Functions
Cyclic system of equations — fixed point analysis
MJAT_TS3_P1
Grade 12
Question:
Let $a$, $b$, $c$ be three non-negative numbers such that:
$$a^3 - a^2 + a = b, \quad b^3 - b^2 + b = c, \quad c^3 - c^2 + c = a$$
The number of possible triplets $(a,b,c)$ is/are:
Step-by-Step Solution
Key Concept: Let $g(x)=x^3-x^2+x$. The system says $g(a)=b$, $g(b)=c$, $g(c)=a$. Trivial solutions: $a=b=c=0$ and $a=b=c=1$. For non-trivial: WLOG $a>1\Rightarrow b>a\Rightarrow c>b\Rightarrow a>c>b>a$ — contradiction.
Only $(0,0,0)$ and $(1,1,1)$ satisfy the system. Answer: $\mathbf{2}$.
Correct Answer: 2