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 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
. Prove that![\[
\sum_{\text{cyc}} \frac{x^3}{y^2 + z^2} \geq \frac{x + y + z}{2}
\]](http://latex.artofproblemsolving.com/9/0/3/903220c471bdd3028dda2fcb13874a4bcf752872.png)
without using the Rearrangement Inequality or Chebyshev's Inequality.
which satisfy the following set of equations
inscribed in 
with 
being the midpoint of 
. The tangents at 
of intersect at 
. Let 
be the projection of 
onto 
. On the perpendicular bisector of , take a point 
that is not on and different from M. Circle 
intersects 
at 
. Lines 
and 
intersect at 
. Prove that 
is an isosceles triangle.
, and 
be positive real numbers, such that 
. Prove the inequality![\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]](http://latex.artofproblemsolving.com/b/e/5/be5819a67c3cd78f2dea35fdccf48688c720ce3c.png)
When does the equality hold?
such that 
divides 
for any positive integers 
and 
.
of real numbers such that 
for all positive integers 
. (Note that 
denotes the greatest integer less than or equal to 
.)
distinct real numbers there exist two pairs 
and 
with 
or 
, such that ![\[ \left| \frac{a-b}{c-d} - 1 \right|< \frac{1}{100000}. \]](http://latex.artofproblemsolving.com/3/b/0/3b03250ffbd03fd657fa00bc2b63a2c17ee75061.png)
is fixed on a plane with largest side . Let 
be an arbitrary diameter of its circumscribed circle, and the point 
lies on the smaller arc 
, and the point 
is on the smaller arc 
. Points 
are feet of perpendiculars dropped from point to the line , from point to the line and from point 
to line . Prove that the center of the circumscribed circle of triangle 
lies on a fixed circle.
, s.t :
is a triangle. 
is a point on the line . 
is the midpoint of . The line through 
parallel to 
meets the line 
at . The line through parallel to meets the line at . The line through parallel to meets the line at 
. The line through parallel to meets the line at . The lines 
and 
meet at 
is the point on the line such that is the midpoint of 
. Show that 
and are collinear.
is a triangle. is a point on the line . is the midpoint of . The line through parallel to meets the line at . The line through parallel to meets the line at . The line through parallel to meets the line at . The line through parallel to meets the line at . The lines and meet at is the point on the line such that is the midpoint of . Show that and are collinear.
is undefined, tried figure with geogebra with no success, as there were two points (
) undefined
parallel to meets the line at .
Prove that
satisfying
Prove
