exam: RMM

problem:
Let $n$ be a positive integer. Initially, a bishop is placed in each square of the top row of a $2^{n} \times 2^{n}$ chessboard; those bishops are numbered from 1 to $2^{n}$, from left to right. A jump is a simultaneous move made by all bishops such that the following conditions are satisfied:

- each bishop moves diagonally, in a straight line, some number of squares, and
- at the end of the jump, the bishops all stand in different squares of the same row.

Find the total number of permutations $\sigma$ of the numbers $1,2, \ldots, 2^{n}$ with the following property: There exists a sequence of jumps such that all bishops end up on the bottom row arranged in the order $\sigma(1), \sigma(2), \ldots, \sigma\left(2^{n}\right)$, from left to right.

problem_match: Problem 1.


problem: Consider an odd prime $p$ and a positive integer $N<50 p$. Let $a_{1}, a_{2}, \ldots, a_{N}$ be a list of positive integers less than $p$ such that any specific value occurs at most $\frac{51}{100} N$ times and $a_{1}+a_{2}+\cdots+a_{N}$ is not divisible by $p$. Prove that there exists a permutation $b_{1}, b_{2}, \ldots, b_{N}$ of the $a_{i}$ such that, for all $k=1,2, \ldots, N$, the sum $b_{1}+b_{2}+\cdots+b_{k}$ is not divisible by $p$.

problem_match: Problem 2.


problem:
Given a positive integer $n$, a set $\mathcal{S}$ is $n$-admissible if

- each element of $\mathcal{S}$ is an unordered triple of integers in $\{1,2, \ldots, n\}$,
- $|\mathcal{S}|=n-2$, and
- for each $1 \leq k \leq n-2$ and each choice of $k$ distinct $A_{1}, A_{2}, \ldots, A_{k} \in \mathcal{S}$,

$$
\left|A_{1} \cup A_{2} \cup \cdots \cup A_{k}\right| \geq k+2
$$

Is it true that, for all $n>3$ and for each $n$-admissible set $\mathcal{S}$, there exist pairwise distinct points $P_{1}, \ldots, P_{n}$ in the plane such that the angles of the triangle $P_{i} P_{j} P_{k}$ are all less than $61^{\circ}$ for any triple $\{i, j, k\}$ in $\mathcal{S}$ ?

Each problem is worth 7 marks.
Time allowed: $4 \frac{1}{2}$ hours.

problem_match: Problem 3.