Maximizing the Area
by steven_zhang123, Mar 29, 2025, 12:45 AM
Given a circle 
and two points 
and 
outside , a quadrilateral 
is defined as "good" if 
are four distinct points on in order, and lines 
and 
intersect at and lines 
and 
intersect at .
For a quadrilateral
, let 
denote its area. If there exists a good quadrilateral, prove that there exists good quadrilateral such that for any good quadrilateral 
, 
.
and two points 
and 
outside , a quadrilateral 
is defined as "good" if 
are four distinct points on in order, and lines 
and 
intersect at and lines 
and 
intersect at .For a quadrilateral
, let 
denote its area. If there exists a good quadrilateral, prove that there exists good quadrilateral such that for any good quadrilateral 
, 
.This post has been edited 3 times. Last edited by steven_zhang123, an hour ago
be an odd integer, 
.
integers 
such that for any 
,
and 
.
for which satisfy 
for some 
,
is greater than 
.
and 
be two infinite sequences of integers. Prove that there exists an infinite sequence of integers 
such that for any positive integer ![\[
\sum_{k|n} k \cdot z_k^{\frac{n}{k}} = \left( \sum_{k|n} k \cdot x_k^{\frac{n}{k}} \right) \cdot \left( \sum_{k|n} k \cdot y_k^{\frac{n}{k}} \right).
\]](http://latex.artofproblemsolving.com/2/5/9/259ee32c89befe7a3ee2797e0d5b1df1c3066cf6.png)
be an infinite sequence and 
be a positive integer such that 
,
for all 
.
is a perfect square for all 
.
.
.
be a polynomial with integer coefficients such that 
.
is divisible 
.
and 
such that 
divides 
and 
divides 
be non-negative real variables with 
Prove that
Equality holds iff 
or 
where 
May 2017
March 2017