1. Given a triangle $A B C$, let $M$ be the midpoint of side $B C$ and let $B ^ { \prime }$ and $C ^ { \prime }$ be two points, respectively, on side $A B$ and on side $A C$, such that $A B ^ { \prime } = \frac { 1 } { 3 } A B$ and $A C ^ { \prime } = \frac { 1 } { 3 } A C$. Prove that, if the segments $M B ^ { \prime }$ and $M C ^ { \prime }$ are congruent to each other, then so are the sides $A B$ and $A C$.

2. Consider the spherical surface with equation $( x - 1 ) ^ { 2 } + ( y - 2 ) ^ { 2 } + z ^ { 2 } = 1$ and the plane $\pi$ with equation $x - 2 y - 2 z + d = 0$. Discuss, as the real parameter $d$ varies, whether the plane $\pi$ is secant, tangent or external to the spherical surface. Determine the value of the parameter $d$ so that $\pi$ divides the sphere into two equal parts.

3. Boccioni's futurist work ``Unique Forms of Continuity in Space'' from 1913, featured on the 20-cent coin, depicts a man advancing rapidly through space. A part of the profile highlighted in the figure, in an appropriate coordinate system, can be approximated by the function
$$f ( x ) = \left\{ \begin{array} { l r } - 4 x ^ { 2 } - 8 x , & - 1 \leq x \leq 0 \\ 1 + \tan \left( x + \frac { 3 } { 4 } \pi \right) , & 0 < x \leq 2 \end{array} \right.$$
Sketch the graph of $f$, after analyzing its continuity and differentiability on the interval $[ - 1 ; 2 ]$. [Figure]

4. Given a function $g$, differentiable on $\mathbb { R }$ and such that $g \left( \frac { \pi } { 4 } \right) = g ^ { \prime } \left( \frac { \pi } { 4 } \right) = 2$, determine the equation of the normal line to the curve $y = g ( x ) \sin ^ { 2 } x$ at its point with abscissa $\frac { \pi } { 4 }$.

5. Determine the value of the real parameter $k$ so that the two curves $y = e ^ { x }$, $y = 6 - k e ^ { - x }$ are tangent to each other, finding the coordinates of the point of tangency.

6. Write a polynomial function $f$ such that the line with equation $y = 2 x + 3$ is tangent to the graph of $f$ at its point with abscissa 0 and $\int _ { 0 } { 3 } f ( x ) d x = 9$.

Ministry of Education and Merit

A002－FINAL STATE EXAMINATION OF THE SECOND CYCLE OF EDUCATION

8. How many anagrams, even without meaning, are there of the word ``STUDIARE''? In how many of these anagrams can the word ``ARTE'' be read consecutively, as for example in ``SUARTEDI''?
How many anagrams, even without meaning, are there of the word ``VACANZA''? ``Mathematics knows no races or geographical boundaries; for mathematics, the cultural world is a single nation''
\footnotetext{Maximum duration of the exam: 6 hours. The use of scientific or graphical calculators is permitted provided they are not equipped with symbolic algebraic processing capability and do not have Internet connectivity. The use of a bilingual dictionary (Italian–language of the country of origin) is permitted for candidates whose native language is not Italian. It is not permitted to leave the Institute before 3 hours have elapsed from the distribution of the exam. }