Round 6: Tossup 18

Description Required. Stanley Tennenbaum proved this fact using a diagram where 2 squares overlap to create the [read slowly] quantity two b minus a squared. Because this fact is true, the formula for half-companion Pell numbers includes a plus or minus one in the numerator. This fact can be shown by stating that [read slowly] a squared is equal to two b squared and then finding a (15[1])(*) contradiction because a and b must be coprime. Applying the rational root test to the equation x squared minus two equals zero can be used to prove this fact. Angry Pythagoreans (10[1])legendarily murdered Hippasus for proving, (10[1]-5[1])for 10 points, what fact (-5[1])that says a number beginning with 1.414 is not a ratio of integers? ■END■ (10[1])

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