Difference between revisions of "2020 AMC 12B Problems/Problem 6"
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MRENTHUSIASM (talk | contribs) (→Solution 2: The original solution violates the condition that n>=9. I am adding explanation why violating it is OK.) |
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This leaves <math>\boxed{\textbf{(D)} \text{ a perfect square}}</math> as the only answer choice left. | This leaves <math>\boxed{\textbf{(D)} \text{ a perfect square}}</math> as the only answer choice left. | ||
− | + | This solution does not consider the condition <math>n \geq 9.</math> The reason is that, with further testing it becomes clear that for all <math>n</math>, <math>(n+2)(n+1)-(n+1) = (n+1)^{2}</math>, proved in Solution 1. We have now revealed that the condition <math>n \geq 9</math> is insignificant. | |
− | ~DBlack2021 | + | ~DBlack2021 (Solution Writing) |
+ | |||
+ | ~MRENTHUSIASM (Edits in Logic) | ||
== Video Solution == | == Video Solution == |
Revision as of 01:31, 23 May 2021
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.
This solution does not consider the condition The reason is that, with further testing it becomes clear that for all , , proved in Solution 1. We have now revealed that the condition is insignificant.
~DBlack2021 (Solution Writing)
~MRENTHUSIASM (Edits in Logic)
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.