Difference between revisions of "PaperMath’s sum"
m (→Proof) |
m (→Proof) |
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Observing that | Observing that | ||
+ | <math>\sum_{i=0}^{n-1} {10^i} = | ||
+ | (10^{n}-1)/9</math> | ||
+ | and | ||
<math>(10^{2n}-1)/9 = 9((10^{n}-1)/9)^2 + 2(10^n -1)/9</math> | <math>(10^{2n}-1)/9 = 9((10^{n}-1)/9)^2 + 2(10^n -1)/9</math> | ||
concludes the proof. | concludes the proof. |
Revision as of 21:32, 1 September 2024
Contents
PaperMath’s sum
Papermath’s sum states,
Or
For all real values of , this equation holds true for all nonnegative values of . When , this reduces to
Proof
First, note that the part is trivial multiplication, associativity, commutativity, and distributivity over addition,
Observing that and concludes the proof.
Problems
AMC 12A Problem 25
For a positive integer and nonzero digits , , and , let be the -digit integer each of whose digits is equal to ; let be the -digit integer each of whose digits is equal to , and let be the -digit (not -digit) integer each of whose digits is equal to . What is the greatest possible value of for which there are at least two values of such that ?
Notes
Papermath’s sum was named by the aops user Papermath. The name is not widely used.