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Baltic Way competition problems

Geometry Difficulty 4.5 AIME Prove it Baltic Way

Problem:

Given an isosceles triangle $A B C$ with $\angle A=90^\circ$ . Let $M$ be the midpoint of $A B$ . The line passing through $A$ and perpendicular to $C M$ intersects the side $B C$ at $P$ . Prove that $\angle A M C=\angle B M P$ .

Solution

Figure 2

Choose the point $K$ such that $A B K C$ is a square. Let $N$ be the point of intersection of $A P$ and $B K$ (see Figure 2). Since the lines $A N$ and $C M$ are perpendicular, $N$ is the midpoint of $B K$ . Moreover, triangles $A M C$ and $B N A$ are congruent, which gives
$\angle A M C=\angle B N A$
Since $|B M|=|B N|$ and $\angle M B P=\angle N B P$ , it follows that triangles $M B P$ and $N B P$ are congruent. This implies that
$\angle B M P=\angle B N P$
Combining (1) and (2) yields the required equality.

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