CMOQR 2020

Show that for all integers $a \ge 1$ , $\lfloor \sqrt{a}+\sqrt{a+1}+\sqrt{a+2}\rfloor = \lfloor \sqrt{9a+8}\rfloor$

Given a set $S$ , of integers, an optimal partition of S into sets T, U is a partition which minimizes the value $|t - u|$ , where $t$ and $u$ are the sum of the elements of $T$ and U respectively. Let $P$ be a set of distinct positive integers such that the sum of the elements of $P$ is $2k$ for a positive integer $k$ , and no subset of $P$ sums to $k$ . Either show that there exists such a $P$ with at least $2020$ different optimal partitions, or show that such a $P$ does not exist.

Let $N$ be a positive integer and $A = a_1, a_2, ... , a_N$ be a sequence of real numbers. Define the sequence $f(A)$ to be $$f(A) = \left( \frac{a_1 + a_2}{2},\frac{a_2 + a_3}{2}, ...,\frac{a_{N-1} + a_N}{2},\frac{a_N + a_1}{2}\right)$$ and for $k$ a positive integer define $f^k (A)$ to be $f$ applied to $A$ consecutively $k$ times (i.e. $f(f(... f(A)))$ ) Find all sequences $A = (a_1, a_2,..., a_N)$ of integers such that $f^k (A)$ contains only integers for all $k$ .

Determine all graphs $G$ with the following two properties: $\bullet$ G contains at least one Hamilton path. $\bullet$ For any pair of vertices, $u, v \in G$ , if there is a Hamilton path from $u$ to $v$ then the edge $uv$ is in the graph $G$