OME 1968

In one night the air temperature remained constant, several degrees below zero, and that of the water of a very extensive cylindrical pond, which formed a layer $10$ cm deep, it reached zero degrees, beginning then to form a layer of ice on the surface. Under these conditions it can be assumed that the thickness of the ice sheet formed is directly proportional to the square root of the time elapsed. At $0$ h, the thickness of the ice was $3$ cm and at $4$ h it was just over to freeze the water in the pond. Calculate at what time the ice sheet began to form, knowing that the density of the ice formed was $0.9$ .

Justify if continuity can be affirmed, denied or cannot be decided in the point $x = 0$ of a real function $f(x)$ of real variable, in each of the three (independent) cases .

a) It is known only that for all natural $n$ : $f\left( \frac{1}{2n}\right)= 1$ and $f\left( \frac{1}{2n+1}\right)= -1$ .

b) It is known that for all nonnegative real $x$ is $f(x) = x^2$ and for negative real $x$ is $f(x) = 0$ .

c) It is only known that for all natural $n$ it is $f\left( \frac{1}{n}\right)= 1$ .

Given a square whose side measures $a$ , consider the set of all points of its plane through which passes a circumference of radius whose circle contains to the quoted square. You are asked to prove that the contour of the figure formed by the points with this property is formed by arcs of circumference, and determine the positions, their centers, their radii and their lengths.

At the two ends $A, B$ of a diameter (of length $2r$ ) of a pavement horizontal circular rise two vertical columns, of equal height h, whose ends support a beam $A' B'$ of length equal to the before mentioned diameter. It forms a covered by placing numerous taut cables (which are admitted to be rectilinear), joining points of the beam $A'B'$ with points of the circumference edge of the pavement, so that the cables are perpendicular to the beam $A'B'$ . You want to find out the volume enclosed between the roof and the pavement.

Find the locus of the center of a rectangle, whose four vertices lies on the sides of a given triangle.

Check and justify , if in every tetrahedron are concurrent: a) The perpendiculars to the faces at their circumcenters. b) The perpendiculars to the faces at their orthocenters. c) The perpendiculars to the faces at their incenters. If so, characterize with some simple geometric property the point in that attend If not, show an example that clearly shows the not concurrency.