What is the product, the sum and the sum of squares of the complex \(\displaystyle m\)th roots of unity?

Difficulty: 2.

\(\displaystyle \binom{n}0+\binom{n}3+\binom{n}6+\ldots=?\)

Difficulty: 3.

Let \(\displaystyle a,b,c\in\CC\). What is the geometric interpretation of

\(\displaystyle \frac12\im\Big((c-a)\cdot\overline{(b-a)}\Big)?\)

Difficulty: 3. Answer (final result) is provided for this problem.

Let \(\displaystyle A_1A_2\ldots A_n\) be the vertices of a regular \(\displaystyle n\)-gon, inscribed into a unit circle, and let \(\displaystyle P\) be another point on the circle. Prove that

\(\displaystyle PA_1\cdot PA_2\cdot\ldots\cdot PA_n\le 2.\)

Difficulty: 3.

Let \(\displaystyle w(z)=\frac 12\left(z+\frac 1z\right)\) be the so called Zhukowksy map. What is the image of

(a) the unit circle? (b) the interior of the unit circle? (c) the exterior of the unit circle?

(d) the circles with center \(\displaystyle 0\)? (e) the lines passing through \(\displaystyle 0\)?

Difficulty: 3. Answer (final result) is provided for this problem.

Sketch the set of those complex number for which

(a) \(\displaystyle \displaystyle\left|\frac{z-1}{z+1}\right|=1\);  (b) \(\displaystyle \displaystyle\left|\frac{z-1}{z+1}\right|=2\);  (c) \(\displaystyle \displaystyle\arg(z+1)=\arg(2z-1)\)  (\(\displaystyle -\pi<\arg z\le\pi\)).

Difficulty: 3.

(a) \(\displaystyle \displaystyle\quad \re(z^2)=4\);    (b) \(\displaystyle \displaystyle\re\frac{z-1}{z+1}=0\);  (c) \(\displaystyle 0< \re (iz) < 2\pi\);  (d)

Difficulty: 3.

Sketch the set of those complex number for which

(a) \(\displaystyle \displaystyle\frac{|z|}{\re z} < K\);  (b) \(\displaystyle |z-1|+|z+1| < 4\);  (c) \(\displaystyle \re\displaystyle\frac{1+z}{1-z}>0\).

Difficulty: 3.

Assume that \(\displaystyle w:\CC\to\CC\) is a distance preserving map. Show that \(\displaystyle w(z)=Az+B\) or \(\displaystyle w(z)=A\bar{z}+B\), where \(\displaystyle |A|=1\).

Difficulty: 4.

What is the product, the sum, and the sum of squares of all primitive \(\displaystyle m\)-th roots of unity?

Difficulty: 5.

Let \(\displaystyle p(z)\in \C[z]\) be of degree at least 1. Prove the following

(a) If all roots of \(\displaystyle p\) have negative real parts, then \(\displaystyle \re\dfrac{p'(z)}{p(z)} >0\).

(b) If the roots of \(\displaystyle p(z)\) all lie in the half plane \(\displaystyle \re z<0\), then the same holds for \(\displaystyle p'(z)\).

(c) (Gauss) If \(\displaystyle p(z)\in \C[z]\), then the roots of \(\displaystyle p'\) are contained in the convex hull of the roots of \(\displaystyle p\).

Difficulty: 5.

Let \(\displaystyle m>1\) and \(\displaystyle a, b:\Z_m\to\CC\) be two functions. Define the sum \(\displaystyle a+b\) and the convolution \(\displaystyle a*b\) of \(\displaystyle a\) and \(\displaystyle b\) as follows

\(\displaystyle (a+b)(n) = a(n)+b(n); \qquad (a* b)(n) = \sum_{k=0}^{m-1}a(k)b(n-k).\)

Prove that this makes the set of complex valued functions on \(\displaystyle \Z_m\) a commutative ring with unit.

Difficulty: 5.

Let \(\displaystyle \varepsilon=\cos\frac{2\pi}m+i\sin\frac{2\pi}m\). Define the Fourier transform of a function \(\displaystyle a: \Z_m\rightarrow \C\) by

\(\displaystyle {\hat a}(n)=\sum_{k=0}^{m-1}a(k)\varepsilon^{nk}.\)

Show that \(\displaystyle \widehat{(a* b)}(n)={\hat a}(n)\cdot{\hat b}(n).\)

Difficulty: 6.

Let \(\displaystyle a_1,a_2,\ldots\) be a decreasing sequence of positive numbers that converges to \(\displaystyle 0\), and let \(\displaystyle b_1,b_2,\ldots\) be a sequence of complex numbers such that the partial sums \(\displaystyle b_1+\ldots+b_n\) are bounded by a constant independent of \(\displaystyle n\). Prove that \(\displaystyle \displaystyle\sum_{n=1}^\infty a_nb_n\) is convergent.

Difficulty: 6.

Let \(\displaystyle f(z)\in \C\) be non-constant. Prove the following

(a) \(\displaystyle \re f\) and \(\displaystyle \im f\) have no local extrema.

(b) If \(\displaystyle |f|\) has a local extremum at \(\displaystyle z_0\), then \(\displaystyle f(z_0)=0\).

(c) Prove the fundamental theorem of algebra.

Difficulty: 7.

Let \(\displaystyle n\ge2\) and \(\displaystyle u_1=1,u_2,\ldots,u_n\) be complex numbers with absolute value at most \(\displaystyle 1\), and let

\(\displaystyle f(z)=(z-u_1)(z-u_2)\ldots(z-u_n). \)

Show that the polynomial \(\displaystyle f'(z)\) has a root with nonnegative real part.

KöMaL A. 430.

Difficulty: 7. Solution is available for this problem.

Let \(\displaystyle k(z)=\dfrac z{(1-z)^2}\) be the so called Koebe map. What is the image of the unit disc under the Koebe map?

Difficulty: 7.

Let \(\displaystyle f\in \C[x]\) and let \(\displaystyle T\) be a rectangle such that \(\displaystyle f\) has no root on the boundary of \(\displaystyle T\). Show that the number of roots of \(\displaystyle f\) inside \(\displaystyle T\) agrees with the winding number about \(\displaystyle 0\) of the image of the boundary of \(\displaystyle T\) under \(\displaystyle f\).

Difficulty: 8.

Find a formula for Fourier inversion in case of the finite Fourier transform.

Difficulty: 8.

Let \(\displaystyle f:\CC\to\CC\) be a continuous function for which \(\displaystyle \displaystyle\lim_{z\to\infty}\frac{f(z)}z=1\) (i.e. \(\displaystyle \displaystyle\frac{f(z)}z\to1\) if \(\displaystyle |z|\to\infty\)). Show that the image of \(\displaystyle f\) is \(\displaystyle \CC\).

Difficulty: 9.

Consider \(\displaystyle \C\) as the \(\displaystyle xy\)-plane in 3-space and pick 2 semicircles in the upper half space whose end points are the complex numbers \(\displaystyle a,b\) and \(\displaystyle c,d\). Show that the two semicircles intersect each other orthogonally if and only if \(\displaystyle (a,b,c,d)=-1\).

(Riesz competition, 1988)

Difficulty: 9.