MathDB
Straightforward vectors

Source: RMO District Round, 9th grade, 2007

January 21, 2008
vectorgeometry proposedgeometry

Problem Statement

Consider △ABC \triangle ABC and points M∈(AB) M \in (AB), N∈(BC) N \in (BC), P∈(CA) P \in (CA), R∈(MN) R \in (MN), S∈(NP) S \in (NP), T∈(PM) T \in (PM) such that \frac {AM}{MB} \equal{} \frac {BN}{NC} \equal{} \frac {CP}{PA} \equal{} k and \frac {MR}{RN} \equal{} \frac {NS}{SP} \equal{} \frac {PT}{TN} \equal{} 1 \minus{} k for some k∈(0,1) k \in (0, 1). Prove that △STR∼△ABC \triangle STR \sim \triangle ABC and, furthermore, determine k k for which the minimum of [STR] [STR] is attained.
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