2018 AMC 12B Problems/Problem 9
Problem
What is
Solution 1
We can start by writing out the first couple of terms:
Looking at the first terms in the parentheses, we can see that
occurs
times. It goes vertically and exists
times horizontally.
Looking at the second terms in the parentheses, we can see that
occurs
times. It goes horizontally and exists
times vertically.
Thus, we have
Solution 2
Recall that the sum of the first positive integers is
It follows that
~Vfire ~MRENTHUSIASM
Solution 3
Recall that the sum of the first positive integers is
Since the nested summation is symmetric with respect to
and
it follows that
~RandomPieKevin ~MRENTHUSIASM
Solution 4
The sum contains terms, and the average value of both
and
is
Therefore, the sum becomes
~Rejas ~MRENTHUSIASM
Solution 5
When we expand the nested summation as shown in Solution 1, note that:
- The term
appears
time.
The term
appears
times.
The term
appears
times.
The term
appears
times.
More generally, the term
appears
times for
- The term
appears
times.
The term
appears
times.
The term
appears
times.
The term
appears
time.
More generally, the term
appears
times for
Together, the nested summation becomes
~MRENTHUSIASM
See Also
2018 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 8 |
Followed by Problem 10 |
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 12 Problems and Solutions |
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