[/quote]
,
be a cyclic quadrilateral. Let 
,
,
be the feet of the perpendiculars from 
to the lines 
,
,
,
if and only if the bisectors of 
and 
are concurrent with 
.
be reals such that 
. Prove that
.
be the feet of the altitudes from 
to 
to 
and 
as the reflections of 
be the circumcircle of 
.
with 
with 
is the circumcenter of 
can be inscribed within a circle.
where 
are lengths of parallel sides and 
are other two sides.
and 
Prove that 
be a triangle. A circle passing through 
intersect at 
,
and 
intersect at 
.
if and only if 
.
and 
.
be reals such that 
Prove that
Prove that
such that 
.
of positive integers satisfying the following two conditions:
,
is in 
,
and 
;
such that 
is not divisible by 
.
?
then
where 
Also, 
and 
is largest real root of the equation
then 
Equality holds iff 
and any cyclic permutations.
and 
be points on segment 
denote the circle passing through 
denote the circle passing through 
.
)
.
Y by Jonathankirk, Adventure10, Mango247
For a sequence 
of real numbers, define the following sets
![\[A=\{a_i | 1\leq i\leq m\}\ \text{and} \ B=\{a_i+2a_j | 1\leq i,j\leq m, i\neq j\}\]](//latex.artofproblemsolving.com/f/e/7/fe792687755e2d5afe8a46f827b6e0bb11f6391e.png)
Let
be a given integer, and 
. For any strictly increasing arithmetic sequence of positive integers, determine, with proof, the minimum number of elements of set 
, where 
of real numbers, define the following sets![\[A=\{a_i | 1\leq i\leq m\}\ \text{and} \ B=\{a_i+2a_j | 1\leq i,j\leq m, i\neq j\}\]](http://latex.artofproblemsolving.com/f/e/7/fe792687755e2d5afe8a46f827b6e0bb11f6391e.png)
Let
be a given integer, and 
. For any strictly increasing arithmetic sequence of positive integers, determine, with proof, the minimum number of elements of set 
, where 
This post has been edited 1 time. Last edited by buzzychaoz, Aug 22, 2015, 12:46 AM
Reason: Typo
Reason: Typo
is 
and 
,
,
,which is obvious when 
.
,
c
a
.
