2021 AMC 10B Problems/Problem 15
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
Solution 3
We can immediately note that the exponents of are an arithmetic sequence, so they are symmetric around the middle term. So,
. We can see that since
,
and therefore
. Continuing from here, we get
, so
. We don't even need to find what
is! This is since
is evidently
, which is our answer.
~sosiaops
Video Solution by OmegaLearn (Algebraic Manipulations and Symmetric Polynomials)
~ pi_is_3.14
2021 AMC 10B (Problems • Answer Key • Resources) | ||
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