Difference between revisions of "2024 SSMO Relay Round 4 Problems/Problem 3"
(Created page with "==Problem== Let <math>T = TNYWR.</math> Given that: \begin{align*} a+b &= -c,\\ a^3 - abc &= 4,\text{ and }\\ b^3 - abc &= T.\\ \end{align*} Then, <math>abc - c^3 = x.</math>...") |
|||
| Line 10: | Line 10: | ||
==Solution== | ==Solution== | ||
| + | Since <math>a+b = -c\implies a+b+c = 0,</math> we have <cmath>a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-ac-bc) = 0.</cmath> So, <math>4+T-x = 0\implies x = 4+T = \boxed{206}.</math> | ||
| + | |||
| + | ~SMO_Team | ||
Latest revision as of 14:47, 10 September 2025
Problem
Let 
Given that:
\begin{align*}
a+b &= -c,\\
a^3 - abc &= 4,\text{ and }\\
b^3 - abc &= T.\\
\end{align*}
Then, 
Find the value of 
Solution
Since 
we have ![\[a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-ac-bc) = 0.\]](http://latex.artofproblemsolving.com/f/2/a/f2a8abbd044285235b93f1d47fd832b32c6c030d.png)
So, 
~SMO_Team
