Difference between revisions of "2010 AMC 10A Problems/Problem 25"
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Revision as of 22:12, 3 August 2012
Problem
Jim starts with a positive integer and creates a sequence of numbers. Each successive number is obtained by subtracting the largest possible integer square less than or equal to the current number until zero is reached. For example, if Jim starts with
, then his sequence contains
numbers:
Let be the smallest number for which Jim’s sequence has
numbers. What is the units digit of
?
Solution
We can find the answer by working backwards. We begin with on the bottom row, then the
goes to the right of the equal's sign in the row above. We find the smallest value
for which
and
, which is
.
We repeat the same procedure except with for the next row and
for the row after that. However, at the fourth row, we see that solving
yields
, in which case it would be incorrect since
is not the greatest perfect square less than or equal to
. So we make it a
and solve
. We continue on using this same method where we increase the perfect square until
can be made bigger than it. When we repeat this until we have
rows, we get:
Hence the solution is the last digit of , which is
.
See also
2010 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
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 |