Art of Problem Solving
Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus
During AMC 10A/12A testing, the AoPS Wiki is in read-only mode and no edits can be made.

2023 WSMO Speed Round Problems/Problem 8: Difference between revisions

Newer edit →
Pinkpig (talk | contribs)
editor
907 edits
Created page with "==Problem== In regular octagon <math>ABCDEFGH</math> of sidelength <math>4,</math> quadrilaterals <math>ACEG</math> and <math>BDFH</math> are drawn. Find the square of the ar..."
 
Pinkpig (talk | contribs)
editor
907 edits
No edit summary
Line 4: Line 4:


==Solution==
==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 <math>X = BH\cap AC</math> and <math>Y = BH\cap AG</math>.

Revision as of 10:43, 12 September 2025

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 $Y = BH\cap AG$.

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2023_WSMO_Speed_Round_Problems/Problem_8&oldid=260414"