Prove that![$$\frac{a}{b}+ \frac{kb^3}{c^3} + \frac{c}{a}\geq 7\sqrt[7]{\frac{k}{729}}$$](http://latex.artofproblemsolving.com/5/d/c/5dcb84c12e880c66ecf09af425d216a37510bbe9.png)
Where 
such that 
different positive integers.
Product of all even divisors of 
If 
dice are rolled, Then number of ways of getting sum at least 
is
are different real numbers. Prove that:
.
are on the sides 
respectively such that 
.
,
?
,
.
is a perfect square. Find the sum of all possible values of 
.
Let 
.
Let 
.
Let 
.
and 
share a common complex root?
![\[17! = 355687ab8096000\]](http://latex.artofproblemsolving.com/a/6/d/a6d147d3866c4f1fe6dac9481e53d1354891388b.png)
for some digits 
and 
.
that is missing above.
be the polynomial with leading coefficent 1 and rational coefficents, such that ![\[p\left(\sqrt{3 + \sqrt{3 + \sqrt{3 + \ldots}}}\right) = 0,\]](http://latex.artofproblemsolving.com/0/f/4/0f472be9f2cede0221e1b4d970d65de94dba01b9.png)
and with the least degree among all such polynomials. Find 
.
such that 
is a positive integral power of a prime positive integer."
for positive integral values of 
be the largest perfect square that divides 
such that 
is divisible by 
.
Y by
In the quadrilateral 
, the angles 
and 
are greater than 
. Let 
be the intersection point of line 
with the line through 
parallel to 
. Let 
be the intersection point of line 
with the line through 
parallel to 
. Prove that 
is parallel 
.
, the angles 
and 
are greater than 
. Let 
be the intersection point of line 
with the line through 
parallel to 
. Let 
be the intersection point of line 
with the line through 
parallel to 
. Prove that 
is parallel 
.This post has been edited 2 times. Last edited by parmenides51, Dec 22, 2022, 1:02 AM
