1. A triangle $A B C$ is given, right-angled at $B$. Prove that this triangle is isosceles if and only if the altitude $B H$ relative to the hypotenuse is congruent to half the hypotenuse.
2. A biased coin is tossed 5 times, giving heads with probability $p$. －What is the probability of obtaining heads exactly 2 times? －For which value of $p$ is the probability of obtaining heads exactly 2 times maximum?
3. In space with orthogonal Cartesian coordinate system $O x y z$, the plane $\pi : 3 x - 2 y + 5 = 0$ is given. －Determine the coordinates of point $H$, the orthogonal projection of $P ( 4,2,1 )$ onto the plane $\pi$; －Determine the intersection of the line $s$ : $\left\{ \begin{array} { l } x - y + 1 = 0 \\ z - 2 = 0 \end{array} \right.$ with the plane $\pi$.
5. Determine the polynomial function of fourth degree $y = p ( x )$ knowing that, in a Cartesian coordinate system, its graph satisfies the following conditions: －it is tangent to the $x$-axis at the origin; －it passes through the point $( 1,0 )$; －it has a stationary point at $( 2 , - 2 )$.
6. Consider the integral function $F ( x ) = \int _ { a } ^ { x } \frac { \cos \left( \frac { 1 } { t } \right) } { t ^ { 2 } } d t , \text{ with } x \geq a$, in which $a$ denotes a positive real parameter. Determine the largest value of $a$ so that $F \left( \frac { 2 } { \pi } \right) = - \frac { 1 } { 2 }$.
7. On July 5 next, the Earth will reach aphelion, the point of its orbit where the distance from the Sun is maximum, approximately $1.52 \cdot 10 ^ { 11 } \mathrm {~m}$. Perihelion is instead the point at minimum distance from the Sun, approximately $1.47 \cdot 10 ^ { 11 } \mathrm {~m}$. Determine, in an appropriate coordinate system, the equation that represents the Earth's trajectory around the Sun.
8. Carlo Emilio Gadda writes in one of the stories from L'Adalgisa—Milanese Sketches: ``The service rooms, the bathroom, the corridors, the antechamber and one of the two toilets were paved with small red tiles: hexagonal [. . .]. The apothem of those tiles measured 5.196 centimeters: whereas the radius of the circumscribed circle reached 60 millimeters''. Express the exact relationship between the radius of the circumscribed circle and the apothem (that is, the radius of the inscribed circle) for a regular hexagon. Verify the result obtained in light of the measurements indicated by the writer. Explain why, using regular hexagonal tiles all congruent to each other, it is possible to tile a plane. With which other regular polygons, congruent to each other, is it possible to tile a plane? Justify your answer. \footnotetext{Maximum duration of the test: 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 test. }