[/quote]
,
be a trapezium inscribed in a circle 
with diameter 
.
be the intersection point of the diagonals 
and 
. The circle with center 
and radius 
meets 
and 
(where 
)
at 
.
is perpendicular to 
.
![\[ x + y^2 = p^m \]](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 
,
lie on a circle 
such that 
.
at 
intersect again at 
.
Equality holds when 
.
determine a triangle 
,
is relative to 
are concurrent at a point 
and 
.
lie in the interiors of segments 
respectively, such that 
Let 
at 
,
intersect 
at 
.
and 
are either parallel or coincide.
and 
.
Where 
such that :
.
be reals such that 
Prove that 
Let 
Prove that 
be a fixed integer. There are 
beads on a circular necklace. You wish to paint the beads using 
colors, such that among any 
consecutive beads every color appears at least once. Find the largest value of 
for which this task is 
possible.
Y by Adventure10
The points 
and 
are placed in the interior of the triangle 
such that 
and similarly for the other 
vertices( and are isogonal conjugates). Let 
and 
be the intersection points of 
and 
with the circumcircle of 
, respectively 
. Similarly the pairs of points 
and 
are defined. Let 
, 
, 
.
Prove the following statements:
Lines 
, 
, 
concur.
, 
, 
and 
are placed in the interior of the triangle 
such that 
and similarly for the other 
vertices( and are isogonal conjugates). Let 
and 
be the intersection points of 
and 
with the circumcircle of 
, respectively 
. Similarly the pairs of points 
and 
are defined. Let 
, 
, 
.Prove the following statements:
Lines 
, 
, 
concur.
, 
, 
This post has been edited 1 time. Last edited by Snakes, Apr 26, 2017, 5:56 PM
the be isogonal conjugates. I made a Geogebra figure and 
doesn't seem to be on AB$, or did I make a mistake in the construction?
Obviously 
and 
are isogonal conjugates. Now applying 
and 
wrt we get that 
and 
are isogonal conjugates, so the result follows immediately.
I'm not too sure about that , but I think the following argument works:
,
and 
obeys homogeneity, cyclicity and bisymmetry, so it must be a triangle center, that is also incident with 
.![[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */import graph; size(200); real labelscalefactor = 0.5; /* changes label-to-point distance */pen dps = linewidth(0.7) + fontsize(0); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -4.3, xmax = 10.38, ymin = -5.86, ymax = 6.3; /* image dimensions */pen zzttqq = rgb(0.6,0.2,0); draw((6.64,5.2)--(0.48,0.24)--(9.08,0.36)--cycle, zzttqq); /* draw figures */draw((6.64,5.2)--(0.48,0.24), zzttqq); draw((0.48,0.24)--(9.08,0.36), zzttqq); draw((9.08,0.36)--(6.64,5.2), zzttqq); draw(circle((4.767151690968777,1.2207954805710357), 4.397911935917431)); draw((6.64,5.2)--(-1.918725664739468,-4.214598231213416)); draw((-1.918725664739468,-4.214598231213416)--(7.673773385432982,1.9933823140236973)); draw((6.64,5.2)--(9.404932126696824,-3.3764253393664996)); draw((7.673773385432982,1.9933823140236973)--(3.44,1.68)); draw((3.44,1.68)--(9.404932126696824,-3.3764253393664996)); draw((9.404932126696824,-3.3764253393664996)--(-1.918725664739468,-4.214598231213416)); draw((6.64,5.2)--(5.063286685109753,0.30395283746664775)); /* dots and labels */dot((6.64,5.2),dotstyle); label("$A$", (6.72,5.4), NE * labelscalefactor); dot((0.48,0.24),dotstyle); label("$B$", (0.56,0.44), NE * labelscalefactor); dot((9.08,0.36),dotstyle); label("$C$", (9.16,0.56), NE * labelscalefactor); dot((3.44,1.68),dotstyle); label("$P$", (3.52,1.88), NE * labelscalefactor); dot((7.673773385432982,1.9933823140236973),linewidth(3pt) + dotstyle); label("$Q$", (7.76,2.12), NE * labelscalefactor); dot((-1.918725664739468,-4.214598231213416),linewidth(3pt) + dotstyle); label("$P_A$", (-1.84,-4.1), NE * labelscalefactor); dot((5.063286685109753,0.30395283746664775),linewidth(3pt) + dotstyle); label("$M_A$", (5.14,0.42), NE * labelscalefactor); dot((9.404932126696824,-3.3764253393664996),linewidth(3pt) + dotstyle); label("$Q_A$", (9.48,-3.26), NE * labelscalefactor); dot((5.556886692716491,1.8366911570118485),linewidth(3pt) + dotstyle); label("$X$", (5.64,1.96), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy]](http://latex.artofproblemsolving.com/d/8/5/d850e14f03e05e7748a0358a95dfd6a376e64c8e.png)
But similarly
and 
passes through the midpoint of 
.
concur on the midpoint of 
