# Mathematical Food for Thought

Serves a Daily Special and an All-You-Can-Eat Course in Problem Solving. Courtesy of me, Jeffrey Wang.

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All Paired Up. Topic: Algebra/NT. Level: AIME. June 20th, 2006

Problem: Find a closed form for the sum

.

Solution: Consider the expansion of

.

It’s not hard to see that if we let the desired sum be , we have

since each pair is counted twice and the squared terms are there. But we can simplify the sums by well-known formulas to get

,

which becomes

after massive simplification. QED.

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Comment: Symmetry was definitely the best way to approach this problem (or as far as I know anyway). You could’ve found a lot of ugly summations to get to the desired expression as well.

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Practice Problem: Can you generalize? In any way, shape, or form.

### 7 Responses to “All Paired Up. Topic: Algebra/NT. Level: AIME.”

1. LiZ Says:

Have fun in boston ^^ *make deep connections so christie says…*

2. Anonymous Says:

mna851 generalized in this thread on AoPS

Well there are other generalizations, too. For instance, the sum of the products taking 3 terms at a time.

4. Anonymous Says:

Hmm how do you do it three at a time?

I would take (1+2+…+n)^3 – 3*(1^2+2^2+…+n^2)(1+2+…+n) + 2(1^3+2^3+…+n^3) and divide by 6… it’s sorta ugly. With a program you could expand f(x) = (x-1)(x-2)…(x-n) and find the coefficient of x^{n-3}.

6. LiZ Says:

fine just ignore me -___- *i know i’m spamming*