OME 1973

Given the sequence $(a_n)$ , in which $a_n =\frac14 n^4 - 10n^2(n - 1)$ , with $n = 0, 1, 2,...$ Determine the smallest term of the sequence.

Determine all solutions of the system $$\begin{cases} 2x - 5y + 11z - 6 = 0 -x + 3y - 16z + 8 = 0 4x - 5y - 83z + 38 = 0 3x + 11y - z + 9 > 0 \end{cases}$$

in which the first three are equations and the last one is a linear inequality.

The sequence $(a_n)$ of complex numbers is considered in the complex plane, in which is: $$a_0 = 1, \,\, a_n = a_{n-1} +\frac{1}{n}(\cos 45^o + i \sin 45^o )^n.$$ Prove that the sequence of the real parts of the terms of $(a_n)$ is convergent and its limit is a number between $0.85$ and $1.15$ .

Let $C$ and $C'$ be two concentric circles of radii $r$ and $r'$ respectively. Determine how much the quotient $r'/r$ must be worth so that in the limited crown (annulus) through $C$ and $C'$ there are eight circles $C_i$ , $i = 1, . . . , 8$ , which are tangent to $C$ and to $C'$ , and also that $C_i$ is tangent to $C_{i+1}$ for $i = 1, . . . ,7$ and $C_8$ tangent to $C_1$ .

Consider the set of all polynomials of degree less than or equal to $4$ with rational coefficients. a) Prove that it has a vector space structure over the field of numbers rational. b) Prove that the polynomials $1, x - 2, (x -2)^2, (x - 2)^3$ and $(x -2)^4$ form a base of this space. c) Express the polynomial $7 + 2x - 45x^2 + 3x^4$ in the previous base.

An equilateral triangle of altitude $1$ is considered. For every point $P$ on the interior of the triangle, denote by $x, y , z$ the distances from the point $P$ to the sides of the triangle. a) Prove that for every point $P$ inside the triangle it is true that $x + y + z = 1$ . b) For which points of the triangle does it hold that the distance to one side is greater than the sum of the distances to the other two? c) We have a bar of length $1$ and we break it into three pieces. find the probability that with these pieces a triangle can be formed.