AoPS Academy
to 
face-down on a table. On each move, the Detective can point to two cards and inquire if the numbers on them are consecutive. The Magician replies truthfully. After a finite number of moves, the Detective points to two cards. She wins if the numbers on these two cards are consecutive, and loses otherwise.
questions.
= 2∠AMP.
is the midpoint of the arc 
, containing 
, in the circumcircle of 
, and let 
be the -excircle's center. Assume that the external angle bisector of 
intersects at 
. Prove that 
is perpendicular to 
, where 
is the incenter of .
with 
and the circle 
of radius 
centered at 
Let 
be the midpoint of 
The line 
intersects a second time at 
Let 
be a point on such that 
Let 
be the intersection of 
and Prove that 
let the numbers 
be determined by the rules 
and 
Find the value of ![\[\sum_{n=1}^{2013}c(n)c(n+2).\]](http://latex.artofproblemsolving.com/3/f/a/3faf9da9b035e02298650c79251370d0c8b61c1f.png)
be expressed as a polynomial in 
with real coefficients?
be a twice differentiable function such that![\[
f(0) = 2, \quad f'(0) = -2, \quad \text{and} \quad f(1) = 1.
\]](http://latex.artofproblemsolving.com/1/e/1/1e178debf8ae0b6b76d2e60ffca95f5a4c4ad4e5.png)
Prove that there exists a point 
such that![\[
f(\xi) \cdot f'(\xi) + f''(\xi) = 0.
\]](http://latex.artofproblemsolving.com/4/6/b/46b8d103981b838ae0aa5afdee104edfae4e4b8f.png)
positive such that 
, then there exists 
positive such that 
converges and 
diverges ?
is continuous, positive with 
, does 
continuous and positive exist on 
such that 
converges and 
diverges ?
is in the 
class, increasing : I chose 
. With this idea, I had the idea to define 
but it is not clear that it is ok, even if is increasing. positive such that , then there exists positive such that converges and diverges ? And if is increasing ? positive increasing such that 
, then 
diverges ? positive increasing such that , then 
converges ? is positive increasing and such that 
does not converge to , can diverge ? only to be continuous ?
such that for every polynomial 
with integer coefficients and for every integer 
the integer![\[p^{(j)}(k)=\left. \frac{d^j}{dx^j}p(x) \right|_{x=k}\]](http://latex.artofproblemsolving.com/a/2/7/a2793d7f46345bcba4ccaa3082cc47afcd35074b.png)
(the -th derivative of at 
) is divisible by 
order, and if the constituents of its first, second, ..., rows are the first 
figurate numbers of the first, second, ..., orders respectively, show that it's value is 
holds inequality 
Let 
and 
is the sum of 
square numbers. Prove that: 
is the sum of square numbers
Let and is the sum of square numbers. Prove that: is the sum of square numbers
Let and is the sum of square numbers. Prove that: is the sum of square numbers
be equal to 
divided by the largest power of 
that divides . Legendre's three-square theorem says that is expressible as the sum of three squares iff is not 
mod 
.
is never mod , so even always works. If is odd, if 
is mod , then 
is even and thus 
is even. Thus 
is odd. Observe that if is odd then 
. Thus if is mod then 
is mod . However, any odd number to an odd power is congruent to itself mod , so this means 
, contradiction.
, which would make it trivial.
,![\[ \dfrac {1}{2ne} < \dfrac {1}{e} - \left( 1 - \dfrac {1}{n} \right)^n < \dfrac {1}{ne}. \]](http://latex.artofproblemsolving.com/1/3/e/13e2e3859e16ca0e4453330eeac03465d2eb8429.png)
