Here is another UKMT Senior Challenge problem from 2017, which has a straight-forward solution:
“The diagram shows a circle of radius 1 touching three sides of a 2 x 4 rectangle. A diagonal of the rectangle intersects the circle at P and Q, as shown.
What is the length of the chord PQ?
__A_√5____B_4/√5____C_√5 – 2/√5____D_5√5/6____E_2”
See the Circle in Slot Problem