Difference between revisions of "2021 AMC 12B Problems/Problem 21"
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<math>\textbf{(A) }S<\sqrt2 \qquad \textbf{(B) }S=\sqrt2 \qquad \textbf{(C) }\sqrt2<S<2\qquad \textbf{(D) }2\le S<6 \qquad \textbf{(E) }S\ge 6</math> | <math>\textbf{(A) }S<\sqrt2 \qquad \textbf{(B) }S=\sqrt2 \qquad \textbf{(C) }\sqrt2<S<2\qquad \textbf{(D) }2\le S<6 \qquad \textbf{(E) }S\ge 6</math> | ||
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+ | Video solution by hippopotamus1: | ||
+ | https://www.youtube.com/watch?v=GjO6C_qC13U&feature=youtu.be | ||
== Solution (Rough Approximation) == | == Solution (Rough Approximation) == |
Revision as of 12:05, 13 February 2021
Contents
Problem
Let be the sum of all positive real numbers for whichWhich of the following statements is true?
Video solution by hippopotamus1:
https://www.youtube.com/watch?v=GjO6C_qC13U&feature=youtu.be
Solution (Rough Approximation)
Note that this solution is not recommended unless you're running out of time.
Upon pure observation, it is obvious that one solution to this equality is . From this, we can deduce that this equality has two solutions, since grows faster than (for greater values of ) and is greater than for and less than for , where is the second solution. Thus, the answer cannot be or . We then start plugging in numbers to roughly approximate the answer. When , , thus the answer cannot be . Then, when , . Therefore, , so the answer is . ~Baolan
Video Solution by OmegaLearn (Logarithmic Tricks)
~ pi_is_3.14
See Also
2021 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 20 |
Followed by Problem 22 |
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.