2014 USAJMO Problems/Problem 2

Problem

Let $\triangle{ABC}$ be a non-equilateral, acute triangle with $\angle A=60\textdegrees$ (Error compiling LaTeX. Unknown error_msg), and let $O$ and $H$ denote the circumcenter and orthocenter of $\triangle{ABC}$ , respectively.

(a) Prove that line $OH$ intersects both segments $AB$ and $AC$ .

(b) Line $OH$ intersects segments $AB$ and $AC$ at $P$ and $Q$ , respectively. Denote by $s$ and $t$ the respective areas of triangle $APQ$ and quadrilateral $BPQC$ . Determine the range of possible values for $s/t$ .

Solution

We draw a diagram to not lose points:

Part a

Part b

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2014 USAJMO Problems/Problem 2

Problem

Solution