2015-16 USAMTS Round 1 Problems

Question

I would like to request that everyone keep an eye out for the USAMTS Round 1 problems, per the ongoing competitions rules. They'll be released tomorrow at this address: here. (Currently the link goes to a placeholder PDF.)

Answer 1

This is just a transcription of the questions from the 2015-16 USAMTS Round 1 problem set. As last year, your benevolent overlords moderators request your help in finding them when they appear on the main site so that they can be locked as per our policy on contest questions.

1. Fill in the spaces of the grid to the right with positive integers so that in each $2 \times 2$ square with top left number $a$, top right number $b$, bottom left number $c$, and bottom right number $d$, either $a+d=b+c$ or $ad=bc$.

   

2. Suppose $a$, $b$, and $c$ are distinct positive real numbers such that $$\begin{align} abc &= 1000, \\ bc(1-a) + a(b+c) &= 110. \end{align}$$ If $a < 1$, show that $10 < c < 100$.

3. Let $P$ be a convex $n$-gon in the plane with vertices labeled $V_1, \ldots , V_n$ in counterclockwise order. A point $Q$ not outside $P$ is called a balancing point of $P$ if, when the triangles $QV_1V_2$, $QV_2V_3$, $\ldots$, $QV_{n-1}V_n$, $QV_nV_1$ are alternately colored blue and green, the total areas of the blue and green regions are the same. Suppose $P$ has exactly one balancing point. Show that the balancing point must be a vertex of $P$.

4. Several players try out for the USAMTS basketball team, and they all have integer heights and weights when measured in centimeters and pounds, respectively. In addition, they all weigh less in pounds than they are tall in centimeters. All of the players weigh at least 190 points and are at most 197 centimeters tall, and there is exactly one player with every possible height-weight combination.

   The USAMTS wants to field a competitive team, so there are some strict requirements.

   1. If person $P$ is on the team, then anyone who is at least as tall and at most as heavy as $P$ must also be on the team.
   2. If person $P$ is on the team, then no one whose weight is the same as $P$'s height can also be on the team.

   Assuming the USAMTS team can have any number of members (including zero), how many different basketball teams can be constructed?

5. Find all positive integers $n$ that have distinct divisors $d_1$, $d_2$, $\ldots$, $d_k$, where $k > 1$, that are in arithmetic progression and $$n = d_1 + d_2 + \cdots + d_k.$$ Note that $d_1$, $d_2$, $\ldots$, $d_k$ do not have to be all the divisors of $n$.

Thanks in advance to all who flag 'em when they see 'em!