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 
according to the diagram. The edges from points 
and 
, respectively are pairwise perpendicular. Prove that ![\[[ABCD]^2+[ABFE]^2+[ADHE]^2=[BCGF]^2+[CDHG]^2+[EFGH]^2,\]](http://latex.artofproblemsolving.com/c/f/b/cfb6dc6726af4711bcfe7827b6531471a75270fb.png)
where ![$[XYZW]$](http://latex.artofproblemsolving.com/9/c/2/9c25ef1b5cfb129136334ed811fb699a84f48a13.png)
denotes the area of quadrilateral 
.
such that
![\[\alpha =1-\cfrac{1}{2a_1-\cfrac{1}{2a_2-\cfrac{1}{2a_3-\cdots}}}\]](http://latex.artofproblemsolving.com/3/0/9/309e0411f481b6d77ccdfe8f18af9852074375aa.png)
where coefficients 
are positive integers, infinitely many of which are greater than 
. Prove that for every positive integer 
at least half of the numbers 
are even.
be an integer. In a configuration of an 
board, each of the 
cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate 
counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of 
, the maximum number of good cells over all possible starting configurations.
be a given positive integer. Luca, the lazy flea sits on one of the vertices of a regular 
-gon. For each jump, Luca picks an axis of symmetry of the polygon, and reflects herself on the chosen axis of symmetry. Let 
denote the number of different ways Luca can make jumps such that she returns to her original position in the end, and does not pick the same axis twice. (It is possible that Luca's jump does not change her position, however, it still counts as a jump.) if is odd. is even, then![\[P(n)=(n-1)!\cdot n!\cdot \sum_{d\mid n}\left(\varphi\left(\frac{n}d\right)\binom{2d}{d}\right).\]](http://latex.artofproblemsolving.com/3/0/9/30992e5982bce4a032d5e96e0fb5fce8ab6ff21a.png)
, interior point 
is chosen such that triangle 
is equilateral. Let 
be the isogonal conjugate of point with respect to triangle . Define point 
on the ray 
such that 
. Similarly, define point 
on the ray 
such that 
. Prove that line 
bisects segment 
.
. Let 
be integers no two of which have their difference divisible by 
. Let 
be nonnegative integers such that 
is divisible by 
. Suppose that for all integers 
, the quantity![\[ (x - a)(x - b)(x - c)[(x - a)^i(x - b)^j(x - c)^k - 1]\]](http://latex.artofproblemsolving.com/7/3/9/739d5dbd62a9ea9bf7d85193daa68f58876a3ec6.png)
. Prove that each of must be divisible by .
has 2 roots 
and equation 
has 2 roots called 
. Prove that 
![$P(x) \in \mathbb{Z}[x]$](http://latex.artofproblemsolving.com/3/b/4/3b4dadc0cf4b02930a03c0ff0ce95e8b7a740ebb.png)
be a polynomial. Suppose that 
for infinitely many pairs 
. Prove that the graph of function 
has a center of symmetry.
is not a finite dimentional vector space.
can it still be a vector space of finite dimensional?
? (there is an easy workaround)
be a non-real algebraic integer of degree two, and let 
be the set of irreducible elements of the ring ![$ \mathbb{Z}[ \alpha] $](http://latex.artofproblemsolving.com/3/1/6/316d00f1eaea5f28346a44deedb58bc83f41657e.png)
. Prove that![\[ \sum_{p\in \mathbb{P}}^{{}}\frac{1}{|p|^{2}}=\infty \]](http://latex.artofproblemsolving.com/d/3/d/d3d48d0ac7bee298334a11a14559e3b05ab0cd1d.png)
may not be real. The original question asks for 
.
![$\mathbb{Z}[\alpha]$](http://latex.artofproblemsolving.com/1/5/d/15d0fe359264077549c9002fcd29e58f021f05dc.png)
into product of two principal ideals has positive density. Great reference is D. A. Cox: Primes of the form 
.
![$ln(\prod_{p \in P} \frac{1}{1-|p|^{-2}})=ln(\sum_{x \in Z[a]} \frac{1}{|x|^2})=+\infty$](http://latex.artofproblemsolving.com/d/b/6/db6abd9b85d6ed2ce44ee1c61cc95bb19ca1f34c.png)
![$\mathbb Z[\alpha]$](http://latex.artofproblemsolving.com/b/7/4/b748e9a87d438730c0c39ef724540917e8d72f44.png)
doesn't need to be UFD.
,![\[
g_r(x) = \frac{1}{\tan^{r+1} x} \cdot \frac{d^r}{dx^r} (\tan x)
\]](http://latex.artofproblemsolving.com/6/e/7/6e7fd00178c7657a478f5d9fda80354f2a78816e.png)
![\[
\lim_{x \to \frac{\pi}{2}} g_r(x)
\]](http://latex.artofproblemsolving.com/3/a/5/3a577519fbeb3b419f6beef525f450fcd1fc9195.png)
![$\sum_{x \in Z[a]} \frac{1}{|x|^2}=+\infty$](http://latex.artofproblemsolving.com/2/2/e/22edcf9355f0e2e3f1b5ebe932b270cf91099ab5.png)
![$\sum_{x \in Z[a]} \frac{1}{|x|^n}<+\infty$](http://latex.artofproblemsolving.com/a/4/6/a4660f6a495743e9f860f6a28a1cf1233828acf2.png)
for 
