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such that 
, 
, and 
. Then, let the circumcenter of 
be 
. Furthermore, let the reflections of 
, 
, and 
across , be 
, 
, and 
respectively. Find ![$[AB'CA'BC']$](http://latex.artofproblemsolving.com/c/5/4/c54d526ffe27fab853f07f87f452887ae61c4331.png)
.![\[\underbrace{\dfrac{v_2(4!)}{v_2(8!)}\cdot\dfrac{v_2(16!)}{v_2(32!)}\dots}_\text{50 terms} = \dfrac{1}{2^n}\]](http://latex.artofproblemsolving.com/b/2/4/b24c2c41eec0340fd57d008ccb07c284b3dd522f.png)
find 
.
take on exactly 
distinct values?
be one of the numbers 
. Suppose that for all positive integers 
, the number 
never has remainder upon division by 
. List all possible values of .
where each 
is chosen from 
. A line is an (unordered) set of three (distinct) cards 
,
,
such that for each , the numbers 
are either all the same or all different. How many different lines are there?
, where 
, is a 
-tangent pair if we have
. Compute the second largest integer that appears in a 
-tangent pair.
has side length 
. For 
, choose 
to be a point on the segment 
uniformly at random, assuming the convention that 
for all integers 
. What is the expected value of the area of hexagon 
?
.-dimensional space passes through the point 
, with 
, 
, and 
all positive. The plane also intersects all three coordinate axes with intercepts greater than zero (i.e. there exist positive numbers 
, 
, 
such that 
, 
, and 
all lie on this plane). Find, in terms of , , , the minimum possible volume of the tetrahedron formed by the origin and these three intercepts.
, when it begins walking to the right along the rubber band as the child begins stretching it. The increasingly tired ant walks at a rate of 
centimeters per second, while the child uniformly stretches the rubber band at a rate of one meter per second. The rubber band is infinitely stretchable and the ant and child are immortal. Compute the time in seconds, if it exists, at which the ant reaches the right end of the rubber band. If the ant never reaches the right end, answer 
.. We say that a point lies at distance d from a line segment if 
is the minimum distance between the point and any point on the line segment. Finally, we say that a lattice point is nearby a line segment if the distance between and the line segment is no greater than the distance between the line segment and any neighbor of . Find the number of lattice points that are nearby the line segment connecting the origin and the point 
.
, comes after 
in the permutation. What is the probability that a random permutation of the first 
integers is valid?
and 
, find the range of all possible values of
.
, 
, and 
. Compute the area of the smallest regular polygon that has three vertices coinciding with the vertices of the given triangle. are there such that for any natural numbers 
, we have 
implies 
?
and 
such that the following limit is finite and nonzero: 
.
.
, and let 
and be independently and uniformly distributed random variables in the interval 
. For the ordered 
-tuple 
, let 
denote the ratio between the area of a sheet with dimension a×b and the area of the horizontal cross-section of the box with dimension 
after the sheet has been folded in halves along each dimension until it occupies the largest possible area that will still fit in the box (because Sean is picky, the sheet must be placed with sides parallel to the box’s sides). Compute the smallest value of b/a that maximizes the expectation 
.
be positive real number satisfy that: 
.Find the minimum: 
be real numbers so that 
and 
Show that ![$a\in[0,\frac{4}{3}]$](http://latex.artofproblemsolving.com/0/e/c/0ecb933d7b008114469c3915dc2f8cbf43e362dc.png)
,![$b\in[0,\frac{2}{3}]$](http://latex.artofproblemsolving.com/8/4/1/841685820e03663b8ba4b65615e7ecf40fc31bcc.png)
,![$c\in[0,\frac{4}{9}]$](http://latex.artofproblemsolving.com/3/3/4/3349e77972554eb938563cef5aed9425367cd48d.png)
is given. Point 
is such that 
and points 
lie in different half-planes wrt line 
. Point 
is the midpoint of side 
. Find the angle 
.
is equivalent to 