[/quote]
,
positive real numbers satisfying 
,![$\sqrt[3]{\frac{a^3+b^3+c^3}{3}}+\frac{a+b+c}{9} \geq \frac{4}{3}$](http://latex.artofproblemsolving.com/c/b/3/cb39e7f838c99ed8a209c6844f3d2810838d1dfd.png)
.
denote the set of natural numbers, and let 
,
,
.
whose product is a perfect cube.
are written on the board.In each step,two random numbers 
and 
are chosen and deleted.Then,the number 
is written instead.What will be the number remained on the board after the last step.
be the unit circle in the complex plane. Let 
be the map given by 
.
and 
for 
.
such that 
is called the period of 
.
of period 
.
)
such that
for all 
.
prove that the triangle must have a right angle.
be positive real numbers. Prove that:
.
Show that for any odd integer 
,
of distinct positive real numbers is called radiant if it satisfies the following property: if 
is also an element of 
?
or 
.
such that any knight, after making 1 or 2 arbitrary moves, does not land on a square occupied by another knight?
is satisfied. Point 
is taken on the circumcircle of 
.
which passes from 
and 
respectively at 
and 
.
passes from circumcenter of 
.
such that![\[
x+f(xy)=f(1+f(y))x
\]](http://latex.artofproblemsolving.com/4/3/3/433f95836dad1224754d480aff8432ef033a37cb.png)
for all 
.
.
,
point on AC, AB respectively such that 
,
.
,
lines intersect at point 
.
,
Y by Adventure10
In a triangle 
, the bisectors of the angles at 
and 
meet the opposite sides at 
and 
, respectively. Suppose that the circumcircle of triangle 
passes through the incenter 
of 
. Given that 
, find all sides of triangle 
.
, the bisectors of the angles at 
and 
meet the opposite sides at 
and 
, respectively. Suppose that the circumcircle of triangle 
passes through the incenter 
of 
. Given that 
, find all sides of triangle 
.This post has been edited 2 times. Last edited by parmenides51, Sep 2, 2024, 9:45 AM
