AoPS Academy
be nonnegative real numbers such that 
. Prove that![\[
\frac{a}{4 - b} + \frac{b}{4 - c} + \frac{c}{4 - a} + \frac{1}{16}(1 - a)^2(1 - b)^2(1 - c)^2 \leq 1,
\]](http://latex.artofproblemsolving.com/5/b/d/5bd3349071e075519bd986c845c500125b7d46f8.png)
and determine all such triples 
where the equality holds.
let 
be the midpoint of 
. Let 
be a circle inside of and is tangent to 
at 
, respectively. The tangents from to meet at 
such that 
and 
lie on the same side of 
. Let 
and 
. If 
prove that 
is tangent to . and 
be the incircle and circumcircle of the acute triangle , respectively. Draw a square 
so that all of its sides are tangent to , and 
, 
are both on . Extend 
and 
, intersecting at and 
, respectively. Prove that 
and 
intersects on .
be real numbers such that 
Prove that if 
is a positive constant, then
is not always true
such that
x>y>0
2f(x)x>0
x,y>0
be the number of positive integers 
such that divides 
and 
divides
where 
denotes the number of positive integers that divide , including 
and . Find the remainder when is divided by 
.
, 
, where are decimal digits. Find the necessary conditions in .
, 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 
, such that 
.
and 
, do not lie in one plane and intersect at a point . Let 
be center of sphere that passes through the points 
and . Prove that 
and
intersect at one point. (P.Kozhevnikov)
, then the circumcenter is at:![\[\frac{(\mathbf{a\cdot a})(\mathbf{b\times c})+(\mathbf{b\cdot b})(\mathbf{c\times a})+(\mathbf{c\cdot c})(\mathbf{a\times b})}{12V}\]](http://latex.artofproblemsolving.com/5/8/6/5867d30918b2da1262f68b5b88ea601b424e13e7.png)
is the volume.
, 
, and 
, where 
so that is at the origin. Then we have:
in space such that 
for three points 
is the line passing through the circumcenter of 
perpendicular to the plane containing .
passing through their midpoints, which is exactly the line described.
, 
lie on a plane 
. Then by the lemma 
for all 
, so 
for 
. This is true for planes 
and 
so in fact the solid created by all the circumcenters is just a rectangular prism. But we're done, since the space diagonals of a rectangular prism obviously intersect the center.
. Let 
be images of points 
respectively. Let spheres 
intersect by circles 
. To prove that lines 
are concurrent it's enough to prove that circles lie on the same sphere.
of planes 
and 
. Line is image of circle , so it's enough to prove that all lines lie on the same plane.
easy to see that lines , 
intersect at point 
. So any two line , lie on the same plane. So all lines lie on the same plane. 
. Prove that
and 
Show that
Let 
Show that
and 
Show that
Let 
Show that
Let 
Show that
