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.
diagonals. A regular pentagon has 
diagonals. 
is the smallest positive integer such that a regular -gon has greater than or equal to 
diagonals. What is the sum of the digits of ?![\[\mathrm a. ~10\qquad \mathrm b. ~11 \qquad \mathrm c. ~12 \qquad\mathrm d. ~13\qquad\mathrm e. ~14\]](http://latex.artofproblemsolving.com/c/4/c/c4cf4ad27f6cab816910e560c00d7b2204709716.png)
, and 
are all polynomials such that ![\[ P(x^5)+xQ(x^5)+x^2R(x^5)=(x^4+x^3+x^2+x+1)S(x),\]](http://latex.artofproblemsolving.com/1/6/a/16a71abc110ece558d427a07d7be2d1f72148024.png)
prove that 
is a factor of 
.
be a prime number and let 
and be positive integers with 
.
.
are positive real numbers, 
then
be the incenter of 
and 
be a point such that 
is perpendicular to 
and 
is parallel to . Let the line parallel to , which is tangent to the incircle of , intersect 
and 
at points 
and 
respectively. Prove that 
.
, 
, 
, 
and 
such that 
is a parallelogram and 
is a cyclic quadrilateral. Let 
be a line passing through . Suppose that intersects the interior of the segment 
at 
and intersects line 
at 
. Suppose also that 
. Prove that is the bisector of angle 
.
touch sides , 
, and at points 
, 
, and 
, respectively. Let 
be the diametrically opposite point of with respect to the incircle. Let lines 
and 
intersect the incircle again at 
and 
, respectively. Prove that the lines 
, 
, and 
are concurrent, i.e., the lines intersect at the same point.
with 
right and 
, 
. On the ray 
we take two points and 
so that 
and 
. Consider the squares 
and 
containing inside 
and 
, respectively. If 
and 
, 
and 
, prove that 
and that 
, 
and 
are collinear.
satisfying the conditions:
for all real numbers 
with ![$ t\in [0,1] . $](http://latex.artofproblemsolving.com/1/2/a/12ac756f5dc034618c1566c89e80dcf52761f105.png)
for any real numbers 
such that 
and 
and any 
real numbers 
the following inequality holds.
and 
are constructed on the sides of
as the bases, outside the triangle. Let 
be a point
such that
and 
.
, and if 
is the projection of onto 
,
.
and are constructed on the sides of as the bases, outside the triangle. Let be a point such that and ., and if is the projection of onto ,.
and 
. Take point different from 
on a line 
so that belongs to the circle with center at 
and radii equal to 
.(Call this circle 
). Take point on a line 
in a similar way.(Call the second circle 
). Then, 
and 
, while 
from triangle 
. It means that points 
are collinear. Assume the point belongs to 
and circle as well(different from 
). Than, 
. It means that 
is cyclic and 
. The last equality means that belongs to the circle as well. So, line 
is the common chord of circles and . So, 
is perpendicular to , the same as , q . e . d . the projection of into the line 
. Easy to notice the similarity of triangles 
and 
. As 
and 
are respective altitudes, 
. It means we have to prove 
. 
. Note the similiraty of triangles 
and 
. (as 
and 
. ) So, 
, while 
. Hence, we have proved , which completes our proof.
and 
.You can also prove that 
are collinear in a more simple way:
points 
are concyclic 
radical axis of the two circles.
such that: 
.
