AoPS Academy
be a trapezoid with parallel sides 
. Points 
and 
lie on the line segments 
and 
, respectively, so that 
. Suppose that there are points 
and 
on the line segment 
satisfying ![\[\angle{APB} = \angle{BCD}\qquad\text{and}\qquad \angle{CQD} = \angle{ABC}.\]](http://latex.artofproblemsolving.com/b/1/a/b1ae208955104081bafe221832509d9dd20eeb83.png)
Prove that the points , , 
and 
are concyclic.
be a positive integer. A board with a format 
is divided in equal squares.Determine all integers ≥3 such that the board can be covered in 
(or 
) pieces so that there is exactly one empty square in each row and each column.
positive integers 
such that 
for any 
, 
such that 
?
and 
be positive real numbers such that![\[
\log_x 64 + \log_{y^2} 16 = \frac{5}{3} \quad \text{and} \quad \log_y 64 + \log_{x^2} 16 = 1
\]](http://latex.artofproblemsolving.com/d/7/9/d79facd90d9c5b53cdc4df9db18923ad34f472df.png)
What is the value of 
? and be positive real numbers such that![\[\log_x64+\log_{y^2}16=\dfrac{5}{3}\text{ and }\log_y64+\log_{x^2}16=1 \]](http://latex.artofproblemsolving.com/6/6/6/666e220af0dd4e1f77283dc5109a59d796c81bda.png)
Find 
be positive integers. How many pairs of 
are there such that 
is rational?
, how many positive integers 
are there such that there are exactly four powers of 2 with exactly 
digits? (Note that 1 is considered a power of 2)
and the difference between their squares is 
, find the value of the sum of their squares in terms of and . (Note that 
are positive.)
. Find the probability of selecting two real numbers ![$a, b \in [0, 10]$](http://latex.artofproblemsolving.com/e/b/7/eb738a5d732aaf4e2ae46ebefe6a57ae8382f86f.png)
such that the equation has at least one real root.
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. 
.let 
be a point on 
,different from its midpoint.
is on the extension of segment 
and 
,
and
intersect at 
respectively.prove that 
are concyclic.
.let be a point on ,different from its midpoint. is on the extension of segment and ,andintersect at respectively.prove that are concyclic.
, and can be chosen independently....
.
is , is a point on side (not the midpoint of side ), is a point on the extension of line segment 
, straight line 
intersects at point 
, and straight line intersects at point 
. Prove that if 
, then the points 
, 
, and 
lie on a circle.
and 
Prove that
be positive reals such that 
,
of 
.
points.
seems wrong with my picture)
is concyclic,
is the polar of 
isn't parallel to 
with 
given. But I can't figure out how to adjust the definition order.