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.
prove that for each 
there exists 
such that either 
or 
is multiple of 
such that 
. The lines 
and 
intersect at a point 
and the line passing through which is parallel to 
, intersects 
at 
. Prove that
(
), point 
is the orthocenter and point 
is the midpoint of the segment . The median 
intersects the circumcircle of triangle at 
. The line 
intersects the perpendicular bisector of at 
and the circumcircle of the triangle again at 
. Point 
lies on circle 
, passing through 
and , such that 
is a trapezoid (
). Prove that 
and 
meet on .![\[
S = \left\{\, x \in \mathbb{Q} : x \text{ has a finite decimal expansion} \,\right\}.
\]](http://latex.artofproblemsolving.com/1/c/e/1ce81426f5d169a9b4c254ef91a1fe34250504b4.png)
For each 
, define![\[
d(x) = \text{the number of digits after the decimal point in the (reduced) decimal form of } x.
\]](http://latex.artofproblemsolving.com/8/1/9/819d07d26274a051a776fcf9f6d7c668ab8d0dab.png)
Find all functions 
such that, whenever both 
and 
,![\[
f(x) + f(y) = f(x+y) + d(x y).
\]](http://latex.artofproblemsolving.com/5/c/a/5ca142ba21a23dc21db6e3c712bc0a76d561647c.png)
be a finite set of points in the plane such that no three of them are on a line. For each convex polygon 
whose vertices are in , let 
be the number of vertices of , and let 
be the number of points of which are outside . A line segment, a point, and the empty set are considered as convex polygons of 
, 
, and 
vertices respectively. Prove that for every real number 
![\[\sum_{P}{x^{a(P)}(1 - x)^{b(P)}} = 1,\]](http://latex.artofproblemsolving.com/3/2/7/3279b2917f21c8f3c2b819e39a32528ad18e4b4b.png)
where the sum is taken over all convex polygons with vertices in .
be a finite point set in the plane and no three points are collinear. A subset 
of will be called round if its elements is the set of vertices of a convex 
gon 
For each round subset let 
be the number of points from which are exterior from the convex gon Subsets with 
and 2 elements are always round, its corresponding polygons are the empty set, a point or a segment, respectively (for which all other points that are not vertices of the polygon are exterior). For each round subset of construct the polynomial![\[ P_A(x) = x^{|A|}(1 - x)^{r(A)}.
\]](http://latex.artofproblemsolving.com/e/b/c/ebc6f8fa2cc303e1062c5a95d4f65c9d2691d0ca.png)
be an odd prime. Define 
over the positive integers as follows:
be chosen such that there exists an ordered pair of positive integers 
where 
such that 
. Prove that there exists at least three integers 
such that 
and 
is a perfect square.
be a positive integer. Aslı and Zehra are playing a game on an 
grid. Initially, 
stones are placed on some of the unit squares of this grid., regardless of the initial placement of the stones, can Aslı guarantee that every square ends up with the same number of stones?
for which all positive divisors of can be put into the cells of a rectangular table under the following constraints:
, let two altitudes 
and 
meet at the orthocenter . Let the tangents of circle 
from 
and 
meet at a point . 
meets 
at . is the midpoint of . Prove that the angle bisector of 
is perpendicular to the angle bisector of 
be triangle with incenter 
. A point in the interior of the triangle satisfies ![\[\angle PBA+\angle PCA = \angle PBC+\angle PCB.\]](http://latex.artofproblemsolving.com/d/6/2/d622094b95a1f317063cf2f02a8b66b31751638f.png)
Show that 
, and that equality holds if and only if 
.
be a positive integer. Define a chameleon to be any sequence of 
letters, with exactly occurrences of each of the letters 
and 
. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon , there exists a chameleon 
such that cannot be changed to using fewer than 
swaps.
such that one can find a number of triangles in the Cartesian plane, the centroid of each triangle is a lattice point, the union of these triangles is a square of side length (the sides of the square are not necessarily parallel to the axis, the vertices of the square are not necessarily lattice points), and the intersection of any two triangles is an empty-set, a common point or a common edge.
such that 
.
and 
to be both integer, 's denominator must divide 
, which in particular means that 
is bounded below
is bounded above (by using the definition of 
for both and 
), which means that is bounded above too.
is unbounded, so from this and the boundedness of we immediately obtain that both sides are zero, so 
.
for some n, so this relation holds for all n, and ultimately the characterisation of sequences are all linear sequences with common difference k.
with no obtuse angles.![$$\frac1{\sqrt3}\cdot\left(\cot A+\cot B+\cot C\right)+\left(2-\sqrt 3\right)\sqrt[3]{\cot A\cot B\cot C}\geq\frac2{\sqrt3}.$$](http://latex.artofproblemsolving.com/d/1/4/d14154281d7ccf60a2dcfe1d937b974ae1d74f97.png)
