Table Legs

Algebra Level pending

Legs $L_1$ , $L_2$ , $L_3$ , $L_4$ of a square table each have length $n$ , where $n$ is a positive integer. For how many ordered 4-tuples $(k_1, k_2, k_3, k_4)$ of nonnegative integers can we cut a piece of length $k_i$ from the end of leg $L_i$ $(i=1,2,3,4)$ and still have a stable table?

(The table is stable if it can be placed so that all four of the leg ends touch the floor. Note that a cut leg of length 0 is permitted.)

Source: 2005 USAMO Problem 4

\frac{2n^3+6n^2+3n+7}{6}

\frac{n^3+6n^2+7n+4}{6}

\frac{n^3+6n^2+5n+6}{6}

\frac{2n^3+6n^2+7n+3}{3}

\frac{2n^3+6n^2+7n+3}{6}

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Table Legs

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