OME 1970

A cylindrical container of revolution is partially filled with a liquid whose density we ignore. Placing it with the axis inclined $30^o$ with respect to the vertical, we observe that when removing liquid so that the level falls $1$ cm, the weight of the contents decreases $40$ g. How much will the weight of that content decrease for each centimeter that lower the level if the axis makes an angle of $45^o$ with the vertical? It is supposed that the horizontal surface of the liquid does not touch any of the bases of the container.

A plant grows in the way we describe below. has a trunk which forks into two branches; each branch of the plant can, in turn, branch off into other two branches, or end in a bud. We will call the load of a branch the total number of buds it bears, that is, the number of buds fed by the sap that passes by that branch; and we will call the distance of a bud the number of bifurcations that it sap has to go through to get from the trunk to that bud. If n is the number of bifurcations that a certain plant of that type has, it is asks a) the number of branches of the plant, b) the number of buds, c) show that the sum of the charges of all the branches is equal to the sum of the clearances of all buds.

An arbitrary triangle $ABC$ is given and a point $P$ lies on the side $AB$ . It is requested to draw through $P$ a line that divides the triangle into two figures of the same area.

Knowing that the polynomials $$2x^5 - 13x^4 + 4x^3 + 61x^2 + 20x-25$$ $$x^5 -4x^4 - 13x^3 + 28x^2 + 85x+50$$ have two common double roots, determine all their roots.

In the sixth-year exams of a Center, they pass Physics at least $70\%$ of the students, Mathematics at least $75\%$ ; Philosophy at least, the $90\%$ and the Language at least, $85\%$ . How many students, at least, pass these four subjects?

Given a circle $\gamma$ and two points $A$ and $B$ in its plane. By $B$ passes a variable secant that intersects $\gamma$ at two points $M$ and $N$ . Determine the locus of the centers of the circles circumscribed to the triangle $AMN$ .