AoPS Academy
are complex numbers with absolute value 
. For 
define 
. Given that the sequence 
converges. Show that,
.
![$f \in C([0,1])$](http://latex.artofproblemsolving.com/6/b/c/6bc9d4b6e9ad5dfba34ad5e30f75f64f1c8888b5.png)
. Is it true that ![$\exists a \in \left[0;\dfrac 13\right] \cup \left[\dfrac 23; 1 \right] , |f(a)| \leq 4 |f(1-a)|$](http://latex.artofproblemsolving.com/d/3/c/d3ce9a8e85762fc0992882206127b514ca99b6f4.png)
?. Is it true that ![$\exists a \in \left[0;\dfrac 12\right], |f(a)| \leq 8 |f(1-a)|$](http://latex.artofproblemsolving.com/6/e/b/6eb21a412549b694d5d68fbe4dee9a2abcd6e75e.png)
?
such that:
be a natural number and 
be a prime number. How many different pairs 
satisfy the equation:
such that 
and 
are both perfect cubes.
different coloured marbles on it and another one is empty
The altitudes 
, intersect
. The tangents at 
and 
of intersect at 
. Let 
be the midpoint of 
. 
intersects 
, 
intersects 
at 
lie on the same circle.
intersects the circle at the second point 
The line intersects at the second point 
. Prove that 
are collinear.
such that 
is an exact 5th power, 
is an exact 6th power, and 
is an exact 7th power.
denote the set of natural numbers and let 
denote 
ordered pairs in 
. Prove that there exist 
distinct elements in the set 
for 
whose product is a perfect cube.
. Prove that
