2021 AMC 12B Problems/Problem 21
Contents
Problem
Let be the sum of all positive real numbers for whichWhich of the following statements is true?
Solution 1
Note that (At this point we see by inspection that is a solution.)
We simplify the RHS, then take the base- logarithm for both sides: The RHS is a line; the LHS is a concave curve that looks like a logarithm and has intercept at
There are at most two solutions, one of which is But note that at we have meaning that the log log curve is above the line, so it must intersect the line again at a point Now we check and see that which means at the line is already above the log log curve. Thus, the second solution lies in the interval The answer is
~ccx09
Solution 2
We rewrite the right side without using square roots, then take the base- logarithm for both sides: By observations, is one solution. Graphing and we conclude that has two solutions, with the smaller solution We construct the following table of values: Let be the larger solution. Since exponential functions outgrow logarithmic functions, we have for all By the Intermediate Value Theorem, we get from which Finally, approximating with results in
The graphs of and are shown below: ~MRENTHUSIASM
Video Solution by OmegaLearn (Logarithmic Tricks)
~ pi_is_3.14
Video Solution by hippopotamus1
https://www.youtube.com/watch?v=GjO6C_qC13U&feature=youtu.be
Video Solution by The Power of Logic
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.