AoPS Academy
of positive integers such that 
is divisible by 
.
be triangle with orthocenter 
and circumcircle 
. A point 
lies on line 
. 
intersects the circumcircle of triangle again at 
. 
intersects circumcircle of triangle 
again at 
and intersects circumcircle of triangle again at 
. Let 
be the intersection of with 
. Let 
lies on 
such that 
. Prove that
and 
do not depend on the choices of .
.
be the midpoint of side in 
, and 
is such that the quadrilateral 
is cyclic and the circumcircle of 
is tangent to the line . If 
is the second common point of the line 
and the circumcircle of , determine the ratio 
.
be an isosceles triangle with 
, and let 
be its incenter. Incircle touches sides 
at 
, respectively. Foot of altitudes from 
to are 
, respectively. Rays 
intersect 
at 
, respectively. Prove that 
touches incircle at 
.
different coloured marbles on it and another one is empty
The altitudes 
, intersect. The tangents at 
and 
of intersect at 
. Let 
be the midpoint of . 
intersects 
, 
intersects 
at 
lie on the same circle. intersects the circle at the second point 
The line intersects at the second point 
. Prove that 
are collinear.
such that 
is an exact 5th power, 
is an exact 6th power, and 
is an exact 7th power.
denote the set of natural numbers and let 
denote 
ordered pairs in 
. Prove that there exist 
distinct elements in the set 
for 
whose product is a perfect cube.
have real roots and real coefficients with 
. What is the maximum value of 
?
such that 
is a perfect cube.
digits are distinct?
Solve for .
, 
, 
, and 
. Let be the point on such that 
, and let be the midpoint of 
. If 
is a point such that 
is a rectangle, compute the area of 
.
has a sidelength of . Points , 
, 
, and are chosen on , , 
, and respectively, such that 
, 
, 
, and 
are length 
. Compute the area of quadrilateral 
.
satisfies for all integers , 
If 
and 
, compute 
.
children have black hair, all but 
have red hair, and all but 
have blond hair. How many children are there in the class? such that the set 
can be partitioned into n Akash subsets.
be defined as follows:
, and for all positive integers 
,![\[
a_n = a_{\left\lfloor \frac{n}{3} \right\rfloor} + a_{\left\lfloor \frac{n}{2} \right\rfloor}.
\]](http://latex.artofproblemsolving.com/8/0/0/8005effb2b72a59ce55f49a5204169ec8ffa3b60.png)
Find the sum of all values 
for which there exists a unique positive integer such that 
.
. Prove that
,
.
.
.
(I won't be proving this here). This new information allows us to utilize the mean inequality chain as follows:
Similarly, applying AM-HM yields 
.
.
:
.
Let 
.