2023 WSMO Speed Round Problems/Problem 8

Problem

In regular octagon $ABCDEFGH$ of sidelength $4,$ quadrilaterals $ACEG$ and $BDFH$ are drawn. Find the square of the area of the overlap of the two quadrilaterals.

Solution

$[asy] pair a = dir(112.5); pair b = dir(67.5); pair c = dir(22.5); pair d = dir(337.5); pair e = dir(292.5); pair f = dir(247.5); pair g = dir(202.5); pair h = dir(157.5); label("$A$", a, NW); label("$B$", b, NE); label("$C$", c, E); label("$D$", d, E); label("$E$", e, SE); label("$F$", f, SW); label("$G$", g, W); label("$H$", h, W); draw(a--b--c--d--e--f--g--h--cycle,black+linewidth(2)); draw(a--c--e--g--cycle,blue+linewidth(1)); draw(b--d--f--h--cycle,blue+linewidth(1)); pair X = intersectionpoint(b--h,a--c); dot(X); label("$X$", X, S); pair Y = intersectionpoint(b--h,a--g); dot(Y); label("$Y$", Y, SE); [/asy]$

Let $X = BH\cap AC$ and $Y = BH\cap AG$ .

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2023 WSMO Speed Round Problems/Problem 8

Problem

Solution