Suppose that $a, b, c$ are real numbers such that $a + b + c> 0$, and so the equation $ax^2 + bx + c = 0$ has no real solutions. Show that $c> 0$.

Suppose that $x$ and $y$ are positive real numbers such that $x^3, y^3$ and $x + y$ are all rational numbers. Show that the numbers $xy, x^2+y^2, x$ and $y$ are also rational

Let $A$ be a set, and let $P (A)$ be the powerset of all non-empty subsets of $A$. (For example, $A = \{1,2,3\}$, then $P (A) = \{\{1\},\{2\} ,\{3\},\{1,2\}, \{1,3\},\{2,3\}, \{1,2,3\}\}$.) A subset $F$ of P $(A)$ is called strong if the following is true: If $B_1$ and $B_2$ are elements of $F$, then $B_1 \cup B_2$ is also an element of $F$. Suppose that $F$ and $G$ are strong subsets of $P (A)$. a) Is the union $F \cup G$ necessarily strong? b) Is the intersection $F \cap G$ necessarily strong?

What is the largest possible area of a quadrilateral with sidelengths $1, 4, 7$ and $8$ ?

The diagonals $AC$ and $BD$ in the convex quadrilateral $ABCD$ intersect in $S$. Let $F_1$ and $F_2$ be the areas of $\vartriangle ABS$ and $\vartriangle CSD$. and let $F$ be the area of the quadrilateral $ABCD$. Show that $\sqrt{ F_1 }+\sqrt{ F_2}\le \sqrt{ F}$

At a two-day team competition in chess, three schools with $15$ pupils each attend. Each student plays one game against each player on the other two teams, ie a total of $30$ chess games per student. a) Is it possible for each student to play exactly $15$ games after the first day? b) Show that it is possible for each student to play exactly $16$ games after the first day. c) Assume that each student has played exactly $16$ games after the first day. Show that there are three students, one from each school, who have played their three parties