Q: airplay doesn't work

See if you meet the requirements:

Requirements for AirPlay Mirroring in OS X Mountain Lion

An Apple TV

• Apple TV (2nd or 3rd generation) with software update v5.0 or later. See How to update Apple TV software.

Supported Macs:  AirPlay Mirroring in OS X Mountain Lion takes advantage of the hardware video encoding capabilities of 2nd and 3rd Generation Intel Core processors. Hardware video encoding allows AirPlay Mirroring to efficiently deliver high frame rates while maintaining optimal system performance. The following Mac models have processors that support AirPlay Mirroring:

• iMac (Mid 2011 or newer)
• Mac mini (Mid 2011 or newer)
• MacBook Air (Mid 2011 or newer)
• MacBook Pro (Early 2011 or newer)

Hope this helps