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

Difference between revisions of "2022 USAJMO Problems/Problem 4"

(→‎Solution)
m
Line 25: Line 25:
  
 
(Lokman GÖKÇE)
 
(Lokman GÖKÇE)
 +
 +
==See Also==
 +
{{USAJMO newbox|year=2022|num-b=3|num-a=5}}
 +
{{MAA Notice}}

Revision as of 18:04, 6 October 2023

Problem

Let $ABCD$ be a rhombus, and let $K$ and $L$ be points such that $K$ lies inside the rhombus, $L$ lies outside the rhombus, and $KA=KB=LC=LD$. Prove that there exist points $X$ and $Y$ on lines $AC$ and $BD$ such that $KXLY$ is also a rhombus.

Solution

(Image of the solution is here [1])

Let's draw ($\ell$) perpendicular bisector of $\overline{KL}$. Let $X, Y$ be intersections of $\ell$ with $AC$ and $BD$, respectively. $KXLY$ is a kite. Let $O$ mid-point of $\overline{KL}$. Let $M$ mid-point of $\overline{BD}$ (and also $M$ is mid-point of $\overline{AC}$). $X, O, Y$ are on the line $\ell$.

$BK=DL$, $BX=XD$, $XK=XL$ and so $\triangle BXK \cong \triangle DXL$ (side-side-side). By spiral similarity, $\triangle BXD \sim\triangle KXL$. Hence, we get \[\angle XBD = \angle XDB = \angle XKL = \angle XLK = b .\]

Similarly, $AK =CL$, $YK = YL$, $YA=YC$ and so $\triangle BXK \cong \triangle DXL$ (side-side-side). From spiral similarity, $\triangle YKL\sim \triangle YAC$. Thus,

\[\angle YAC = \angle YCA = \angle YKL = \angle YLK = a .\]

If we can show that $a=b$, then the kite $KXLY$ will be a rhombus.

By spiral similarities, $\dfrac{BX}{BD} = \dfrac{XK}{KL}$ and $\dfrac{YA}{AC} = \dfrac{YK}{KL}$. Then, $KL = \dfrac{BD \cdot XK}{BX} = \dfrac{AC \cdot YK}{YA}$.

$\dfrac{XK}{YK} = \dfrac{AC \cdot BX}{AY \cdot BD}$. Then, $\dfrac{YK}{XK} = \dfrac{(BD/2) \cdot AY}{(AC/2) \cdot BX} = \dfrac{\sin b}{\sin a}$. Also, in the right triangles $\triangle KXO$ and $\triangle KYO$, $\dfrac{YK}{XK} = \dfrac{OK/\cos a}{OK/\cos b} = \dfrac{\cos b}{\cos a}$. Therefore, \[\dfrac{\sin b}{\sin a} = \dfrac{\cos b}{\cos a}.\]

$\sin a \cos b = \sin b \cos a \implies \sin a \cos b - \sin b \cos a = 0 \implies \sin(a-b) = 0$ and we get $a=b$.

(Lokman GÖKÇE)

See Also

2022 USAJMO (Problems • Resources)
Preceded by
Problem 3 		Followed by
Problem 5
1 • 2 • 3 • 4 • 5 • 6
All USAJMO Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2022_USAJMO_Problems/Problem_4&oldid=199237"