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These have all been created by a variety of well-known constructors. There may be one or two that we might question, but they're still fun. Please explain all answers.
-1. Tempt with selection of authentic emeralds (6)
-2. Creep has a second way to pay (2,4)
-3. Conservative Party turned crafty (6)
-4. Divine school of math (6)
-5. Hammers small footholds on a cliff (7)
-6. Contrary boy told fibs and wasted time (7)
-7. Filled with dread, chicken goes a little overboard? (5,3)
-8. Set one's cap for 500 - and get 500 - for orchestra member (8)
-9. Prepare awful tragedy, taking ending from Shakespeare (3,5)
-10. Advertising slogan of Cat Chow improved by R&D (9)
-11. Pass into storms and white water (6)
-12. Good sound quality provided in spoken greeting (2-2)
-13. Spare devil admitted into the heavens (6)
-14. After losing opener, pitcher pitches a few times (6)
-15. Overhear where to find wine merchant (6)
-16. Extensive hunger (4)