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.
and 
be the sets of integers and rationals respectively. into three non-empty subsets 
such that the sets 
are disjoint? into three non-empty subsets such that the sets are disjoint?
denotes the set 
, for 
and for 
.
be positive real numbers such that 
. Show that![\[ \frac{x^2-yz}{x^2+x}+\frac{y^2-zx}{y^2+y}+\frac{z^2-xy}{z^2+z}\leq 0. \]](http://latex.artofproblemsolving.com/e/a/1/ea1f56cc594862673040411c3ad322cb00cfe904.png)
grid (
), we color some of its cells black.A coloring is called balanced if each row and each cell contains exactly 
black cells.Detemine the number of balanced colorings.
be a quadrilateral inscribed in a circle with center 
and 
be the intersection of segments 
and 
. Let 
be the circumcircle of 
and 
be the circumcircle of 
. The tangent to at 
and the tangent to at 
meet at 
. The tangent to at 
and the tangent to at 
meet at 
. Show that 
.
, let 
denote the number of 
s in the binary representation of . Prove that for any positive integer ,![\[\sum_{i=1}^{n}(-1)^{s(3i)} > 0.\]](http://latex.artofproblemsolving.com/d/e/9/de975cc2632685d7a3f71c023115535d0aeb5dbb.png)
Holden Mui
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.
for which there exist real numbers 
satisfying 
, 
and
for 
.
be an integer. Find the smallest integer 
with the property that there exists a set of distinct real numbers such that each of its elements can be written as a sum of other distinct elements of the set.
be a regular 
-gon. Assign integers between and to the vertices of such that each integer appears exactly once. (If two assignments coincide under rotation, treat them as the same. ) An operation is a swap of the integers assigned to a pair of adjacent vertices of . Find the smallest integer such that one can achieve every other assignment from a given one with no more than operations., the point is the midpoint of and is a point on the side such that 
. The cotangents of the angles 
, 
, and 
are positive integers 
.
is also an integer and equals 
.
, some of which consisting of Fibonacci numbers.
such that for all positive real numbers 
and 
:
and be two isogonal conjugate with respect to triangle . Let 
and 
be two points lying on the circle 
such that 
and 
are perpendicular and parallel to bisector of , respectively. Prove that 
and 
bisect two arcs containing and not containing , respectively, of 
.
be a triangle such that 
and 
. The tangent at to the circumcircle 
of triangle meets the line at . Let be the reflection of in the line , let 
be the foot of the perpendicular from to 
, and let 
be the midpoint of the segment 
. Let the line 
intersect the circle again at 
. is tangent to the circumcircle of triangle 
.
so that:
with all real numbers 
satisfies 
so that:with all real numbers satisfies 
at 
.
