Mathematics: (This is an adapted question from a previous AMC 12 problem) Suppose a, b, c are real numbers such that a + b + c = 7 and a² + b² + c² = 33. Let f(a) = sin(a - 2) - 202020/202021, where a is bounded by the parameters found above. Given that ⌊a³⌋ has only 1 possible value when f(a) ≥ 0, find ⌊a³⌋.

55745 rounded to the nearest thousand is 56,000....

(This is an adapted question from a previous AMC

NikkiZoeller

23.02.2021

So, (a+b+c)2=a2+b2+c2+2(ab+bc+ca)

⟹ab+bc+ca=−

So a,b,c are the roots of the equation t3−

t−u=0 where u=abc.

Using discriminant of the cubic equation fro real roots,

△=−4⋅1⋅(−

)3−27(−u)2=

−27u2≥0⟺u2≤

Step-by-step explanation:

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