Prove that exist infinite integers $n$ so that $n^2$ divides $2^n+3^n$. Thanks

Show that for every polynomial $f(x)$ with integer coefficients, there exists a integer $C$ such that the set $\{n \in Z :$ the sum of digits of $f(n)$ is $C\}$ is not finite.

Show that if equiangular hexagon has sides $a, b, c, d, e, f$ in order then $a - d = e - b = c - f$.

Show that if $a \ge b \ge c \ge 0$ then $$a^2b(a - b) + b^2c(b - c) + c^2a(c - a) \ge 0.$$

Consider a cyclic quadrilateral $ABCD$ and define points $X = AB \cap CD$, $Y = AD \cap BC$, and suppose that there exists a circle with center $Z$ inscribed in $ABCD$. Show that the $Z$ belongs to the circle with diameter $XY$ , which is orthogonal to circumcircle of $ABCD$.

Suppose that every integer is colored using one of $4$ colors. Let $m, n$ be distinct odd integers such that $m + n \ne 0$. Prove that there exist integers $a$, $ b$ of the same color such that $ a - b$ equals one of the numbers $m$, $n$, $m - n$, $m + n$.

1 and only round Hi all forumers, after some years of break, let's see if someone want to partecipate again.. It's the third edition of a sort of individual, and telematic contest, with training problems levels for national and imo (difficulty is subjective and depends also from states to states); imho it can be useful to a lot of forumers here. I'll try to sum up again all details about it, please be carefully 1- The contest is made by: unique round of 6 problems, starting in the morning of 23 october, and ending at 23:59 of 25 october (substantially, 3 days) 2- Every problem will have some points, as usual from 0 to 7: specifically, 6 for all correct solution, and 7 only for clear and original ones. 3- Everyone can send his solution, there are no age contrasints: the unique limit is that non-olympic argument cannot be used, otherwise the solution is not going to be valid. 4- It's enough to send the solution by mail (that you can find below, from which you'll receive a confirme answer): Enrollment is not necessary. 5- I’ll take in consideration time of arriving solutions only for ex-aequo positions. 8- Way of sending solutions. - You need to send solutions to email: $\text{ leonetti.paolo (AT) gmail.com }$ ; - Solution need to be in a unique pdf file (so, with attachment) of a reasonable size; - The pdf file must be written with Latex (or a way that it's easy to understand); - You have to rename the pdf file with your nickname (the same of yours in ML); - You need to write nothing else in the email; - Try to be clear in solutions! it will be appreciated. - {If (and only if) you are not able to create a pdf in latex model, you can send me a email in .doc format with solutions, as latex or in a way that i (and all correctors) can understand}. You can find also problems here: { Ed. 2008, round 1, round 2, round 3 } , { Ed. 2009, round 1, round 2 } . I'll put below this topic the list of problems on the morning of 23 october 2012: Later, I'll make a list still below this topic where i'll write all names of who sent me solutions in right way. Any comment and/or suggestion will be welcome. Thanks for the attentions And good luck Paolo Leonetti Italian version Nb: problem 6 has been modified from $a+b$ to $a-b$