Anja has to write $2014$ integers on the board such that arithmetic mean of any of the three numbers is among those $2014$ numbers. Show tha...

1. Anja has to write $2014$ integers on the board such that arithmetic mean of any of the three numbers is among those $2014$ numbers. Show that this is possible only when she writes nothing but $2014$ equal integers.
2. The $100$ vertices of a prism, whose base is a $50$-gon, are labeled with numbers $1, 2, 3, \ldots, 100$ in any order. Prove that there are two vertices, which are connected by an edge of the prism, with labels differing by not more than $48$. (In all the triangles the three vertices do not lie on a straight line.)
3. A regular hexagon with side length $1$ is given. Using a ruler construct points in such a way that among the given and constructed points there are two such points that the distance between them is $\sqrt7$. (''Using a ruler construct points $\ldots$'' means: Newly constructed points arise only as the intersection of straight lines connecting two points that are given or already constructed. In particular, no length can be measured by the ruler.)
4. Find all postive integers $n$ for which the number $\dfrac{4n+1}{n(2n-1)}$ has a terminating decimal expansion.
5. Show that for all positive integers $n$, the number $2^{3^n}+1$ is divisible by $3^{n+1}$.
6. For all positive integers $m$ and $k$ with $m\ge k$, define $a_{m,k}=\binom{m}{k-1}-3^{m-k}$. Determine all sequences of real numbers $\{x_1, x_2, x_3, \ldots\}$, such that each positive integer $n$ satisfies the equation $a_{n,1}x_1+ a_{n,2}x_2+ \cdots + a_{n,n}x_n = 0$
7. A line $g$ is given in a plane. $n$ distinct points are chosen arbitrarily from $g$ and are named as $A_1, A_2, \ldots, A_n$. For each pair of points $A_i,A_j$, a semicircle is drawn with $A_i$ and $A_j$ as its endpoints. All semicircles lie on the same side of $g$. Determine the maximum number of points (which are not lying in $g$) of intersection of semicircles as a function of $n$.
8. Three non-collinear points $A_1, A_2, A_3$ are given in a plane. For $n = 4, 5, 6, \ldots$, $A_n$ be the centroid of the triangle $A_{n-3}A_{n-2}A_{n-1}$.
a) Show that there is exactly one point $S$, which lies in the interior of the triangle $A_{n-3}A_{n-2}A_{n-1}$ for all $n\ge 4$.
b) Let $T$ be the intersection of the line $A_1A_2$ with $SA_3$. Determine the two ratios, $A_1T : TA_2$ and $TS : SA_3$.

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