Someone from the LogMeIn support gave me the solution. Here it is:
Why does my internet connection fail when Hamachi is enabled?
After installing Hamachi you may encounter connectivity issues due to connection Metrics. Windows uses Metrics to determine the order it will attempt to acquire internet access. The Hamachi adapter may be auto assigned a lower Metric than the adapter that provides your internet connection. You will need to manually assign the Metric to resolve.
To manually assign a Metric:
Windows XP- Open the Control Panel.
- Go into Network Connections.
- Right Click on the Hamachi adapter, and click Properties.
- Highlight Internet Protocol (TCP/IP), and click the Properties button.
In a 64bit OS, it will be listed as Internet Protocol Version 4 (TCP/IPv4). - Now choose the Advanced button.
- At the bottom of this page, uncheck the box labeled Automatic Metric, and enter 9000 as the Interface Metric.
- Open the Control Panel.
- Go into Network and Sharing Center.
- In the left pane, click Manage network connections.
- Right Click on the Hamachi adapter, and click Properties.
- Highlight Internet Protocol (TCP/IP), and click the Properties button.
In a 64 bit OS, it will be listed as Internet Protocol Version 4 (TCP/IPv4). - Now choose the Advanced button.
- At the bottom of this page, uncheck the box labeled Automatic Metric, and enter 9000 as the Interface Metric.
- Open the Control Panel.
- Click into Network and Internet > Network and Sharing Center.
- Click Change adapter settings on the left pane.
- Right-click on the Hamachi adapter, and click Properties.
- Click Internet Protocol (TCP/IP) to highlight it, and click Properties.
In a 64-bit OS, it is listed as Internet Protocol Version 4 (TCP/IPv4). - Click Advanced.
- At the bottom of this page, uncheck the Automatic metric option, and enter 9000 into the Interface metricfield.
Performing the above steps will assign Hamachi a very high metric, making it the very last thing Windows use to attempt an internet connection. If that does not resolve it, perform the same steps on the adapter you want your internet connection to be on, but assign it a 100 metric.