[/quote]
,
that satisfy the equation
with incenter 
.
at 
.
and 
be points on the side BC such that 
and 
,
and 
be the incenter of triangles 
and 
,
is the Euler line of triangle 
.
,
,
,
,
are odd primes and![\[2^ap^b=(p+2)^c-1.\]](http://latex.artofproblemsolving.com/5/8/5/58518a8ebed14836b72af743ff100eae2efba5e2.png)
we have 
.
.
be reals such that 
and 
Prove that
the intersection of 
be the reflection of 
.
concyclic.
. Prove that: 
and ray 
at 
. The circumcircle of triangle 
intersects the circumcircle 
of triangle 
distinct from point 
and 
intersect 
and 
respectively.
such that 
for all real number 
and 
is an equilateral . Let 
and 
at 
and 
at 
,
on 
satisfies 
and 
.
be the point where the perpendicular from 
to line 
meets the angle bisector of 
.
holds.
be a positive integer such that 
.
.
Y by
Let 
be a convex quadrilateral with 
. Points 
and 
are the midpoints of the sides 
and 
, respectively.
a) If
and 
are points, such that 
and 
are parallelograms, prove that 
.
b) Let
, 
, 
. Prove that the triangle 
is iscosceles.
be a convex quadrilateral with 
. Points 
and 
are the midpoints of the sides 
and 
, respectively.a) If
and 
are points, such that 
and 
are parallelograms, prove that 
.b) Let
, 
, 
. Prove that the triangle 
is iscosceles.
![[asy]
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pen dotstyle = black; /* point style */
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/* draw figures */
draw((-41.978885139048835,25.38086667773564)--(-45.886946363223274,-15.870890688551102), linewidth(2) + dtsfsf);
draw((-41.978885139048835,25.38086667773564)--(7.520399117474955,10.363826472549839), linewidth(2) + wrwrwr);
draw((-45.886946363223274,-15.870890688551102)--(38.35348446898129,-17.318320771578705), linewidth(2) + wrwrwr);
draw((38.35348446898129,-17.318320771578705)--(7.520399117474955,10.363826472549839), linewidth(2) + dtsfsf);
draw((-45.886946363223274,-15.870890688551102)--(-7.005264007725278,-50.779121369777656), linewidth(2) + wrwrwr);
draw((-7.005264007725278,-50.779121369777656)--(-3.7667309471209904,-16.594605730064902), linewidth(2) + dtsfsf);
draw((-3.7667309471209904,-16.594605730064902)--(-41.978885139048835,25.38086667773564), linewidth(2) + wrwrwr);
draw((7.520399117474955,10.363826472549839)--(-3.7667309471209904,-16.594605730064902), linewidth(2) + wrwrwr);
draw((-3.7667309471209904,-16.594605730064902)--(27.066354404385347,-44.27675297419344), linewidth(2) + dtsfsf);
draw((27.066354404385347,-44.27675297419344)--(38.35348446898129,-17.318320771578705), linewidth(2) + wrwrwr);
draw(circle((-17.229243010786938,17.87234657514274), 44.270419271452326), linewidth(2) + wrwrwr);
/* dots and labels */
dot((-41.978885139048835,25.38086667773564),dotstyle);
label("D", (-41.49010212257189,26.51942389351229), NE * labelscalefactor);
dot((-45.886946363223274,-15.870890688551102),dotstyle);
label("A", (-45.433638275016165,-14.655732992303895), NE * labelscalefactor);
dot((38.35348446898129,-17.318320771578705),dotstyle);
label("B", (38.77245721541152,-16.16355563882674), NE * labelscalefactor);
dot((7.520399117474955,10.363826472549839),dotstyle);
label("C", (8.036072497831167,11.557183785708661), NE * labelscalefactor);
dot((-17.229243010786938,17.87234657514274),linewidth(4pt) + dotstyle);
label("$N$", (-16.78500799108278,18.748337946048387), NE * labelscalefactor);
dot((-3.7667309471209904,-16.594605730064902),linewidth(4pt) + dotstyle);
label("$M$", (-9.186372248599626,-14.075801205179724), NE * labelscalefactor);
dot((-7.005264007725278,-50.779121369777656),linewidth(4pt) + dotstyle);
label("$E$", (-15.941812903017718,-52.69925822764955), NE * labelscalefactor);
dot((27.066354404385347,-44.27675297419344),linewidth(4pt) + dotstyle);
label("$F$", (26.82586240065387,-46.783953998983), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */[/asy]](http://latex.artofproblemsolving.com/f/4/4/f446bdf8166a6fee4cf698ead2727a1e2572e356.png)
