[/quote]
,
Prove that
Let 
Prove that
Let 
Prove that
-
,
,
be positive real numbers such that 
.![$$\sqrt[3]{(a^2+1)(b^2+1)(c^2+1)} \le \left(\frac{a+b+c}{3}\right)^2 + 1.$$](http://latex.artofproblemsolving.com/2/0/7/207934bca53047858a3820f37705171ff0abfd45.png)
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 
points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with rational area.
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 
Y by Adventure10, Mango247
Given a trapezoid 
with bases 
and 
, with 
. Let 
be the midpoint of 
be the intersection point of the sides 
and 
, 
be the intersection point of 
and 
be the intersection point of 
and . In what ratio, point 
divides the segment ?
with bases 
and 
, with 
. Let 
be the midpoint of 
be the intersection point of the sides 
and 
, 
be the intersection point of 
and 
be the intersection point of 
and . In what ratio, point 
divides the segment ?This post has been edited 1 time. Last edited by parmenides51, Jun 21, 2022, 1:37 AM
are all mid-points,
is parallelogram, 
and hence, 
are both medians,
