1. **Stating the problem:** We are given the function $$E(x) = \left(\frac{x}{\sqrt{2}} - \sqrt{2}\right)^2 - x\left(\frac{x}{2} - \sqrt{2}\right) - \sqrt{2}(1 - \sqrt{2})x$$ and the value $$E(0) = 2$$. We also have the expression $$N = E(n) + 2 \cdot E(2n) + 1485$$. Additionally, we have values for $$x$$ and $$y$$: $$x = \left(\frac{1}{2} + \frac{1}{3} - \frac{1}{6}\right) \cdot \frac{3}{2}$$ $$y = 16^2 : (2^2)^3 : 2$$ Finally, it is given that $$x = 1$$. 2. **Calculate the value of $$x$$ from the fraction expression:** $$x = \left(\frac{1}{2} + \frac{1}{3} - \frac{1}{6}\right) \cdot \frac{3}{2}$$ Calculate inside the parentheses: $$\frac{1}{2} + \frac{1}{3} - \frac{1}{6} = \frac{3}{6} + \frac{2}{6} - \frac{1}{6} = \frac{4}{6} = \frac{2}{3}$$ Multiply by $$\frac{3}{2}$$: $$x = \frac{2}{3} \cdot \frac{3}{2} = \cancel{\frac{2}{\cancel{3}}} \cdot \cancel{\frac{3}{2}} = 1$$ This confirms the given $$x=1$$. 3. **Calculate the value of $$y$$:** $$y = 16^2 : (2^2)^3 : 2$$ Calculate each term: $$16^2 = 256$$ $$(2^2)^3 = 4^3 = 64$$ So, $$y = 256 : 64 : 2$$ Division is left to right: $$\frac{256}{64} = 4$$ $$\frac{4}{2} = 2$$ Therefore, $$y = 2$$. 4. **Simplify the function $$E(x)$$:** Start with: $$E(x) = \left(\frac{x}{\sqrt{2}} - \sqrt{2}\right)^2 - x\left(\frac{x}{2} - \sqrt{2}\right) - \sqrt{2}(1 - \sqrt{2})x$$ Expand the first square: $$\left(\frac{x}{\sqrt{2}} - \sqrt{2}\right)^2 = \left(\frac{x}{\sqrt{2}}\right)^2 - 2 \cdot \frac{x}{\sqrt{2}} \cdot \sqrt{2} + (\sqrt{2})^2 = \frac{x^2}{2} - 2x + 2$$ Expand the second term: $$- x \left(\frac{x}{2} - \sqrt{2}\right) = - \frac{x^2}{2} + x \sqrt{2}$$ Expand the third term: $$- \sqrt{2}(1 - \sqrt{2})x = - \sqrt{2}x + 2x$$ 5. **Combine all terms:** $$E(x) = \left(\frac{x^2}{2} - 2x + 2\right) + \left(- \frac{x^2}{2} + x \sqrt{2}\right) + \left(- \sqrt{2}x + 2x\right)$$ Group like terms: - For $$x^2$$: $$\frac{x^2}{2} - \frac{x^2}{2} = 0$$ - For $$x$$: $$-2x + x \sqrt{2} - \sqrt{2}x + 2x = (-2x + 2x) + (x \sqrt{2} - \sqrt{2}x) = 0 + 0 = 0$$ - Constant term: $$2$$ 6. **Final simplified form:** $$E(x) = 2$$ 7. **Check given value:** Given $$E(0) = 2$$ matches our simplified result. 8. **Calculate $$N$$:** Since $$E(n) = 2$$ and $$E(2n) = 2$$ for any $$n$$, $$N = E(n) + 2 \cdot E(2n) + 1485 = 2 + 2 \cdot 2 + 1485 = 2 + 4 + 1485 = 1491$$ **Final answer:** $$N = 1491$$