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2025 SSMO Speed Round Problems/Problem 2

Problem

Let $A$ and $B$ be points such that $AB = 50$. Points $M$ and $N$ lie on $\overline{AB}$ such that $M$ lies between points $A$ and $N$ and $N$ lies between points $B$ and $M$. Given that $MN = 20$ and $AN \cdot BM$ is maximized, find the length of $AM$.

Solution

[asy] pair A,B,M,N; A=(0,0); B=(50,0); N=(35,0); M=(15,0); draw(A--B); dot(A,linewidth(4)); dot(B,linewidth(4)); dot(M,linewidth(4)); dot(N,linewidth(4)); label("$A$",A,dir(90)); label("$B$",B,dir(90)); label("$M$",M,dir(90)); label("$N$",N,dir(90)); [/asy]

Note that $AN+BM = AB + MN = 70$. By AM-GM, the value of $AN\cdot BM$ is at a maximum when $AN = BM = 35$. Thus, $AM = AN-MN = \boxed{15}$.

~Sedro

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