OME 1964

Given the equation $x^2+ax+1=0$ , determine:

a) The interval of possible values for $a$ where the solutions to the previous equation are not real. b) The loci of the roots of the polynomial, when $a$ is in the previous interval.

The RTP tax is a function $f(x)$ , where $x$ is the total of the annual profits (in pesetas). Knowing that:

a) $f(x)$ is a continuous function b) The derivative $\frac{df(x)}{dx}$ on the interval $0 \leq 6000$ is constant and equals zero; in the interval $6000< x < P$ is constant and equals $1$ ; and when $x>P$ is constant and equal 0.14. c) $f(0)=0$ and $f(140000)=14000$ .

Determine the value of the amount $P$ (in pesetas) and represent graphically the function $y=f(x)$ .

A convex polygon of $n$ sides is considered. All its diagonals are drawn and we suppose that any three of them can only intersect on a vertex and that there is no pair of parallel diagonals. Under these conditions, we wish to compute

a) The total number of intersection points of these diagonals, excluding the vertices.

b) How many points, of these intersections, lie inside the polygon and how many lie outside.

We are given an equilateral triangle $ABC$ , of side $a$ , inside its circumscribed circle. We consider the smallest of the two portions of circle limited between $AB$ and the circumference. If we consider parallel lines to $BC$ , some of them cut the portion of circle in a segment. Which is the maximum possible length for one of the segments?

Given a regular pentagon, its five diagonals are drawn. How many triangles do appear in the figure? Classify the set of triangles in classes of equal triangles.

Make a graphical representation of the function $y=\vert \vert \vert x-1 \vert -2 \vert -3 \vert$ on the interval $-8 \leq x \leq 8$ .