(a) Prove the parallelogram law that says that in a parallelogram the sum of the squares of the lengths of the four sides equals the sum of the squares of the lengths of the two diagonals. (b) The edges of a tetrahedron have lengths $a, b, c, d, e$ and $f$. The three line segments connecting the centers of intersecting edges have lengths $x, y$ and $z$. Prove that $$4 (x^2 + y^2 + z^2) = a^2 + b^2 + c^2 + d^2 + e^2 + f^2$$

In Miss Lies' class there are only students who never lie and students who always lie. All students know which category they belong to. During the day in a class discussion, every student in the class says about every other student or he or she a liar or not. In total, it is said $320$ times that someone is not lying. The next day, one of the students who always lies is sick. There will be one again organize such a class discussion in which no mention is made of the sick pupil. Now it is said $300$ times that someone does lie. How many liars are there in the Miss Lies' class ?

Let $PQRS$ be a quadrilateral with $| P Q | = | QR | = | RS |$, $\angle Q= 110^o$ and $\angle R = 130^o$ . Determine $\angle P$ and $\angle S$ .

Let $P(x)$ be a polynomial of degree $5$ and suppose that a and b are real numbers different from zero. Suppose the remainder when $P(x)$ is divided by $x^3 + ax + b$ equals the remainder when $P(x)$ is divided by $x^3 + ax^2 + b$. Then determine $a + b$.