\[ \mathrm{x}+\mathrm{y}=6 \text {, } \] It is known that the solutions are integers. \( \mathrm{x}^{y}=\mathrm{y}^{\mathrm{x}} \) \( \quad \) a. \( \{(1,5),(5,1)\} \) b. \( \{(5,5),(-10,16),(16,-10)\} \) c. \( \{(3,3),(4,2),(2,4)\} \) d. \( \{ \) all integers divisible by 3 and 2\( \} \) e. None of the above

The ring \( \mathbb{Z} \) has no nontrivial idempotent elements, whereas \( \mathbb{Z} \times \mathbb{Z} \) has the idernpotent elements \( (1,0) \) and \( (0,1) \).

Describe the steps involved in finding the inverse of a function algebraically.

Bonus Question It is given that the general term \( T_{n} \) of a sequence is \( (-1)^{n} \times 2 n \). Find the value of \( T_{50}-T_{49}-T_{48}-\cdots-T_{2}-T_{1} \).

\( ( a + b ) ( a - b ) = a ^ { r } - b ^ { r } \)

596 Per quale valore di \( n \in \mathrm{~N} \) il sistema: \( \left\{\begin{array}{l}x^{3} y^{2}+x^{2} y=1 \\ x^{n+1} y^{n}+x^{2} y^{2 n+2}=3\end{array}\right. \) ha grado uguale a \( 100 ? \)