Difference between revisions of "2021 AMC 10B Problems/Problem 15"
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+ | == Video Solution by OmegaLearn (Algebraic Manipulations and Symmetric Polynomials) == | ||
+ | https://youtu.be/hzcSPVGFbC8 | ||
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+ | ~ pi_is_3.14 | ||
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{{AMC10 box|year=2021|ab=B|num-b=14|num-a=16}} | {{AMC10 box|year=2021|ab=B|num-b=14|num-a=16}} |
Revision as of 15:02, 12 February 2021
Contents
Problem
The real number satisfies the equation
. What is the value of
Solution 1
We square to get
. We subtract 2 on both sides for
and square again, and see that
so
. We can divide our original expression of
by
to get that it is equal to
. Therefore because
is 7, it is equal to
.
Solution 2
Multiplying both sides by and using the quadratic formula, we get
. We can assume that it is
, and notice that this is also a solution the equation
, i.e. we have
. Repeatedly using this on the given (you can also just note Fibonacci numbers),
~Lcz
Video Solution by OmegaLearn (Algebraic Manipulations and Symmetric Polynomials)
~ pi_is_3.14
2021 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Problem 16 | |
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All AMC 10 Problems and Solutions |