[/quote]
,
,
on 
satisfies 
and 
.
be the point where the perpendicular from 
to line 
meets the angle bisector of 
.
holds.
be a positive integer, and let its positive divisors be![\[
d_1 < d_2 < \cdots < d_k.
\]](http://latex.artofproblemsolving.com/6/3/e/63ec82aba0d6c4eece89af6a1247a89498616281.png)
Define 
to be the number of ordered pairs 
with 
such that 
.
.![\[
n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k},
\]](http://latex.artofproblemsolving.com/2/7/2/272dc2cc33ebbf695311cb61567fa0297e1299b6.png)
where the 
are distinct primes and the 
are positive integers.
such that 
.
that verify the following equation for all 
:
be a positive integer. Determine all sequences 
of positive integers such that
for all positive integers 
.
and 
be two circles with respective centres 
and 
intersecting in two distinct points 
and 
is an obtuse angle. Let the circumcircle of 
intersect 
and 
.
intersect 
; let the line 
intersect 
.
are concyclic.
be an acute scalene triangle with circumcenter 
be the feet of the altitudes from 
to 
respectively. Let 
.
be the circle with diameter 
.
be the intersection of 
and 
.
be the orthocenter of 
be the intersection of 
and 
be the lines through 
tangent to 
respectively. Let 
be the intersection of 
and 
.
be the intersection of 
and 
.
be the line through 
and let 
be the reflection of 
is tangent to 
.
where 
are integers in 
are each labeled with a positive integer less than or equal to 
such that every label is used exactly 
and 
such that the following property holds: For any two lattice points 
and 
that are both labeled 
,
and 
are both divisible by 
be an odd prime. Define 
over the positive integers as follows:
where 
such that 
.
such that 
and 
is a perfect square.
denote the set of all positive integers less than 
that are relatively prime to 
.
reducible over the rational numbers, given that it has integer coefficients?
students, each pair of students have watched a movie together. Let the set of movies watched by a student be his movie collection. If every student has watched every movie at most once, at least how many different movie collections can these students have?
be non-negative real numbers such that 
Prove that
and 
.
Y by Adventure10
In a circle 
, there are six points, 
in a counterclockwise order. 
, and 
are concurrent. Let the perpendicular from 
to 
be 
, and the perpendicular from 
to 
be 
. Prove that 
.
, there are six points, 
in a counterclockwise order. 
, and 
are concurrent. Let the perpendicular from 
to 
be 
, and the perpendicular from 
to 
be 
. Prove that 
.This post has been edited 1 time. Last edited by parmenides51, May 1, 2019, 8:08 AM
Reason: typo, CF,BE,AD instead of CF,BE,AC as correctly mentioned in next comment
Reason: typo, CF,BE,AD instead of CF,BE,AC as correctly mentioned in next comment
,
,
be the intersection of 
be the intersection of 
and 
,
and 
are cyclic. Since 
,
and 
.
,
,
.
implies that 
.![[asy]
unitsize(1.25inch);
pair B = dir(140); pair C = dir(190); pair F = dir(35); pair G = foot(B, C, F); pair D = dir(265); pair E = dir(350); pair H = extension(B, E, C, F); pair A = H - 0.735D+0.02; pair X = extension(B, D, C, F);
pair M = foot(B, A, C); pair N = foot(D, C, E);
draw(C--F, heavygreen); draw(B--D, heavygreen); draw(circumcircle(B, C, F), lightblue+dashed);
draw(B--E); draw(D--A);
draw(A--C, blue); draw(C--E, blue);
draw(D--N, heavygreen); draw(B--M, heavygreen);
draw(X--M); draw(X--N); draw(M--N, heavymagenta); draw(A--E, heavymagenta);
dot("A", A, dir(A)); dot("B", B, dir(B)); dot("C", C, dir(C)); dot("D", D, dir(D)); dot("E", E, dir(E)); dot("F", F, dir(F)); dot("M", M, -dir(M)); dot("N", N, -dir(N)-0.3); dot("X", X, dir(X)); dot("Y", H, dir(H)+.4);
[/asy]](http://latex.artofproblemsolving.com/e/7/b/e7bb9fb6114bd19e57597b9fbaadca1db3535ae7.png)
