AoPS Academy
be real numbers with 
that satisfy the equations
} can be written in the form 
, where 
, and 
are positive integers. What is 
?
to be an infinite polynomial where ![\[P(x) = \lim_{n\to \infty}\prod_{k=0}^n \left(x-\left(\cos\left(\dfrac{2\pi k}{n}\right)+i\sin\left(\dfrac{2\pi k}{n}\right)\right)\right).\]](http://latex.artofproblemsolving.com/c/a/8/ca8d0747b7186eb097eb7bd71a95f75d2e38af32.png)
, 
. 
, and 
terms. and , independently at random from the range 
to 
. He then selects an operation out of 
, 
, and 
and performs that operation on them. Find the expected value of the answer.
has a right angle at 
. 
and 
. Point 
lies on 
such that 
and point 
lies on 
such that 
= 
. Find the perimeter of 
.
be an infinite, nonconstant sequence of rational numbers, meaning it is not the case that 
Suppose that 
is also an infinite, nonconstant sequence of rational numbers with the property that 
is an integer for all 
and 
. Prove that there exists a rational number 
such that 
and 
are integers for all and .
denote the set of positive integers. Find all functions 
such that for positive integers 
and 
![\[f(a^2 + b^2) = f(a)f(b) \text{ and } f(a^2) = f(a)^2.\]](http://latex.artofproblemsolving.com/c/1/f/c1f8ffe04cfcc45b498f3931a4796d5c56dc04d0.png)
and 
be positive integers. Prove that there exists a positive integer 
such that for every odd integer 
, the digits in the base-
representation of 
are all greater than .
be an equilateral triangle, and point 
on its circumcircle. Let 
and 
intersect at 
, 
and 
intersect at 
, and 
and 
intersect at 
. Prove that the area of 
is twice the area of 
.
, where 
and 
are relatively prime positive integers. Find 
.
for which 
is a perfect square.
. Three points 
are on the sides 
respectively such that 
. If 
, then find the length of 
?
have lengths 
and choose points 
above such that:
:
(i);
(ii);
(iii)
(via Wolfram Alpha). Hence:
.
.
.
.
.
.
from base 
shares angles 
with 
through AA Postulate. 
,
and 
.
,
(as both are right angles), and 
,
through SAS postulate. 
,
.
.
