USAMTS 2022-23, third round

Question

Previous rounds of this years' USAMTS were brought up recently on meta.¹ This post is to serve as a reminder that there is an ongoing contest.

The relevant links: the USAMTS site, the page with the problems and the PDF files with problems for round 3.

The deadline is January 3, 2023.

Perhaps it might be useful to collect the text of the problems here - so that it's easier for the users of this site to check them out.²

¹It was mentioned in the following post: Someone is cheating on current USAMTS problems. It was even mentioned that probably it might be better to keep such posts locked a few days after deadline: USAMTS Discussion Policy

²Similarly as in the past, such as here: 2015-16 USAMTS Round 1 Problems

Answer 1

This is a community wiki answer with the text of the problems - do not hesitate to edit it, if needed.

1/3/34. In the $8 × 8$ grid below, label $8$ squares with $X$ and $8$ squares with $Y$ such that:

1. No square can be labeled with both an $X$ and a $Y$.
2. Each row and each column must contain exactly one square labeled $X$ and one square labeled $Y$.
3. Any square marked with a $\ast$ or a $\heartsuit$ cannot be labeled with an $X$ or a $Y$.
4. We say that a square marked with a $\ast$ or a $\heartsuit$ sees a label ($X$ or $Y$) if one can move in a straight line horizontally or vertically from the marked square to the square with the label, without crossing any other squares with $X$'s or $Y$'s. It is OK to cross other squares marked with a $\ast$ or $\heartsuit$. Using this definition:
   (a) Each square marked with a $\ast$ must see exactly 2 $X$'s and 1 $Y$.
   (b) Each square marked with a $\heartsuit$ must see exactly 1 $X$ and 2 $Y$’s.

There is a unique solution, but you do not need to prove that your answer is the only one possible. You merely need to find an answer that satisfies the conditions of the problem. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.)

2/3/34. Let $\mathbb Z^+$ denote the set of positive integers. Determine, with proof, if there exists a function $f \colon \mathbb Z^+ \to \mathbb Z^+$ such that $$f(f(f(f(f(n))))) = 2022n$$ for all positive integers $n$.

3/3/34. A positive integer $N$ is called googolicious if there are exactly $10^{100}$ positive integers $x$ that satisfy the equation $$\left\lfloor \frac{N}{\left\lfloor \frac Nx \right\rfloor} \right\rfloor=x.$$ where $\lfloor z\rfloor$ denotes the greatest integer less than or equal to $z$. Find, with proof, all googolicious integers.

4/3/34. Let $\omega$ be a circle with center $O$ and radius $10$, and let $H$ be a point such that $OH = 6$. A point $P$ is called snug if, for all triangles $ABC$ with circumcircle $\omega$ and orthocenter $H$, we have that $P$ lies on $\triangle ABC$ or in the interior of $\triangle ABC$. Find the area of the region consisting of all snug points.

5/3/34. A lattice point is a point on the coordinate plane with integer coefficients. Prove or disprove: there exists a finite set $S$ of lattice points such that for every line $\ell$ in the plane with slope $0$, $1$, $−1$, or undefined, either $\ell$ and $S$ intersect at exactly $2022$ points, or they do not intersect.

Answer 2

This is a community wiki answer - do not hesitate to edit it, if you have suitable additions.

We could even prepare some searches (as suggested in a recent discussion) which should be able to return the problems from this contest at least in cases when the OP copy-pasted the whole text.

Since we're interested in posts made after the contest started, it makes sense to look at the recent or recently active questions (in the case of built-in search) or restrict the date for the search results (in the case of an external search engine).

• Problem 1. grid label square sees: recently active, newest, Google, Google Images. (In this case, it might make sense to try Google images too - the problem is quite likely to contain a picture of the grid, which should be relatively easy to spot among the search results.)
• Problem 2. "f(f(f(f(f(n)))))": recently active, newest, Google
• Problem 2. "f^5(n)=2022n": recently active, newest, Google
• Problem 3. googolicious positive integers: recently active, newest, Google
• Problem 4. snug circumcircle orthocenter: recently active, newest, Google
• Problem 5. finite set lattice points intersect exactly: recently active, newest, Google