![\[ x + y^2 = p^m \]](http://latex.artofproblemsolving.com/e/c/f/ecf553036f22327dd457dc06611b6bc1b0ef072c.png)
![\[ x^2 + y = p^n \]](http://latex.artofproblemsolving.com/5/b/d/5bd672934a395aabd6865ca4b50d8da75c0fe07e.png)
For 
positive integers and 
being a prime.
and 
.
Equality holds when 
or 
,
and 
such that 
.
at 
and 
.
intersect again at 
.
.
Equality holds when 
Y by Adventure10, Mango247
The circle whose diameter is the altitude dropped from the vertex 
of the triangle 
intersects the sides
and 
at 
and 
, respectively 
. Show that the circumcenter of lies on the altitude drawn from the vertex of the triangle 
, or on its extension.
of the triangle 
intersects the sides
and 
at 
and 
, respectively 
. Show that the circumcenter of lies on the altitude drawn from the vertex of the triangle 
, or on its extension.This post has been edited 1 time. Last edited by parmenides51, Jun 21, 2022, 2:07 AM
![[asy]
unitsize(0.3inches);
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */import graph; size(0cm); real labelscalefactor = 0.5; /* changes label-to-point distance */pen dps = linewidth(0.7) + fontsize(0); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -15.18, xmax = 15.58, ymin = -10.66, ymax = 10.56; /* image dimensions */pen wrwrwr = rgb(0.3803921568627451,0.3803921568627451,0.3803921568627451); /* draw figures */draw((-7.48,7.74)--(-9.38,-1.74), linewidth(2) + wrwrwr); draw((-9.38,-1.74)--(-0.46,-0.66), linewidth(2) + wrwrwr); draw((-0.46,-0.66)--(-7.48,7.74), linewidth(2) + wrwrwr); draw(circle((-6.928115462364739,3.1818425224939553), 4.591445974050567), linewidth(2) + wrwrwr); draw((-9.193925685041476,-0.8115871022069423)--(-2.540388301406585,1.8293535230506144), linewidth(2) + wrwrwr); draw((-9.193925685041476,-0.8115871022069423)--(-6.376230924729478,-1.3763149550120894), linewidth(2) + wrwrwr); draw((-6.376230924729478,-1.3763149550120894)--(-2.540388301406585,1.8293535230506144), linewidth(2) + wrwrwr); draw((-7.48,7.74)--(-6.376230924729478,-1.3763149550120894), linewidth(2) + wrwrwr); draw(circle((-5.353873876494373,2.383476831786847), 5.76305064567867), linewidth(2) + wrwrwr); draw((-5.353873876494373,2.383476831786847)--(-7.48,7.74), linewidth(2) + wrwrwr); /* dots and labels */dot((-7.48,7.74),dotstyle); label("$A$", (-7.4,7.94), NE * labelscalefactor); dot((-9.38,-1.74),dotstyle); label("$B$", (-9.78,-2.24), NE * labelscalefactor); dot((-0.46,-0.66),dotstyle); label("$C$", (-0.38,-0.46), NE * labelscalefactor); dot((-6.376230924729478,-1.3763149550120894),linewidth(4pt) + dotstyle); label("$P$", (-6.52,-2.04), NE * labelscalefactor); dot((-2.540388301406585,1.8293535230506144),linewidth(4pt) + dotstyle); label("$E$", (-2.36,1.84), NE * labelscalefactor); dot((-9.193925685041476,-0.8115871022069423),linewidth(4pt) + dotstyle); label("$D$", (-9.7,-1.16), NE * labelscalefactor); dot((-5.353873876494373,2.383476831786847),linewidth(4pt) + dotstyle); label("$O$", (-5.28,2.54), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */
[/asy]](http://latex.artofproblemsolving.com/3/f/3/3f394ad37e6b901a2bef0ab18e02fa9d1fa9eafe.png)
be the altitude from 
.
.
,
,
.
