4

Part of 2024 Portugal MO

Problems(1)

2024 Portuguese MO Finals Day 2 P4

Source: 2024 Portuguese MO Finals Day 2 P4

A circle inscribed in the square

ABCD

10

cm, intersects sides

BC

AD

M

N

respectively. The point

I

is the intersection of

AM

with the circle different from

M

P

is the orthogonal projection of

I

MN

. Find the value of segment

PI