AoPS Academy
be an infinite sequence of positive integers. Prove that there exists a unique integer 
such that![\[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\]](http://latex.artofproblemsolving.com/9/4/f/94fc5a51b7e68588123c5b527fe75183bc4c4937.png)
+1 w
with 
, 
and 
for all positive integers 
.
,
if 
is a root of 
and solve the equation.
are constructed on diameter 
of a semicircle of radius 
. The centers of the small semicircles divide 
into four line segments of equal length, as shown. What is the area of the shaded region that lies within the large semicircle but outside the smaller semicircles?
be an infinite geometric sequence where all the terms are positive integers. Amanda lists down the two-digit numbers formed by the last two digits of each term in the sequence in order, adding a leading zero if necessary. When she is about to write a two-digit number that she has already written before, she stops. For example, if the sequence was 
, Amanda will list the numbers 
, then stop.
and 
. If 
, how many possible values are there for 
?
regions of a three-set Venn Diagram with 
colors such that each color is used at least once? Two colorings are considered the same if one can be reached from the other by rotation and reflection.
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?
are all real positive number that: 
. Prove that:
![\[ 4 \geq AC \cdot BD = AB \cdot CD + AD \cdot BC \geq 2 \sqrt{AB \cdot BC \cdot CD \cdot DA} \geq 4 \]](http://latex.artofproblemsolving.com/3/b/b/3bba516895f19c3bf01d208d0554635885b8ceae.png)
(since 
and 
are not longer than a diameter of the circle). and 
are diameters, 
is a rectangle, and also 
which implies that 
, so is a square, and we're done.
with equality iff and are diameters.
with equality iff 
.
,
concave on ![$[0, \pi]$](http://latex.artofproblemsolving.com/0/a/1/0a151f5b93ce5735f72cdd79c46051f52b9d820d.png)
?
then AM-GM then jensen.
,
,
,
.
.
and etc, Hence:![\[ AB \cdot BC \cdot CD \cdot DA=16R^2 \cdot \sin{a} \sin{b} \sin{c} \sin{d} \]](http://latex.artofproblemsolving.com/4/a/e/4ae772824bec8a40897191acf9ce19461da38024.png)
.
.
