It's that time of year again: USAMTS 2016-17 Round 1 problem set is out!

The 2016-17 USAMTS Round 1 problem set is out, and some problems from it have been found on the site already.

This is yet another request for all users to be on the lookout for these problems, and to flag them for moderator attention when you see them so that we can lock them as per our policy on ongoing contest problems. Participants can submit solutions until 17 October 2016, and so the moderators will keep these questions locked until at least a UTC-day or two later.

The current problem set can be found here, and I transcribe the problems below.

Again, a big thanks to the users who help us keep these questions off the site until the contest concludes.

1. Fill in each cell of the grid with one of the numbers $1$, $2$, or $3$. After all numbers are filled in, if a row, column, or any diagonal has a number of cells equal to a multiple of $3$, then it must have the same amount of $1$'s, $2$'s, and $3$'s. (There are 10 such diagonals, and they are all marked in the grid by a gray dashed line.) Some numbers have been given to you.

You do not need to prove that your answer is the only one possible; you merely need to find an answer that satisfies the constraints above. (Note: in any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.)

2. A tower of height $h$ is a stack of contiguous rows of squares of height $h$ such that

1. the bottom row of the tower has $h$ squares,
2. each row above the bottom row has one fewer square that the row below it,
3. the squares in any given row all lie directly above a square in the row below.

A tower is called balanced if when the squares of the tower are colored black and white in a checkerboard fashion, the number of black squares is equal to the number of white squares. For example, the figure above shows a tower of height $5$ that is not balanced, since there are $7$ white squares and $8$ black squares.

How many balanced towers are there of height $2016$?

3. Find all positive integers $n$ for which $(x^n+y^n+z^n)/2$ is a perfect square whenever $x$, $y$, and $z$ are integers such that $x+y+z = 0$.

4. Find all functions $f(x)$ from the nonnegative reals to the nonnegative reals such that $f(f(x)) = x^4$ and $f(x) \leq Cx^2$ for some constant $C$.

5. Let $ABCD$ be a convex quadrilateral with perimeter $\frac{5}{2}$ and $AC = BD = 1$. Determine the maximum possible area of $ABCD$.

• saw one of those yesterday, number 3. – Will Jagy Sep 13 '16 at 17:22
• – Will Jagy Sep 13 '16 at 17:40
• in disguised language math.stackexchange.com/questions/1924950/… – Will Jagy Sep 13 '16 at 17:44
• @WillJagy re: the "disguised language" question. I am a bit uncomfortable locking that question per our policy. I think it is unquestionably "inspired" by the USAMTS question, but I don't think that our policy covers questions "inspired" by contest questions. I'll throw up a question here on meta UTC-tomorrow (don't have the energy to right now) to get some feedback about this. (Unless, of course, someone beats me to the punch.) – user642796 Sep 13 '16 at 19:21
• arjafi, thanks for taking a look, and for letting me know you agree about the "inspired" nature of the post. – Will Jagy Sep 13 '16 at 19:37
• What would count as "inspired"? Like for example a problem like Let $ABCD$ be a convex quadrilateral with perimeter 5 and $AC=BD=2$. Determine the maximum possible area of $ABCD$. would be obviously a disguised form of a contest question. – suomynonA Sep 14 '16 at 2:16
• @Anonymous I think part of it has to be how "trivially" a solution to the "inspired" question can be used to solve the original. For Will Jagy's example, I don't think an answer carries over very well at all, since it is concerned with when $x^n+y^n+z^n$ is a perfect square, and if this number is a perfect square then half of it (which is the concern of the USAMTS question) cannot be a perfect square. (cont.) – user642796 Sep 14 '16 at 8:58
• And this is ignoring that the USAMTS problem requests all $n$ such that something holds, whereas the "inspired" question is essentially only looking for an upper bound on the $n$s such that something holds. So unless I am terribly mistaken, the two questions are really different and while an answer to one might give ideas for solving the other, that's about it. That is to say, if the question was asked to quickly get a solution to the USAMTS problem, then it is a very poor question indeed. (cont.) – user642796 Sep 14 '16 at 8:58
• The "inspired" question from your comment is decidedly not like this; an answer to it will with very little effort yield an answer to the respective USAMTS problem. Similarly we sometimes see users simply change the variable names (e.g., $x$ becomes $a$, $y$ becomes $c$, ...). – user642796 Sep 14 '16 at 8:58
• Did anyone notice that the shape of the numbers given in the grid make a "28" for 28 years of USAMTS? :) – Yeah.. Oct 11 '16 at 5:57
• Arjafi please look into this question. It is a modification on the geometry question as @suomynonA said. math.stackexchange.com/questions/1965788/… – N.S.JOHN Oct 13 '16 at 3:28
• @N.S.JOHN People seem to like modifying that question the most; I've seen it posted a couple of times now... – suomynonA Oct 13 '16 at 3:33
• It's been a couple UTC days. Shall we start allowing these questions? – Richard Rast Oct 20 '16 at 16:37