AoPS Academy
is inscribed in circle 
. The tangent to at 
intersects line 
at point 
. The perpendicular from to line 
intersects line 
at point 
. Let 
be the circumcenter of 
. The perpendicular from to line 
meets it at point 
. Prove that 
and 
are either equal or supplementary.
with the following property: For any six non-negative real numbers 
summing to 
, there exist six non-negative real numbers 
summing to such that for any 
,![\[(a_i + a_j + a_k)(b_i + b_j + b_k) \leq M.\]](http://latex.artofproblemsolving.com/3/e/a/3ea74e39bd863244cb937144fdf0c2d329361248.png)
Proposed by Wu Zhuo and Xu Wenchang
satisfies that for any positive integer 
,![\[f(f(x)) + f(x) \geq 301 + x.\]](http://latex.artofproblemsolving.com/e/e/7/ee7d6ea65f3b290774f02f4be80570711552ecf3.png)
Find the minimum possible value of 
.
, find the smallest real number 
such that no matter how each integer point in the coordinate plane 
is colored with one of different colors, there always exist three non-collinear points 
of the same color such that the area of 
is at most .
as 
. The difference of the cubes is 
, which is prime, and we can show that
mod 6.
and 
be natural numbers such that 
is a prime number. What is the remainder of the number 
?
so 
it's trivial that 
whit 
and we get the following equation: 
.
thus 
.
.
.
.
![\[ m^3-n^3=k(let) \]](http://latex.artofproblemsolving.com/5/e/e/5ee4325743044d8699308a977fc7648a884f2cf3.png)
is a prime and ![\[ m-n \mid k \implies m-n=1 \]](http://latex.artofproblemsolving.com/0/a/d/0addad94c68f54aa01489d778b6fc3afdc85f0e3.png)
then ![\[ m^3-n^3\equiv m-n \equiv 1 \mod 6 \]](http://latex.artofproblemsolving.com/8/0/c/80c85b1e7434dd22fff39e0e370f73391c37c56c.png)