AoPS Academy
, 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 
If a problem has not been solved within days, then others are eligible to post a new probkem. Not only simple one-variable equations, but also systems of equations are allowed. The difficulty of these equations should be no less than that of typical quadratic one-variable equations. If the problem involves higher degrees or more variables, please ensure that the problem is solvable (i.e., has a definite solution, rather than an approximate one). Please indicate the domain of the solution to the equation (e.g., solve in 
, solve in 
).
be real numbers such that 
Prove that
Let be real numbers such that 
Prove that
are located 
different points that do not belong to the segment ![$[AB]$](http://latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
. Prove that the sum of the distances from point 
to these points is not equal to the sum of the distances from point 
to these points.
are located different points that do not belong to the segment . Prove that the sum of the distances from point to these points is not equal to the sum of the distances from point to these points.
and 
Prove that
points be closer 
and the ones closer to 
.
.
.
which is clearly false.
then suppose there are 
points closer to 
.
points closer to 
.
however this is a contradiction so we can't have the sums being the same.