1988 AIME Problems

Problem 1

One commercially available ten-button lock may be opened by depressing -- in any order -- the correct five buttons. The sample shown below has as its combination. Suppose that these locks are redesigned so that sets of as many as nine buttons or as few as one button could serve as combinations. How many additional combinations would this allow?

Solution

Problem 2

For any positive integer , let denote the square of the sum of the digits of . For , let . Find .

Solution

Problem 3

Solution

Problem 4

Solution

Problem 5

Solution

Problem 6

Solution

Problem 7

Solution

Problem 8

Solution

Problem 9

Find the smallest positive integer whose cube ends in 888.

Solution

Problem 10

Solution

Problem 11

Solution

Problem 12

Solution

Problem 13

Solution

Problem 14

Solution

Problem 15

Solution

See also

• American Invitational Mathematics Examination
• AIME Problems and Solutions
• Mathematics competition resources