2018 USAJMO Problems/Problem 3

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Problem

($*$) Let $ABCD$ be a quadrilateral inscribed in circle $\omega$ with $\overline{AC} \perp \overline{BD}$. Let $E$ and $F$ be the reflections of $D$ over lines $BA$ and $BC$, respectively, and let $P$ be the intersection of lines $BD$ and $EF$. Suppose that the circumcircle of $\triangle EPD$ meets $\omega$ at $D$ and $Q$, and the circumcircle of $\triangle FPD$ meets $\omega$ at $D$ and $R$. Show that $EQ = FR$.

[asy] unitsize(3cm);  real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */  pen dotstyle = black; /* point style */  pen wrwrwr = rgb(0.3803921568627451,0.3803921568627451,0.3803921568627451); pen rvwvcq = rgb(0.08235294117647059,0.396078431372549,0.7529411764705882);   draw((0.5922362871684147,0.8057643453026269)--(0.8660254037844386,-0.5)--(0.06064145095757084,-0.9981596137020172)--(-0.9987724554622847,0.04953364724950819)--cycle, linewidth(2) + rvwvcq);  /* draw figures */ draw(circle((0,0), 1), linewidth(2) + wrwrwr);  draw((0.8660254037844386,-0.5)--(-0.9987724554622847,0.04953364724950819), linewidth(2) + wrwrwr);  draw((0.5922362871684147,0.8057643453026269)--(0.8660254037844386,-0.5), linewidth(2) + rvwvcq);  draw((0.8660254037844386,-0.5)--(0.06064145095757084,-0.9981596137020172), linewidth(2) + rvwvcq);  draw((0.06064145095757084,-0.9981596137020172)--(-0.9987724554622847,0.04953364724950819), linewidth(2) + rvwvcq);  draw((-0.9987724554622847,0.04953364724950819)--(0.5922362871684147,0.8057643453026269), linewidth(2) + rvwvcq);  draw((0.5922362871684147,0.8057643453026269)--(0.06064145095757084,-0.9981596137020172), linewidth(2) + wrwrwr);  draw((0.5249726058304045,-2.4139334560841545)--(2.3530139989292476,0.7523271151020314), linewidth(2) + wrwrwr);  draw((-0.9987724554622847,0.04953364724950819)--(1.5189031419104242,-0.6923952683993904), linewidth(2) + wrwrwr);  draw((-0.9987724554622847,0.04953364724950819)--(0.5249726058304045,-2.4139334560841545), linewidth(2) + wrwrwr);  draw((-0.9987724554622847,0.04953364724950819)--(2.3530139989292476,0.7523271151020314), linewidth(2) + wrwrwr);  draw((0.8660254037844386,-0.5)--(2.3530139989292476,0.7523271151020314), linewidth(2) + wrwrwr);  draw((0.8660254037844386,-0.5)--(0.5249726058304045,-2.4139334560841545), linewidth(2) + wrwrwr);  draw(circle((0.5922362871684147,0.8057643453026269), 1.7615883990890795), linewidth(2) + linetype("4 4") + wrwrwr);  draw(circle((0.06064145095757076,-0.9981596137020177), 1.4899728165839203), linewidth(2) + linetype("4 4") + wrwrwr);  draw((0.5249726058304045,-2.4139334560841545)--(0.9854301844182564,0.17008042696736536), linewidth(2) + wrwrwr);  draw((2.3530139989292476,0.7523271151020314)--(0.3454211217688861,-0.9384477868458769), linewidth(2) + wrwrwr);  /* dots and labels */ dot((0,0),dotstyle);  label("$O$", (0.03388760411534265,0.08889671794036069), NE * labelscalefactor);  dot((0.8660254037844386,-0.5),dotstyle);  label("$B$", (0.9043736119372844,-0.4159851665963708), NE * labelscalefactor);  dot((-0.9987724554622847,0.04953364724950819),dotstyle);  label("$D$", (-0.9671713048798903,0.13242101833145825), NE * labelscalefactor);  dot((0.5922362871684147,0.8057643453026269),dotstyle);  label("$A$", (0.625818089434263,0.8897438451365556), NE * labelscalefactor);  dot((0.06064145095757084,-0.9981596137020172),linewidth(4pt) + dotstyle);  label("$C$", (0.09482162466287856,-0.9295719112113219), NE * labelscalefactor);  dot((2.3530139989292476,0.7523271151020314),dotstyle);  label("$E$", (2.3841998252345853,0.8375146846672384), NE * labelscalefactor);  dot((0.5249726058304045,-2.4139334560841545),dotstyle);  label("$F$", (0.5561792088085077,-2.3310543838046627), NE * labelscalefactor);  dot((1.5189031419104242,-0.6923952683993904),linewidth(4pt) + dotstyle);  label("$P$", (1.5572381178037407,-0.6249018084736391), NE * labelscalefactor);  dot((0.3454211217688861,-0.9384477868458769),linewidth(4pt) + dotstyle);  label("$Q$", (0.38208200724411934,-0.8686378906637853), NE * labelscalefactor);  dot((0.9854301844182564,0.17008042696736536),linewidth(4pt) + dotstyle);  label("$R$", (1.0175367929541368,0.23687933927009236), NE * labelscalefactor); [/asy]

Solution 1

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See also

2018 USAJMO (ProblemsResources)
Preceded by
Problem 2
Followed by
Problem 4
1 2 3 4 5 6
All USAJMO Problems and Solutions
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