The integers $1, 2, 3,. . . , 2016$ are written in a board. You can choose any pair of numbers in the board and replace them with their average. For example, you can replace $1$ and $2$ with $1.5$, or you can replace $1$ and $3$ with a second copy of $2$. After such replacements, the board will have only one number. (a) Prove that there is a sequence of substitutions that will make let the final number be $2$. (b) Prove that there is a sequence of substitutions that will make let the final number be $1000$.

Determine all $6$-digit numbers $(abcdef)$ such that $(abcdef) = (def)^2$ where $(x_1x_2...x_n)$ is not a multiplication but a number of $n$ digits.

Let $ABCD$ be a cyclic quadrilateral. Let $ P$ be the intersection of the lines $BC$ and $AD$. Line $AC$ cuts the circumscribed circle of the triangle $BDP$ in $S$ and $T$, with $S$ between $ A$ and $C$. The line $BD$ intersects the circumscribed circle of the triangle $ACP$ in $U$ and $V$, with $U$ between $ B$ and $D$. Prove that $PS = PT = PU = PV$.

The integers $1, 2,. . . , n$ are arranged in order so that each value is strictly larger than all values above or is strictly less than all values previous. In how many ways can this be done?

$ABCD$ is a quadrilateral, $E, F, G, H$ are the midpoints of $AB$, $BC$, $CD$, $DA$ respectively. Find the point $P$ such that area $(PHAE) =$ area $(PEBF) =$ area $(PFCG) =$ area $(PGDH).$

$N$ denotes the set of all natural numbers. Define a function $T: N \to N$ such that $T (2k) = k$ and $T (2k + 1) = 2k + 2$. We write $T^2 (n) = T (T (n))$ and in general $T^k (n) = T^{k-1} (T (n))$ for all $k> 1$. (a) Prove that for every $n \in N$, there exists $k$ such that $T^k (n) = 1$. (b) For $k \in N$, $c_k$ denotes the number of elements in the set $\{n: T^k (n) = 1\}$. Prove that $c_{k + 2} = c_{k + 1} + c_k$, for $1 \le k$.