2024 SSMO Relay Round 4 Problems/Problem 3

Problem

Let $T = TNYWR.$ Given that: \begin{align*} a+b &= -c,\\ a^3 - abc &= 4,\text{ and }\\ b^3 - abc &= T.\\ \end{align*} Then, $abc - c^3 = x.$ Find the value of $x.$

Solution

Since $a+b = -c\implies a+b+c = 0,$ we have $a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-ac-bc) = 0.$ So, $4+T-x = 0\implies x = 4+T = \boxed{206}.$

~SMO_Team

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2024 SSMO Relay Round 4 Problems/Problem 3

Problem

Solution