OME 1971

Calculate $$\sum_{k=5}^{k=49}\frac{11_(k}{2\sqrt[3]{1331_(k}}$$ knowing that the numbers $11$ and $1331$ are written in base $k \ge 4$ .

In a certain geometry we operate with two types of elements, points and lines, related to each other by the following axioms:I. Given two points $A$ and $B$ , there is a unique line $(AB)$ that passes through both.**II.**There are at least two points on a line. There are three points not situated on a straight line.III. When a point $B$ is located between $A$ and $C$ , then $B$ is also between $C$ and $A$ . ( $A, B, C$ are three different points on a line.)IV. Given two points $A$ and $C$ , there exists at least one point $B$ on the line $(AC)$ of the form that C is between $A$ and $B$ .V. Among three points located on the same straight line, one at most is between the other two.VI. If $A, B, C$ are three points not lying on the same line and a is a line that does not contain any of the three, when the line passes through a point on segment [AB] , then it goes through one of the $[BC]$ , or it goes through one of the [AC] . (We designate by [AB] the set of points that lie between $A$ and $B$ .)

From the previous axioms, prove the following propositions: Theorem 1. Between points A and C there is at least one point $B$ . Theorem 2. Among three points located on a line, one is always between the two others.

If $0 < p$ , $0 < q$ and $p +q < 1$ prove $$(px + qy)^2 \le px^2 + qy^2$$

Prove that in every triangle with sides $a, b, c$ and opposite angles $A, B, C$ , is fulfilled (measuring the angles in radians) $$\frac{a A+bB+cC}{a+b+c} \ge \frac{\pi}{3}$$

Prove that whatever the complex number $z$ is, it is true that $$(1 + z^{2^n})(1-z^{2^n})= 1- z^{2^{n+1}}.$$ Writing the equalities that result from giving $n$ the values $0, 1, 2, . . .$ and multiplying them, show that for $|z| < 1$ holds $$\frac{1}{1-z}= \lim_{k\to \infty}(1 + z)(1 + z^2)(1 + z^{2^2})...(1 + z^{2^k}).$$

The velocities of a submerged and surfaced submarine are, respectively, $v$ and $kv$ . It is situated at a point $P$ at $30$ miles from the center $O$ of a circle of $60$ mile radius. The surveillance of an enemy squadron forces him to navigate submerged while inside the circle. Discuss, according to the values of $k$ , the fastest path to move to the opposite end of the diameter that passes through $P$ . (Consider the case particular $k =\sqrt5$ .)