[/quote]
,
.
such that no sum of three distinct elements of 
.
.
,
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 
such that 
is an obtuse angle. Let the circumcircle of 
intersect 
and 
.
intersect 
intersect 
.
are concyclic.
.
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.
that satisfy the equation 
.
Y by Adventure10
Let 
be the point on the arc 
of the circumcircle circumscribed around the isosceles triangle 
with 
that does not contain the vertex 
. Let 
be the intersection point of the segment 
and the line perpendicular to , which passes through the vertex . Prove that 
be the point on the arc 
of the circumcircle circumscribed around the isosceles triangle 
with 
that does not contain the vertex 
. Let 
be the intersection point of the segment 
and the line perpendicular to , which passes through the vertex . Prove that 
This post has been edited 2 times. Last edited by parmenides51, Jul 23, 2021, 9:25 PM
Reason: title
Reason: title
![[asy]
size(8cm);
pair A = dir(90), B = dir(230), C = dir(310), D = dir(237), E = foot(A,C,D), F = 2E-C;
dot("$A$", A, dir(90));
dot("$B$", B, dir(230));
dot("$C$", C, dir(310));
dot("$D$", D, dir(237));
dot("$E$", E, dir(280));
dot("$F$", F, dir(326));
draw(A--B--D--C--cycle);
draw(A--E);
draw(A--F, blue+dashed);
draw(A--D, blue+dotted);
draw(unitcircle);
[/asy]](http://latex.artofproblemsolving.com/f/8/4/f8417bbfe4200ebfae8ae2179d10ab0bcf930023.png)
over 
.
and 
so 
.
as desired.
