Školjka

%V0 The numbers $0$, $1$, $\dots$, $n$ ($n \ge 2$) are written on a blackboard. In each step we erase an integer which is the arithmetic mean of two different numbers which are still left on the blackboard. We make such steps until no further integer can be erased. Let $g(n)$ be the smallest possible number of integers left on the blackboard at the end. Find $g(n)$ for every $n$.

%V0 Let $x_1$, $x_2$, $\ldots$, $x_n$ be real numbers satisfying the conditions: $ |x_1 + x_2 + \dots + x_n| = 1 $ and $|x_i| \leq \frac{n+1}{2}$, for $i = 1, 2, \dots, n$ Show that there exists a permutation $y_1$, $y_2$, $\ldots$, $y_n$ of $x_1$, $x_2$, $\ldots$, $x_n$ such that $$| y_1 + 2 y_2 + \cdots + n y_n | \leq \frac {n + 1}{2}.$$

%V0 For each finite set $U$ of nonzero vectors in the plane we define $l(U)$ to be the length of the vector that is the sum of all vectors in $U.$ Given a finite set $V$ of nonzero vectors in the plane, a subset $B$ of $V$ is said to be maximal if $l(B)$ is greater than or equal to $l(A)$ for each nonempty subset $A$ of $V.$ (a) Construct sets of 4 and 5 vectors that have 8 and 10 maximal subsets respectively. (b) Show that, for any set $V$ consisting of $n \geq 1$ vectors the number of maximal subsets is less than or equal to $2n.$

%V0 Two students $A$ and $B$ are playing the following game: Each of them writes down on a sheet of paper a positive integer and gives the sheet to the referee. The referee writes down on a blackboard two integers, one of which is the sum of the integers written by the players. After that, the referee asks student $A:$ “Can you tell the integer written by the other student?” If A answers “no,” the referee puts the same question to student $B.$ If $B$ answers “no,” the referee puts the question back to $A,$ and so on. Assume that both students are intelligent and truthful. Prove that after a finite number of questions, one of the students will answer “yes.”

%V0 Let $n$ be a positive integer having at least two different prime factors. Show that there exists a permutation $a_1, a_2, \dots , a_n$ of the integers $1, 2, \dots , n$ such that $$\sum_{k=1}^{n} k \cdot \cos \frac{2 \pi a_k}{n}=0.$$

%V0 Prove that we can find an infinite set of positive integers of the from $2^n-3$ (where $n$ is a positive integer) every pair of which are relatively prime.