5. In the office, each computer was connected by wires to exactly 5 other computers. After some computers were infected by a virus, all wires from the infected computers were disconnected (a total of 26 wires were disconnected). Now, each of the uninfected computers is connected by wires to only 3 others. How many computers were infected by the virus?
# 5. Answer: 8.

Let $\mathrm{m}$ be the number of infected computers, and $\mathrm{n}$ be the number of uninfected computers. Then, before the infection, there were $5(\mathrm{~m}+\mathrm{n}) / 2$ cables, and after the disconnection, there were $3 \mathrm{n} / 2$ cables (from which it follows that $\mathrm{n}$ is even). The difference between these numbers is 26, leading to the equation $5 \mathrm{~m}+2 \mathrm{n}=52$. This equation has two solutions in natural numbers where $\mathrm{n}$ is even (this can be proven by enumeration): $\mathrm{m}=4, \mathrm{n}=16$ and $\mathrm{m}=8, \mathrm{n}=6$. The first solution does not work: even if all infected computers were connected only to healthy ones, a maximum of $4 \cdot 5=20$ cables would have to be disconnected, not 26.