Školjka

%V0 Given seven points in the plane, some of them are connected by segments such that: (i) among any three of the given points, two are connected by a segment; (ii) the number of segments is minimal. How many segments does a figure satisfying (i) and (ii) have? Give an example of such a figure.

%V0 The set $\{a_0, a_1, \ldots, a_n\}$ of real numbers satisfies the following conditions: (i) $a_0 = a_n = 0,$ (ii) for $1 \leq k \leq n - 1,$ $$a_k = c + \sum^{n-1}_{i=k} a_{i-k} \cdot \left(a_i + a_{i+1} \right)$$ Prove that $c \leq \frac{1}{4n}.$

%V0 Let $a, b, c, d,m, n \in \mathbb{Z}^+$ such that $$a^2+b^2+c^2+d^2 = 1989,$$ $$a+b+c+d = m^2,$$ and the largest of $a, b, c, d$ is $n^2.$ Determine, with proof, the values of $m$ and $n.$

%V0 There are n cars waiting at distinct points of a circular race track. At the starting signal each car starts. Each car may choose arbitrarily which of the two possible directions to go. Each car has the same constant speed. Whenever two cars meet they both change direction (but not speed). Show that at some time each car is back at its starting point.

%V0 Prove that $\forall n > 1, n \in \mathbb{N}$ the equation $$\sum^n_{k=1} \frac{x^k}{k!} + 1 = 0$$ has no rational roots.

%V0 Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he finds that if he lays it flat on the floor of either of his storerooms, then each corner of the carpet touches a different wall of that room. He knows that the sides of the carpet are integral numbers of feet and that his two storerooms have the same (unknown) length, but widths of 38 feet and 50 feet respectively. What are the carpet dimensions?