[/quote]
,
with 
.![\[
\frac{a^5}{b^2 + 2c^3} + \frac{2b^5}{3c + a^6} + \frac{c^7}{a + b^4} \geq 2.
\]](http://latex.artofproblemsolving.com/f/f/5/ff5043705025592aeefc55a764a8b7170e13ff4b.png)
be an integer. Prove that there is a set of 
be a triangle with circumcircle 
.
on 
respectively. Circumcircle 
of triangle 
intersects 
.
intersects 
at 
.
cuts 
respectively. Construct parallelogram 
.
are collinear.
be real numbers such that 
.
be the unit circle in the complex plane. Let 
be the map given by 
.
and 
for 
.
is called the period of 
.
of period 
.
)
,
.![\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]](http://latex.artofproblemsolving.com/b/e/5/be5819a67c3cd78f2dea35fdccf48688c720ce3c.png)
When does the equality hold?
be a positive integer such that ![\[
1 + \sum_{i=1}^{n-1} i^{n-1}.
\]](http://latex.artofproblemsolving.com/e/2/d/e2daf7800da3f984cd761ed1fecf10083444db25.png)
Prove that 
.
Where 
,
holds for all positive reals 
and for every positive real 
,
,
.
at 
intersects 
at 
.
intersect 
.
intersect 
at 
,
Y by
A sequence of integers 
is call 
if it satisfies the following properties:
and 
for all indices 
.
.
Find the smallest integer
for which: Every sequence, there always exist two terms whose diffence is not less than 
. (where is given positive integer)
is call 
if it satisfies the following properties:
and 
for all indices 
.
.Find the smallest integer
for which: Every sequence, there always exist two terms whose diffence is not less than 
. (where is given positive integer)This post has been edited 1 time. Last edited by TUAN2k8, Apr 19, 2025, 8:24 AM
Reason: some mistake
Reason: some mistake
