NL-NWO 1975

Are the following statements true? $x^7 \in Q \land x^{12} \in Q \Rightarrow x \in Q$ , and $x^9 \in \land x^{12} \in Q \Rightarrow x \in Q$ .

Let $T = \{n \in N|$ n consists of $2$ digits $\}$ and $$P = \{x|x = n(n + 1)... (n + 7); n,n + 1,..., n + 7 \in T\}.$$ Determine the gcd of the elements of $P$ .

Given are the real numbers $x_1,x_2,...,x_n$ and $t_1,t_2,...,t_n$ for which holds: $\sum_{i=1}^n x_i = 0$ . Prove that $$\sum_{i=1}^n \left( \sum_{j=1}^n (t_i-t_j)^2x_ix_j \right)\le 0.$$

Given is a rectangular plane coordinate system.

(a) Prove that it is impossible to find an equilateral triangle whose vertices have integer coordinates.

(b) In the plane the vertices $A, B$ and $C$ lie with integer coordinates in such a way that $AB = AC$ . Prove that $\frac{d(A,BC)}{BC}$ is rational.

Describe a method to convert any triangle into a rectangle with side 1 and area equal to the original triangle by dividing that triangle into finitely many subtriangles.