CR-MO 2016

Find all solutions of the system $\sqrt[3]{\frac{yz^4}{x^2}}+2wx=0$ $\sqrt[3]{\frac{xz^4}{y}}+5wy=0$ $\sqrt[3]{\frac{xy}{x}}+7wz^{-1/3}=0$ $x^{12}+\frac{125}{4}y^5+\frac{343}{2}z^4=16$ where $x, y, z \ge 0$ and $w \in R$ PSI attached the system, in case I have any typos

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The initial number of inhabitants of a city of more than $150$ inhabitants is a perfect square. With an increase of $1000$ inhabitants it becomes a perfect square plus a unit. After from another increase of $1000$ inhabitants it is again a perfect square. Determine the quantity of inhabitants that are initially in the city.

Let $a, b$ and $c$ be real numbers, and let $f (x) = ax^2 + bx + c$ and $g (x) = cx^2 + bx + a$ functions such that $| f (-1) | \le 1$ , $| f (0) | \le 1$ and $| f (1) | \le 1$ . Show that if $-1 \le x \le 1$ , then $| f (x) | \le \frac54$ and $| g (x) | \le 2$ .

Sea $f: R^+ \to R$ defined as $$f (x) = \frac{1}{\sqrt[3]{x^2 + 6x + 9} + \sqrt[3]{x^2 + 4x + 3} + \sqrt[3]{x^2 + 2x + 1}}$$ Calculate $$f (1) + f (2) + f (3) + ... + f (2016).$$

Let $f: Z^+ \to Z^+ \cup \{0\}$ a function that meets the following conditions: a) $f (a b) = f (a) + f (b)$ , b) $f (a) = 0$ provided that the digits of the unit of $a$ are $7$ , c) $f (10) = 0$ . Find $f (2016).$