This is a summation identities for decomposition or reconstruction of summations. Papermath’s sum states,
For all real values of , this equation holds true for all nonnegative values of . When , this reduces to
We will first prove a easier variant of Papermath’s sum,
This is the exact same as
But everything is multiplied by .
Notice that this is the exact same as saying
Substituting this into yields
Adding on both sides yields
As you can see,
Is true since the RHS and LHS are equal
This equation holds true for any values of . Since this is true, we can divide by on both sides to get
And then multiply both sides to get
Which proves Papermath’s sum
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 ?
Papermath’s sum was discovered by the aops user Papermath, as the name implies.