of positive integers such that 
is divisible by 
.
be triangle with orthocenter 
and circumcircle 
.
lies on line 
.
intersects the circumcircle of triangle 
.
intersects circumcircle of triangle 
again at 
and intersects circumcircle of triangle 
.
be the intersection of 
.
lies on 
such that 
.
and 
do not depend on the choices of 
.
be the midpoint of side 
,
is such that the quadrilateral 
is cyclic and the circumcircle of 
is tangent to the line 
is the second common point of the line 
and the circumcircle of 
.
,
be its incenter. Incircle touches sides 
at 
,
to 
, respectively. Rays 
intersect 
at 
,
touches incircle at 
.
different coloured marbles on it and another one is empty
The altitudes 
, intersect
and 
o
.
b
intersects 
intersects 
at 
lie on the same circle.
The line 
.
are collinear.
such that 
is an exact 5th power, 
is an exact 6th power, and 
is an exact 7th power.
denote the set of natural numbers and let 
denote 
ordered pairs in 
.
distinct elements in the set 
for 
whose product is a perfect cube.
. Prove that
have real roots and real coefficients with 
.
?
,
.
.
.
(I won't be proving this here). This new information allows us to utilize the mean inequality chain as follows:
Similarly, applying AM-HM yields 
.
.
such that 
is a perfect cube.
digits are distinct?
Solve for 
,
,
,
.
,
.
is a point such that 
is a rectangle, compute the area of 
.
has a sidelength of 
,
,
,
,
,
,
are length 
.
.
satisfies for all integers 
If 
and 
,
.
children have black hair, all but 
have red hair, and all but 
have blond hair. How many children are there in the class?
can be partitioned into n Akash subsets.
:
.
Let 
.
,
.
,
,
,
Y by Crazy_LittleBoy, Grotex, Havister, Adventure10, Modern_Hunter, IraeVid13, Mango247, SunnyEvan, MS_asdfgzxcvb, and 2 other users
Let 
and 
are two sets of nonnegative quantities. Now, if we have all the following conditions fulfilled:
(generally the
-th elementary symmetric functions 
for any natural number with 
)
then
for 
.
and 
are two sets of nonnegative quantities. Now, if we have all the following conditions fulfilled:
(generally the
-th elementary symmetric functions 
for any natural number with 
)then
for 
.
