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 
denote three distinct integers, and let 
denote a polynomial having integer coefficients. Show that it is impossible that 
and 
.
+1 w
is a minuan pair if it satisfies the following two conditions:
is even.
are all the positive divisors of , ordered such that 
, then the set of all positive divisors of 
is precisely
.
mod 3 can be written as the sum of one square and 2 cubes.
, let 
be a point on the segment 
and let 
be the intersection point of 
with the diagonal 
. The line perpendicular to the segment through intersects the side 
at point 
. Let 
be a point on the segment 
such that 
. Let 
be the intersection point of the diagonal 
and the segment 
.
and 
are concyclic.
, 
and 
be the respective intersections with BC of the internal angle bisector, external angle bisector, and the median from A. The circumcircle of 
intersects 
a second time at point 
different from A. Define 
and 
analogously. Prove that the circumcenter of 
lies on the Euler line of ABC.
be a polynomial of degree 
with integer coefficients and let 
be a positive integer. Consider the polynomial 
, where occurs times. Prove that there are at most integers 
such that 
.
, 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 one to one maps. Show that the function 
defined by 
, for all 
, cannot be a surjective function.
be a surjective function. Show that there exist surjective functions 
such that 
, for all .
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 a quadrilateral inscribed in a circle 
such that 
. Let and 
be the midpoints of and respectively. The line 
meets again at . Prove that the tangent at to , the line and the line 
are concurrent.
, so 
is harmonic. This implies that the tangents at and to and are concurrent. Hence it suffices to show that , the tangent at , and are concurrent. be the intersection of and . 
and the fact that 
and are parallel tell us 
, so 
. This then gives 
as 
. Hence 
as 
. Therefore, 
is tangent to , so is our desired concurrence point.
intersects point , where is the intersection of the tangent lines to .
by LoC twice on 
and 
where 
)
,
.
,
.
are collinear. 