OME 1966

To a manufacturer of three products whose unit prices are $50$ , $70$ , and $65$ pta, a retailer asks him for $100$ units, remitting him $6850$ pta as payment, on the condition that you send as many of the higher-priced product as possible and the rest of the other two. How many of each product should he send to serve the request?

A three-digit number is written $xyz$ in the base $7$ system and $zyx$ in the base $9$ system . What is the number?

Given a regular pentagon, consider the convex pentagon limited by its diagonals. You are asked to calculate: a) The similarity relation between the two convex pentagons. b) The relationship of their areas. c) The ratio of the homothety that transforms the first into the second.

You want to hang a weight $P$ so that it is $7$ m below a ceiling. To do this, it is suspended by means of a vertical cable attached to the midpoint $M$ of a chain hung by its ends from two points on the ceiling $A$ and $B$ distant from each other $4$ m. The price of the cable $PM$ is $p$ pta/m and that of the chain $AMB$ is $q$ pta/m. It is requested: a) Determine the lengths of the cable and the chain to obtain the lowest price cost of installation. b) Discuss the solution for the different values of the relation $p/q$ of both prices. (It is assumed that the weight is large enough to be considered rectile lines the chain segments $AM$ and $MB$ ).

The length of the hypotenuse $BC$ of a right triangle $ABC$ is $a$ , and on it the points $M$ and $N$ are taken such that $BM = NC = k$ , with $k < a/2$ . Assuming that (only) the data $a$ and $k$ are known, calculate: a) The value of the sum of the squares of the lengths $AM$ and $AN$ . b) The ratio of the areas of triangles $ABC$ and $AMN$ . c) The area enclosed by the circle that passes through the points $A, M' , N'$ , where $M'$ is the orthogonal projection of $M$ onto $AC$ and $N'$ that of $N$ onto $AB$ .

They tell us that a married couple has $5$ children. Calculate the probability that among them there are at least two men and at least one woman. Probability of being born male is considered $1/2$ .