AoPS Academy
have real roots and real coefficients with 
. What is the maximum value of 
?
complex numbers with 
such that
Find the value of 
.
and 
be complex numbers. For a complex number 
where 
, define the norm 
to be the distance of 
from the origin, or 
. Let 
be the minimum value and 
be the maximum value of 
for all complex numbers where 
. Find 
.
such that,
be 
, and 
. What is the sum 
?
through 
, shuffle them, and then place them in a stack facedown, turning over the top card next to the stack. They then take turns either drawing the card at the top of the stack into their hand, showing the drawn card to the other player, or drawing the card that is faceup, replacing it with the card on the top of the pile. This is repeated until all cards are drawn, and the player with the largest sum for their cards wins. What is the probability that the player who goes second wins, assuming optimal play?
such that 
is also prime.
vertices of an octagon each red, black, and white, such that no two adjacent sides are the same color?
be the number of positive integers such that divides 
and 
divides
where 
denotes the number of positive integers that divide , including and . Find the remainder when is divided by 
.
and 
are points on the segments 
and 
, respectively, such that 
and 
. If the area of triangle 
is 
, then what is the area of triangle 
?
consists of the union of disjoint pieces. Compute the distance between the endpoints of any one piece
has a maximum value of and a minimum value of in the interval ![$[0,1]$](http://latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
. Confirm whether the value of 
depends on 
or 
.
be a concave-up parabola, then we have the following cases:![$f'(x) \le 0, [0,1] \subset \left(-\infty, -\frac{a}{2}\right]$](http://latex.artofproblemsolving.com/0/4/3/0438b1e5eca71d9f06f34be256c6e787dd544f2f.png)
):
and 
.
): and 
.
):
and 
.![$f'(x) > 0, [0,1] \subset \left(-\frac{a}{2}, \infty\right)$](http://latex.artofproblemsolving.com/4/b/b/4bbc4cd5b009e7c6dc63d92c8c94cd24e7bceb1e.png)
): and 
.
it always depends on the coefficient .
,
.
have roots 
.
(I won't be proving this here). This new information allows us to utilize the mean inequality chain as follows:
Similarly, applying AM-HM yields 
.
.
.
Let 
.
From here we must have,
Observe that thus far the RHS of the divisibility statement has been strictly positive, following from the fact that 
for 
.
to be,
,
.
.
,
