AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
be a triangle and let 
be the midpoint 
. On the exterior of the triangle, consider the parallelogram 
such that 
and 
. Prove that line 
passes through the midpoint of segment 
.
, where the coefficients 
and 
are all irrational numbers.
and a minimum value of 
over the interval ![$[-1, 1]$](http://latex.artofproblemsolving.com/5/c/3/5c3818b9565a33fd3aadba10026d32c5e3eea90f.png)
?
,
triangle with 
. We take , 
point on AC, AB respectively such that 
, 
. 
, 
lines intersect at point 
. If 
, find 
. Prove that
Let 
. Prove that
Let 
. Prove that
Let . Prove that
be positive reals such that 
. Prove that ![\[(a-b)^2 + a + b \geq 2\]](http://latex.artofproblemsolving.com/c/f/a/cfa01a5b54a72afe6746f547390e65dbbb82ea64.png)
and 
we define 
as the set of all products of an element of and an element of . For example, if 
and 
then 
We call a set 
of positive integers good if there do not exist sets 
of positive integers, each with at least two elements and such that the sets 
and are the same. Prove that the set of perfect powers greater than or equal to 
is good.
, 
, no two elements are equal, but any two or three of these sets may have common elements. A perfect power is an integer of the form 
, where 
and 
are integers.)
baskets, each containing at least one candy. The total number of candies is 
. Asad and Sevinch make moves alternatingly, with Asad going first. On one move, Asad can take all the candies from 
consecutive non-empty baskets, while Sevinch can take all the candies from a single non-empty basket that has at least one empty neighboring basket. Prove that Asad can take overall at least 
candies, regardless of the initial distribution of candies and Sevinch's actions.
be a convex quadrilateral with ![\[\angle ADC = 90^\circ, \ \ \angle BCD = \angle ABC > 90^\circ, \mbox{ and } AB = 2CD.\]](http://latex.artofproblemsolving.com/b/4/1/b416d5c0b29c3a69a3e87cd22b6000aa8e70d456.png)
The line through 
, parallel to 
, intersects the external angle bisector of 
at point 
. Prove that the angles 
, 
, 
, 
, 
can be divided into two groups, so that the angles in each group have a sum of 
.
be a function satisfying 
. Assume also that 
satisfies equations (A) and (B) below. 
for all integers 
.
.
such that 
, prove that the set 
is infinite.
for which the following statement holds: there exists at least one triple 
of integers such that
and all triples of real numbers, satisfying the equations, are such that 
are integers.
be a point outside circle 
centered at 
. Tangents from to touch at 
. Line 
intersects bigger 
arc at 
.The line drawn from parallel to 
intersects 
at 
. Ray 
intersects small arc at 
. Prove that the circumcircle of 
is tangent to .
be a right triangle with 
and 
. Let 
be the point on side such that 
, be the point on such that 
and 
, and be the point on segment such that 
. Find the degree measure of 
.
Prove that
!