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.
such that p is a prime, 
and 
is a divisor of 
.
such that![\[ \frac {2^n + 1}{n^2}
\]](http://latex.artofproblemsolving.com/9/2/2/922444e7669c326efcad5fde3ea00f8c2d9fa5d8.png)
is an integer.
on a circle and points 
and 
on segment , such that 
Find the ratio 
, where 
is a point on arc , such that 
is maximized.
let 
and 
be the altitudes. Let the points 
and 
fulfil the following requirements:
is the circumcentre of triangle .
and 
are equal to the circumradius of triangle .
has the same direction as the oriented segment 
. Similarly, 
has the same direction as 
, and 
has the same direction as . is the incentre of triangle .
be a triangle with circumcenter 
. Let be a point on the perpendicular bisector of (see figure) and 
, 
be the intersections of the perpendicular bisectors of 
and 
, respectively, with 
and 
. Prove that the midpoints of 
and and the point are collinear.
such that 
is a divisor of 
.
, the following inequality holds:![\[
\sqrt{a + b} + \sqrt{b + c} + \sqrt{c + a} \geq \frac{4(ab + bc + ca)}{\sqrt{(a + b)(b + c)(c + a)}}
\]](http://latex.artofproblemsolving.com/e/7/2/e72fd0f134289c359bc8766440babc52ae131961.png)
be a positive integer, 
, and 
. Prove that 
be an acute triangle with circumcenter . Draw altitude 
, with on side . The parallel line to 
passing through intersects line 
at point 
. Prove that point and the midpoints of sides and are collinear.
be a positive integer. Claire writes distinct positive real numbers 
in a row on a blackboard. In a 
William can erase a number 
and replace it with either 
or 
at the same location. His goal is to perform a sequence of moves such that after he is done, the number are strictly increasing from left to right.
independent of 
such that William can always reach his goal in at most 
moves.
independent of such that Claire can choose the initial numbers such that William cannot attain his goal in less than 
moves.
and 
, where is not the zero polynomial, we define 
to be the unique polynomial 
such that we can write 
, where 
is a polynomial such that either 
or the degree of is less than the degree of . Find all polynomials 
with real coefficients such that 
on the legs 
and 
(
), the points 
and 
are taken so that 
, 
. Prove that the angle between the segments 
and 
is 
.
satisfy that: 
satisfied 
Find the minimum of 
