AoPS Academy
with circumcenter 
and orthocenter 
, let 
be an arbitrary point on the circumcircle of triangle such that does not lie on line 
and that line 
is not parallel to line 
. Let 
be the point on the circumcircle of triangle such that 
is perpendicular to , and let 
be the point on line 
such that 
. Let 
and 
be the points on the circumcircle of triangle such that 
is a diameter, and 
and 
are parallel. Let 
be the midpoint of 
.
and 
are perpendicular.
and 
are perpendicular.
be positive integers and 
be an odd prime such that:![\[
a^p \equiv 1 \pmod{p^n}.
\]](http://latex.artofproblemsolving.com/3/4/8/34815b8211ded2f37b0653525dc4b436356e6eee.png)
Prove that:![\[
a \equiv 1 \pmod{p^{n-1}}.
\]](http://latex.artofproblemsolving.com/0/7/a/07a4ed2ab3c94f41028080f7789cfb6e2973955c.png)
be a cyclic quadrilateral. A circle centered at passes through 
and and meets lines 
and again at points and (distinct from 
). Let denote the orthocenter of triangle 
Prove that if lines 
are concurrent, then triangle and 
are similar.
kilometers while other countries use miles per gallon. Suppose that 
kilometer equals 
miles, and gallon equals 
liters. Which of the following gives the fuel efficiency in liters per kilometers for a car that gets 
miles per gallon?
, meaning BC and CA have closer angles together A and B than the abs value of the difference between EF and FD, or angles D and E. However, part of it is true. Note that a>b>c since the angles subtending them in the triangle are in this order as well. Then 
, so 
. We know that 
, meaning that 
, and because 
, the ratio between angles in DEF is closer to 1 (which represents an equilateral triangle) than in ABC, meaning that angle D is closer to 60 degrees than angle A. Continuing analogously on other cases for pairwise angles A,B,C vs. D,E,F, the result follows. Note that if A is obtuse, the sine angle formulas to angle inequalities still hold because both of angles A and D will either be obtuse or we will be done, with angle A obtuse and angle D nonobtuse.
satisfying 
be the lengths of the sides 
respectively. Consider a triangle 
with the side lengths 
,
,
.
implies 
.
doesn't tell you anything about the angles necessarily because we're working with wildly different triangles. 
doesn't matter at all! Do you see why?