AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
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.
satisfying![\[\begin{aligned}
\begin{cases}
a+b+c+d=0,\\
a^2+b^2+c^2+d^2=12,\\
abcd=-3.\\
\end{cases}
\end{aligned}\]](http://latex.artofproblemsolving.com/8/b/1/8b1923e0303c0aa79332ee4540ee1885a9cf3f08.png)
crosses are placed. A set of 2025 cells we will call balanced if no two of them are in the same row or column. It is known that any balanced set has at least 
crosses. for which it is always possible to color crosses in two colors such that any balanced set has crosses of both colors.
be an acute triangle with 
, and let be the projection from 
onto . Let be a point on the extension of 
past such that 
. Let 
be on the perpendicular bisector of 
such that and 
are on the same side of and![\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]](http://latex.artofproblemsolving.com/c/6/3/c63eeb053f1fa8a29bef93516814ae121382219e.png)
Let the reflection of across 
and 
be 
and 
, respectively. Let 
and 
such that 
. Let 
and 
intersect the circumcircles of 
and 
at 
and 
, respectively. Let 
and 
intersect 
and 
at 
and 
. Let 
intersect at . Prove that 
.
![\[
C(p-1, k-1) \equiv (-1)^{k-1} \pmod{p}
\]](http://latex.artofproblemsolving.com/4/0/7/4072235afbaa1f2d8d4c9ca044615a113a4306ce.png)
for 
, where 
is the binomial coefficient 
choose 
.
satisfy that: 
and 
. Find the maximum: 
be a convex quadrilateral. Suppose that the circles with diameters and 
intersect at points 
and . Let 
and 
. Prove that the points , , and are concyclic. and are not the diagnols)
be a function such that for every regular 
-gon 
we have 
. Prove that 
for all reals 
.
, 
be midpoints of 
. Let 
be . By DIT on 
with line 
, we see that 
, 
, 
are involutions ( is parallel to ). Projecting this through 
, 
, 
, and 
are pairs of an involution.
and 
are corresponding midpoints so 
and 
are isogonal in 
. Clearly, 
and 
are also isogonal, hence this characterizes our involution, and 
, 
are isogonal as desired.
satisfying 
such that 
parallel to 
of 
.
,
.
,
.
and 
,
.
.
,
.![[asy]
import olympiad;import geometry;
size(6cm);defaultpen(fontsize(10pt));pen org=magenta;pen med=mediummagenta;pen light=pink;pen deep=deepmagenta;
pair O,A,B,C,D,mb,mc,P,Q,M,T;
O=(0,0);A=dir(120);B=dir(200);C=dir(340);path w=circumcircle(A,B,C);mc=midpoint(A--B);mb=midpoint(A--C);
P=intersectionpoints(w,mb--100mc-99mb)[0];Q=intersectionpoints(w,mb--100mb-99mc)[0];
T=intersectionpoint(line(B,C),perpendicular(A,line(A,O)));
M=extension(A,T,P,Q);
draw(A--B--C--cycle,deep);draw(w,deep);draw(A--T--B,deep);draw(M--Q,deep);
dot("$A$",A,dir(A));
dot("$B$",B,dir(B));
dot("$C$",C,dir(C));
dot("$P$",P,dir(P));
dot("$Q$",Q,dir(Q));
dot(mb);
dot(mc);
dot("$M$",M,dir(M));
dot("$T$",T,dir(T));
[/asy]](http://latex.artofproblemsolving.com/9/f/6/9f6f29c08c005290e335360abfeca9c13972c930.png)
