AoPS Academy
be a quadrilateral with 
. Denote by 
the midpoint of the side 
, and assume that 
. Let 
be the foot of the perpendicular from the point to the line 
. The line 
meets 
at 
, and the line 
meets 
at 
. Show that 
and 
.
, 
, 
, 
lie on a semicircle 
in this order with diameter 
, and is not parallel to 
. Points 
and 
lie on segments and respectively such that 
and 
. A circle 
passes through and is tangent to , and intersects again at 
. Prove that the lines 
, and 
are concurrent.
, determine the smallest integer 
, such that it is possible to place numbers 
around a circle so that the sum of every consecutive numbers takes one of at most values.
a grid of 9*9, we choose a little square in of this grid three times, we can choose three times the same.
and its prime fatorization 
, its false derived is defined by 
Prove that there exist infinitely many naturals such that 
.
that satisfy![\[f(xf(y) + f(x)) = f(x)f(y) + 2f(x) + f(y) - 1,\]](http://latex.artofproblemsolving.com/4/7/f/47f094ce57fe35d339ec3b93152ec3459a22ec48.png)
for every 
, and 
for every 
and 
, such that 
.
be a set of 
points in a plane that lie within a disk of radius 
billion. Define a 
as picking a point 
and reflecting it across 's centroid. Does there always exist a sequence of at most 
moves after which all points of are contained in a disk of radius ?
such that for every real number 
, 
is a perfect square if and only if 
is a perfect square. Prove that 
is the square of a polynomial with real coefficients.
be positive integers such that 
and 
Prove that if 
and 
have the same prime divisors, then 
is a power of 2.
be non-empty subsets of the positive integers, and let 
denote their union. Prove that![\[\sum_{i=1}^m \sum_{(a,b)\in H_i^2}\gcd(a,b)\ge\frac1m \sum_{(a,b)\in S^2}\gcd(a,b).\]](http://latex.artofproblemsolving.com/1/c/d/1cdf561d3af92f524628e0724e0c5ee24d8d6a09.png)
Proposed by Dávid Matolcsi, Berkeley
. Initially, the numbers 
are written on a board. In each move, two numbers are selected such that their difference is also present on the board. This difference is then erased from the board. (For example, if the numbers 
and 
are on the board, then 
can be erased as 
, or 
as 
, or 
as 
.) is it possible to end with only one number remaining on the board?
and 
be the incircle and incenter of triangle 
. Let , 
, 
be the tangency points of to 
, 
, 
and let 
be the reflection of about . Assume 
intersects the tangents to at and at points and . Show that 
.
,
.
.
.
,
from the perpendiculars.
be the unit circle
to get that 
and 
angles sum up to 
we prove that 
and using the fact that 
we get that 
