The most difficult math problem in the senior high school entrance examination

Turn BCP clockwise around B to make BC and AB coincide.

Point p falls on point q and connects QP.

So BQ=BP=2, AQ=PC=3.

Because CBP angle = ABQ angle, QBP angle =90 degrees.

So QP=2* root number 2, angle QPB=45 degrees.

In the triangle APQ, AP= 1, AQ=3, QP=2 and root number 2.

That is, AP Square +QP Square =AQ Square (not difficult to see)

So APQ angle =90 degrees.

So angle APB= angle APQ+ angle QPB=90+45= 135 degrees.

The final answer is 135 degrees.