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New, exact closed-form expressions are derived for the probability density function and the cumulative distribution function of the end-to-end signal-to-noise ratio (SNR) of opportunistic dual-hop amplify-and-forward (AF) relaying systems with relay selection. The expressions are used to obtain the first exact integral solutions for the ergodic capacity and average symbol error probability, and the first exact closed-form solution for outage probability of an opportunistic AF relaying system where the best node is selected from a number of candidate intermediate nodes to relay the data signal between the source and the destination. The selection follows a maximum end-to-end SNR policy, based on the available channel state information. The results are precise for any number of candidate relays and Rayleigh, Nakagami-m or Rician fading distributions. The effects of different channel fading parameters and the number of relays in the relay selection pool are studied. The system performance is compared to that of dual-hop AF systems without relay selection and to dual-hop AF relaying systems with maximum relay-to-destination SNR relay selection. The adopted selection method provides diversity gain over dual-hop AF relaying systems without relay selection and over maximum relay-to-destination SNR relay selection. The diversity gain is proportional to the relay selection pool size.