Round 3: Tossup 5

Summing the first n natural numbers raised to this power gives the square of the nth triangular number, per (15[1])Nicomachus’s theorem. The nth taxicab number is the smallest that can be written (15[1])as the sum of two numbers raised to this power (15[1])in n different ways. (15[1])Multiplying by i this number of times is equivalent to a rotation through 270 (-5[1])degrees. (15[1])This is the (15[1])lowest number n for which (15[1])the equation “a to the n plus b to the n equals c to the n” (-5[1])has (*) no (10[1])integer solution, (10[1])per Fermat’s last theorem. (-5[1])The derivative of x to this power is proportional to “x squared.” (10[1])For 10 points, the volume (-5[1])of a cube is found by raising the side length (10[2])to what power? ■END■ (10[4])

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