2005 AMC 10A Problems/Problem 11
Contents
Problem
A wooden cube units on a side is painted red on all six faces and then cut into
unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is
?
Solution 1
Since there are little faces on each face of the big wooden cube, there are a total of
little faces painted red. Moreover, as each unit cube has
faces, there are
little faces in total.
Accordingly, as one-fourth of the little faces are painted red, we have
Solution 2
We recall that when a cube of side length has its entire surface painted and is then split into
unit cubes, there will be exactly
unit cubes with
faces painted,
unit cubes with
face painted,
unit cubes with
faces painted, and
unit cubes with
faces painted.
(Observe that these form the terms of the binomial expansion of .)
Hence the total number of faces painted red is and now we may proceed as in Solution 1, again giving
See also
2005 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions.