2002 AMC 12P Problems/Problem 13
Problem
What is the maximum value of for which there is a set of distinct positive integers
for which
Solution
Note that
When ,
.
When ,
.
Therefore, we know .
Now we must show that works. We replace some integer
within the set $\{1, 2, ... 17}$ with an integer$ (Error compiling LaTeX. Unknown error_msg)a > 17
2002
2002-1785 = 217$.
Essentially, this boils down to writing$ (Error compiling LaTeX. Unknown error_msg)217a
b
a > 17
b \leq 17
a^2 - b^2 = 217$.
We can rewrite this as$ (Error compiling LaTeX. Unknown error_msg)(a+b)(a-b) = 217217 = (7)(31)
a+b = 217
a-b = 1
a+b = 31
a-b = 7$. We analyze each case separately.
Case 1:$ (Error compiling LaTeX. Unknown error_msg)a+b = 217a-b = 1
a = 109
b = 108
108 > 17$, so this case does not yield a solution.
Case 2:$ (Error compiling LaTeX. Unknown error_msg)a+b = 31a-b = 7
a = 19
b = 12$. This satisfies all the requirements of the problem.
The list$ (Error compiling LaTeX. Unknown error_msg)1, 2 ... 11, 13, 14 ... 17, 1917
2002
\boxed {\text{(D) }17}$.
See also
2002 AMC 12P (Problems • Answer Key • Resources) | |
Preceded by Problem 12 |
Followed by Problem 14 |
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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