Let it be known that all the roots of some equation \(x^3 + px^2 + qx + r = 0\) are positive. What additional condition must be satisfied by its coefficients \(p, q\) and \(r\) in order for it to be possible to form a triangle from segments whose lengths are equal to these roots?
Let \(z = x + iy\), \(w = u + iv\). Find a) \(z + w\); b) \(zw\); c) \(z/w\).
Prove the equalities:
a) \(\overline{z+w} = \overline{z} + \overline{w}\); b) \(\overline{zw} = \overline{z} \overline{w}\); c) \(\overline{\frac{z}{w}} = \frac{\overline{z}}{\overline{w}}\); d) \(|\overline{z}| = |z|\); d) \(\overline{\overline{z}} = z\).
Prove the equalities:
a) \(z + \overline {z} = 2 \operatorname{Re} z\);
b) \(z - \overline {z} = 2i \operatorname{Im} z\);
c) \(\overline {z} z = |z|^2\).
It is known that \(\cos \alpha^{\circ} = 1/3\). Is \(\alpha\) a rational number?
Let \(a, b\) be positive integers and \((a, b) = 1\). Prove that the quantity cannot be a real number except in the following cases \((a, b) = (1, 1)\), \((1,3)\), \((3,1)\).
Let \(f (x)\) be a polynomial of degree \(n\) with roots \(\alpha_1, \dots , \alpha_n\). We define the polygon \(M\) as the convex hull of the points \(\alpha_1, \dots , \alpha_n\) on the complex plane. Prove that the roots of the derivative of this polynomial lie inside the polygon \(M\).
For what values of \(n\) does the polynomial \((x+1)^n - x^n - 1\) divide by:
a) \(x^2 + x + 1\); b) \((x^2 + x + 1)^2\); c) \((x^2 + x + 1)^3\)?
a) Using geometric considerations, prove that the base and the side of an isosceles triangle with an angle of \(36^{\circ}\) at the vertex are incommensurable.
b) Invent a geometric proof of the irrationality of \(\sqrt{2}\).
Let \(z_1\) and \(z_2\) be fixed points of a complex plane. Give a geometric description of the sets of all points \(z\) that satisfy the conditions:
a) \(\operatorname{arg} \frac{z - z_1}{z - z_2} = 0\);
b) \(\operatorname{arg} \frac{z_1 - z}{z - z_2} = 0\).