# PaperMath’s sum

## Contents

## PaperMath’s sum

This is a summation identities for decomposition or reconstruction of summations. Papermath’s sum states,

Or

For all real values of , this equation holds true for all nonnegative values of . When , this reduces to

## Proof

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

Notice that

Substituting this into yields

Adding on both sides yields

Notice that

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

Or

Which proves Papermath’s sum

## 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 discovered by the aops user Papermath, as the name implies.