2020 AMC 12B Problems/Problem 6
Problem
For all integers the value of
is always which of the following?
Solution 1
We first expand the expression:
We can now divide out a common factor of from each term of this expression:
Factoring out , we get
which proves that the answer is .
Solution 2
Factor out an to get:
Now, without loss of generality, test values of
until only one answer choice is left valid:
, knocking out
,
, and
.
, knocking out
.
This leaves as the only answer choice left.
With further testing it becomes clear that for all ,
, proved in Solution 1.
~DBlack2021
Video Solution
https://youtu.be/ba6w1OhXqOQ?t=2234
~ pi_is_3.14
Video Solution
~IceMatrix
See Also
2020 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 5 |
Followed by Problem 7 |
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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