OME 1965

We consider an equilateral triangle with its circumscribed circle, of center $O$ , and radius $4$ cm. We rotate the triangle $90º$ around $O$ . Compute the common area that was covered by the previous position of the triangle and is also covered by the new one.

How many numbers of $3$ digits have their central digit greater than any of the other two? How many of them have also three different digits?

A disk in a record turntable makes $100$ revolutions per minute and it plays during $24$ minutes and $30$ seconds. The recorded line over the disk is a spiral with a diameter that decreases uniformly from $29$ cm to $11.5$ cm. Compute the length of the recorded line.

Find all the intervals $I$ where any element of the interval $x \in I$ satisfies $$\cos x +\sin x >1.$$ Do the same computation when $x$ satisfies $$\cos x +\vert \sin x \vert>1.$$

It is well-known that if $\frac{p}{q}=\frac{r}{s}$ , both of the expressions are also equal to $\frac{p-r}{q-s}$ . Now we write the equality $$\frac{3x-b}{3x-5b}=\frac{3a-4b}{3a-8b}.$$ The previous property shows that both fractions should be equal to $$\frac{3x-b-3a+4b}{3x-5b-3a+8b}=\frac{3x-3a+3b}{3x-3a+3b}=1.$$ However, the initial fractions given may not be equal to $1$ . Explain what is going on.

We have an empty equilateral triangle with length of a side $l$ . We put the triangle, horizontally, over a sphere of radius $r$ . Clearly, if the triangle is small enough, the triangle is held by the sphere. Which is the distance between any vertex of the triangle and the centre of the sphere (as a function of $l$ and $r$ )?