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 real numbers such that 
Prove that
Let be real numbers such that 
Prove that
If a problem has not been solved within 
days, then others are eligible to post a new probkem. Not only simple one-variable equations, but also systems of equations are allowed. The difficulty of these equations should be no less than that of typical quadratic one-variable equations. If the problem involves higher degrees or more variables, please ensure that the problem is solvable (i.e., has a definite solution, rather than an approximate one). Please indicate the domain of the solution to the equation (e.g., solve in 
, solve in 
).
be a triangle inscribed in a circle, where the tangents to the circle at points 
and 
intersect at the point 
. Let 
be a point on the arc 
(not containing ) such that 
and 
. Let the lines 
and 
intersect at point 
. Let 
be the reflection of with respect to the line . The lines 
and 
intersect at point 
, and intersects the circumcircle of again at point 
. lies on the circumcircle of 
.
be circumscribed about circle 
, and let 
be the orthocenter of . The circle touches line at 
. The tangent to the circle 
at meets at 
. Let 
be the midpoint of 
, and let the line 
meet again at 
. The tangent to parallel to meets the line 
at 
. Prove that 
is tangent to .
+1 w
Prove that 
When does equality hold?
.
, inscribed in the circle 
, satisfies 
and 
, and 
is the intersection point of the diagonals and 
. The circle with center 
and radius 
intersects in two points 
and 
. Prove that the line 
is tangent to the circles with diameters 
and 
.
such that 
and 
for any real numbers 
and 
.
, let 
denote the sum of squares of digits of . Prove that the sequence 
is eventually periodic.
grid, where is a composite integer.
unit squares are divided up into 
disjoint sets of 
unit squares arbitrarily such that 
. Denote this family of sets as . unit squares are again divided up into 
disjoint sets of 
unit squares arbitrarily such that 
. Denote this family of sets as .
unit squares such that no two unit squares are in the same set of or the same set of ?
, the following inequality holds:![\[
\sqrt{a + b} + \sqrt{b + c} + \sqrt{c + a} \geq \frac{4(ab + bc + ca)}{\sqrt{(a + b)(b + c)(c + a)}}
\]](http://latex.artofproblemsolving.com/e/7/2/e72fd0f134289c359bc8766440babc52ae131961.png)
![\[\left(-\cos(1^{\circ})\right)\left(-\cos(2^{\circ})\right)\left(-\cos(4^{\circ})\right) \cdots \left(-\cos{(256^{\circ})}\right)\]](http://latex.artofproblemsolving.com/4/4/e/44e5bbdf821aed7bb2d855efcf39af763ebaea48.png)
on a blackboard. Then, he erases each term on the blackboard with probability 
. What is the expected value of the remaining expression?
It can't seem to be simplified further.
i thought that after you could try multiplying by 
but it doesn't work 
and 
Prove that
satisfied 
Find the minimum of 
