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 
consists of the union of disjoint pieces. Compute the distance between the endpoints of any one piece
so that any two different points in the graph of the quadratic function 
are not symmetrical to the straight line 
such that
x>y>0
2f(x)x>0
x,y>0
, points and lie on the side , with 
. Also, is the midpoint of 
. Show that the centroid of lies on 
.
be non-zero real numbers. Define 
, then the number of all integers which lies in the range of is equal to.
is a non-negative quantity. Also by triangle inequality, we can easily see that 
Hence 
But
which contradicts the fact that are non-zero real numbers. Hence we conclude that 
So the set of all integers lying in the range of is a subset of 
is the required set.
. Prove that
be positive reals such that 
.
![\[
M = \frac{1 - x^2}{x^2} + 2y^2 + 3x + \frac{24}{y} + 2025.
\]](http://latex.artofproblemsolving.com/6/b/d/6bd846ab0f95b1cf4766c7bf0356a439f3bfc93a.png)
. Prove that
and 
Show that
Let 
Show that
and 
Show that
Let 
Show that
Let 
Show that
it is evident if 
or 
be real numbers. 
Let 