4. Find all functions $\mathrm{f}$ defined on the set of real numbers and taking real values, such that $2 \mathrm{f}(\mathrm{x})+2 \mathrm{f}(\mathrm{y})-\mathrm{f}(\mathrm{x}) \mathrm{f}(\mathrm{y}) \geq 4$ for all real $\mathrm{x}, \mathrm{y}$.
Answer: $\mathrm{f}(\mathrm{x})=2$ (a constant function, all values of which are equal to 2).

Hint. By setting $\mathrm{y}=\mathrm{x}$ in the inequality and simplifying, we get $\mathrm{f}^{2}(\mathrm{x})-4 \mathrm{f}(\mathrm{x})+4 \leq 0$ or $(\mathrm{f}(\mathrm{x})-2)^{2} \leq 0$. From this, we have that $\mathrm{f}(\mathrm{x})=2$. By verification, we confirm that this function fits.

Criteria. If there is no verification, one point will be deducted.

5. The three-segment broken line ABCD is such that the angles between adjacent segments are not less than $120^{\circ}$. Prove that the length of the broken line is less than the sum of the lengths of segments AC and DB.

Hints. Let $\mathrm{AB}=\mathrm{a}, \mathrm{BC}=\mathrm{b}, \mathrm{CD}=\mathrm{c}, \mathrm{AC}=\mathrm{m}, \mathrm{BD}=\mathrm{n}, \angle \mathrm{ABC}=\beta \geq 120^{\circ}$, $\angle \mathrm{BCD}=\gamma \geq 120^{\circ}$. By the cosine theorem for triangle ABC, we have $m^{2}=a^{2}+b^{2}-2 a b \cos \beta \geq a^{2}+b^{2}-2 a b \cos 120^{\circ}=a^{2}+b^{2}+a b>(a+b / 2)^{2}$. Therefore, $m>a+b / 2$. Similarly, it can be shown that $n>c+b / 2$. Adding these inequalities, we get $n+m>a+b+c$.

Criteria. Considering special cases: 0 points.