If a³ + b³ + c³ = a⁴ + b⁴ + c⁴ then prove that (a/a²+b³+c³) +(b/a³+b²+c³)+ (c/a³+b³+c²) ≥1

Let's start by analyzing the given equation and the expression we need to prove.

**Given:**
\[ a^3 + b^3 + c^3 = a^4 + b^4 + c^4 \]

We need to prove:
\[ \frac{a}{a^2 + b^3 + c^3} + \frac{b}{a^3 + b^2 + c^3} + \frac{c}{a^3 + b^3 + c^2} \geq 1 \]

### Step 1: Simplify the given condition
Let's rewrite the condition \( a^3 + b^3 + c^3 = a^4 + b^4 + c^4 \).

We can factorize each term as:
\[ a^3(1 - a) + b^3(1 - b) + c^3

Explore how to prove the inequality involving sums of powers in algebra. This problem deals with a specific mathematical condition and its proof, suitable for advanced high school students.

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