AoPS Academy
be an acute triangle. Define 
as its incenter. Let 
and 
be the incircle's tangent points to 
and 
, respectively. Let 
be the midpoint of 
. Let 
be the intersection point of a perpendicular line passing through to 
. Line 
intersects the circumcircle of 
at 
. The circumcircle of 
intersects line 
at 
. Prove that quadrilateral 
is cyclic.
for which there exists a sequence of distinct positive integers 
, 
, 
, 
satisfying ![\[s_1^{s_2}=s_2^{s_3}=\cdots=s_{n-1}^{s_n}.\]](http://latex.artofproblemsolving.com/a/f/5/af594ff07aa2f89cb70ae669263558049556cf0e.png)
be an integer and 
is a sequence of real numbers for which 
and:
Prove that all elements of the sequence are positive.
be a positive integer, and let 
be a subset of 
. An -partition of into 
parts is a representation of n as a sum 
, where the parts 
belong to and are not necessarily distinct. The number of different parts in such a partition is the number of (distinct) elements in the set 
.-partition of into parts is optimal if there is no -partition of into 
parts with 
. Prove that any optimal -partition of contains at most ![$\sqrt[3]{6n}$](http://latex.artofproblemsolving.com/2/0/1/201489b1f95ad200036698a606f712da7bb465b9.png)
different parts.
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?
such that 
.
, find the value of x that minimizes 
.
and 
such that the only 
-intercept of 
equals its 
-intercept. Compute 
. digit numbers 
(with 
) such that 
, 
, and 
. What’s the size of this set?
be the midpoint of 
in 
. A line perpendicular to D intersects 
at 
. If the area of is four times that of the area of 
, what is 
in degrees?
with 
and 
for 
. Find 
.
.
. Let 
(x) be the th derivative of 
. What is the largest integer 
such that 
divides 
?
be the set of vectors 
where 
are all real numbers. Let 
denote 
. Let 
be the set in 
given by 
If a point 
is uniformly at random from , what is ![$E[||z||^2]$](http://latex.artofproblemsolving.com/e/1/a/e1aa966627c05d1f317981d82094e07d1b1ce0e1.png)
?
be the unique integer between 
and 
, inclusive, that is equivalent modulo to 
. Let be the set of primes between 
and 
, inclusive. Find 
.
,
,
,
. We pick an integer between and 
, inclusive, uniformly at random. We shoot a puck from in the direction of 
and the puck bounces perfectly around the box (angle in equals angle out, no friction) until it hits one of the four vertices of the box. What is the expected number of times it will hit an edge or vertex of the box, including both when it starts at and when it ends at some vertex of the box?
. She can only buy full packs of each of the following items. A pack of pens is 
, a pack of pencils is 
, and any type of notebook or stapler is 
. Sarah buys at least pack of pencils. She will either buy stapler or no stapler. She will buy at most college-ruled notebooks and at most 
graph paper notebooks. How many ways can she buy school supplies?
be the center of the circumcircle of right triangle 
with 
. Let 
be the midpoint of minor arc 
and let 
be a point on line such that 
. Let 
be the intersection of line 
and the Circle and let 
be the intersection of line 
and 
. If 
and 
, compute the radius of the Circle .
.
to be the maximum possible least-common-multiple of any sequence of positive integers which sum to . Find the sum of all possible odd 
but I am looking for the exact value. Does anyone know how to solve this problem?
continuous function with 
for all real 
. exist that satisfy 
for all real and the domain of f is strictly within the set of 
, the range being 
?
s.t. 
and 
be positiv integer numbers.
![\[ 4 \geq AC \cdot BD = AB \cdot CD + AD \cdot BC \geq 2 \sqrt{AB \cdot BC \cdot CD \cdot DA} \geq 4 \]](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)
.
.
