Difference between revisions of "PaperMath’s sum"
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− | == | + | PapreMath's sum is a thereom discovered by the APS user Papermath on October 8th, 2023. |
− | + | ||
+ | == Statement == | ||
+ | '''PaperMath’s sum''' states, | ||
<math>\sum_{i=0}^{2n-1} {(10^ix^2)}=(\sum_{j=0}^{n-1}{(10^j3x)})^2 + \sum_{k=0}^{n-1} {(10^k2x^2)}</math> | <math>\sum_{i=0}^{2n-1} {(10^ix^2)}=(\sum_{j=0}^{n-1}{(10^j3x)})^2 + \sum_{k=0}^{n-1} {(10^k2x^2)}</math> | ||
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<math>\sum_{i=0}^{2n-1} {10^i}=(\sum_{j=0}^{n -1}{(3 \times 10^j)})^2 + \sum_{k=0}^{n-1} {(2 \times 10^k)}</math> | <math>\sum_{i=0}^{2n-1} {10^i}=(\sum_{j=0}^{n -1}{(3 \times 10^j)})^2 + \sum_{k=0}^{n-1} {(2 \times 10^k)}</math> | ||
− | ==Proof== | + | == Proof == |
First, note that the <math>x^2</math> part is trivial multiplication, associativity, commutativity, and distributivity over addition, | First, note that the <math>x^2</math> part is trivial multiplication, associativity, commutativity, and distributivity over addition, | ||
Observing that | Observing that | ||
− | <math>\sum_{i=0}^{n-1} {10^i} = | + | <math>\sum_{i=0}^{n-1} {10^i} = \frac{10^{n}-1}{9}</math> and <math>(10^{2n}-1)/9 = 9((10^{n}-1)/9)^2 + 2(10^n -1)/9</math> concludes the proof. |
− | |||
− | and | ||
− | <math>(10^{2n}-1)/9 = 9((10^{n}-1)/9)^2 + 2(10^n -1)/9</math> | ||
− | concludes the proof. | ||
− | ==Problems== | + | == Problems == |
− | |||
For a positive integer <math>n</math> and nonzero digits <math>a</math>, <math>b</math>, and <math>c</math>, let <math>A_n</math> be the <math>n</math>-digit integer each of whose digits is equal to <math>a</math>; let <math>B_n</math> be the <math>n</math>-digit integer each of whose digits is equal to <math>b</math>, and let <math>C_n</math> be the <math>2n</math>-digit (not <math>n</math>-digit) integer each of whose digits is equal to <math>c</math>. What is the greatest possible value of <math>a + b + c</math> for which there are at least two values of <math>n</math> such that <math>C_n - B_n = A_n^2</math>? | For a positive integer <math>n</math> and nonzero digits <math>a</math>, <math>b</math>, and <math>c</math>, let <math>A_n</math> be the <math>n</math>-digit integer each of whose digits is equal to <math>a</math>; let <math>B_n</math> be the <math>n</math>-digit integer each of whose digits is equal to <math>b</math>, and let <math>C_n</math> be the <math>2n</math>-digit (not <math>n</math>-digit) integer each of whose digits is equal to <math>c</math>. What is the greatest possible value of <math>a + b + c</math> for which there are at least two values of <math>n</math> such that <math>C_n - B_n = A_n^2</math>? | ||
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<math>\textbf{(A) } 12 \qquad \textbf{(B) } 14 \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 18 \qquad \textbf{(E) } 20</math> | <math>\textbf{(A) } 12 \qquad \textbf{(B) } 14 \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 18 \qquad \textbf{(E) } 20</math> | ||
− | ==Notes== | + | ([[2018 AMC 10A Problems/Problem 25|Source]]) |
− | + | ||
+ | == Notes == | ||
+ | |||
+ | PaperMath’s sum was named by the aops user PaperMath. | ||
+ | PaperMath is very orz. | ||
==See also== | ==See also== | ||
+ | |||
*[[Cyclic sum]] | *[[Cyclic sum]] | ||
*[[Summation]] | *[[Summation]] | ||
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[[Category:Algebra]] | [[Category:Algebra]] | ||
[[Category:Theorems]] | [[Category:Theorems]] | ||
+ | {{Stub}} |
Latest revision as of 22:28, 22 June 2025
PapreMath's sum is a thereom discovered by the APS user Papermath on October 8th, 2023.
Contents
Statement
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
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
?
(Source)
Notes
PaperMath’s sum was named by the aops user PaperMath. PaperMath is very orz.
See also
This article is a stub. Help us out by expanding it.