Problems

Let $z=x+iy$, $w=u+iv$. Find a) $z+w$; b) $zw$; c) $z/w$.

Prove the equalities:

a) $\overline{z+w}=\bar{z}+\bar{w}$; b) $\overline{zw}=\bar{z}\bar{w}$; c) $\overline{\frac{z}{w}}=\frac{\bar{z}}{\bar{w}}$; d) $|\bar{z}|=|z|$; d) $\overline{\stackrel{―}{z}}=z$.

Prove the equalities:

a) $z+\bar{z}=2\mathrm{Re}z$;

b) $z-\bar{z}=2i\mathrm{Im}z$;

c) $\bar{z}z=|z{|}^2$.

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) $\arg \frac{z-z_1}{z-z_2}=0$;

b) $\arg \frac{z_1-z}{z-z_2}=0$.

Find the largest and smallest values of the functions

a) $f_1(x)=a\cos x+b\sin x$; b) $f_2(x)=a{\cos }^{2}x+b\cos x\sin x+c{\sin }^{2}x$.