the crate and floor to be 0.37 and show that the crate does not move. (b)

A second man helps by pulling up on the crate. Calculate the minimum vertical

force he must apply so that the crate starts to move across the floor. (c) If the

second man applies a horizontal rather than a vertical force, what minimum force,

in addition to the 100 N force of the first man, must he exert to get the crate

started?

I saw the solution to subquestion (b), and the solution implies that if the frictional force is equals to the force applied, the crate will start to move, but isn't it by convention that if the applied force is still equals to the static friction, the object will not move?

The solution for subquestion (b) is:

For the box to move, Frictional force = Applied Force = 100N

Fn = Frictional force/coefficient of static friction = 100/0.37 = 270N

The current normal force = 343N

So in order for the crate to move, the second man has to lift the crate with a force of (343-270)N = 73N