Miscellaneous Tricks
On this page I have compiled some tricks that can help you solve a problem fast, and some traps that you should watch out for.
The WLOG Trick
Don't we all love the sound of "Without loss of generality..."! it can give us a special, usually easy case to solve, without worrying too much about overcomplicated cases.
For example, When you have percentage involved, such as Problem 19 and Problem 21 of the 1985 AJHSME, use WLOG and set up some easy number can quickly give you the result you want. Here is a list problems that you COULD use WLOG to solve quickly (but I hope you do know how to solve it properly using algebra!):
Watch Out for the OBOE
OBOE (Off By One Error) happens a lot when counting the number of terms in a sequence. Be mindful whenever you run into those problems, and do not let your guard down!
1992 AJHSME Problem 19
The "Squeezing"
When there seems to be more than one answer (or when there seems to be a range of valid answers), finding the minimum (lower bound) and maximum (upper bound) might give us a better view of the nature of the problem
1987 AJHSME: Problem 8 Problem 11
1989 AJHSME: Problem 14 Problem 17
1990 AJHSME: Problem 1
1992 AJHSME: Problem 19
1994 AJHSME: Problem 23
The Grid
This trick only applies to the simplest type of counting / probability problems. When there are two variables, each with limited cases, sometimes it is actually faster to just make a table, with one variable as the row and the other as the column, and simply enumerate the results one by one. The "spinning wheel" probability problems are the best examples here.
1989 AJHSME Problem 25
1991 AJHSME Problem 14 The total number of possible combinations (27) might be intimidating, but notice we need a number large enough to beat everyone else, therefore, we can start enumerating from 5, 5, 5, and it is clear that at 5, 3, 3, the sum is no longer guaranteed to be the largest.
1991 AJHSME Problem 22
1994 AJHSME Problem 22