Smart Arithmetic
Often appearing at the beginning of AMC 8 /10/ 12, these problems are not hard to tackle using brute force approach, but almost always have a nice "shortcut" method that is faster, simpler, and more elegant.
Some of the most frequently used tools are:
Number Sense
Tips: be on the lookout for numbers that make easy calculations (8 * 125, 4 * 25, etc.)
Associative/Communicative property
Tips: which numbers to work on first makes a big difference
Distributive property
The principle of (for the lack of a better word) simplifying before computing
Tips: refrain from jumping into computation right away. Check if any simlifying can be done to make your life easier