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We have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain.

1729 can be expressed as the sum of two perfect cubes in two distinct ways.

How many such numbers exist?

**Hint:** Given one set of such numbers, how could you construct another set?

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If \(a\) and \(b\) are integers such that \(a^3+b^3=2015\), then what is \(a+b\)?

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\[\LARGE{2 + 2 = 2 \times 2 \\ 1 + 2 + 3 = 1 \times 2 \times 3}\]

These are sets of two and three positive integers whose sum equals their product.

Find a set of four positive integers whose sum equals their product. What is their sum?

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Find the positive integer \(n \) such that \(n^3 + 2n^2 + 9n + 8 \) is a perfect cube.

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\[ a + b + c + d + e = a \times b \times c \times d \times e \]

How many unordered 5-tuples of positive integers are there which satisfy the above equation?

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