Prove that the equation $$x^2 + y^2 + z^2 = x + y + z + 1$$ has no rational solutions. In a triangle $ABC$ with $BC = CA + \frac 12 AB$, p...

1. Prove that the equation $$x^2 + y^2 + z^2 = x + y + z + 1$$ has no rational solutions.
2. In a triangle $ABC$ with $BC = CA + \frac 12 AB$, point $P$ is given on side $AB$ such that $BP : PA = 1 : 3$. Prove that $\angle CAP = 2 \angle CPA.$
3. Let $a, b, c$ be non-negative numbers with $a+b+c = 3$. Prove the inequality $\frac{a}{b^2+1}+\frac{b}{c^2+1}+\frac{c}{a^2+1} \geq \frac 32.$
4. Consider a system of infinitely many spheres made of metal, with centers at points $(a, b, c) \in \mathbb Z^3$. We say that the system is stable if the temperature of each sphere equals the average temperature of the six closest spheres. Assuming that all spheres in a stable system have temperatures between $0^\circ C$ and $1^\circ C$, prove that all the spheres have the same temperature.

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