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IMC 2013 Problem 6

August 11, 2013 Leave a comment

Problem 6. Let {z} be a complex number with {|z+1|>2}. Prove that {|z^3+1|>1}.

Solution: Write {z} in its trigonometric form:

\displaystyle z=r(\cos \theta + i \sin \theta).

Then the condition {|z+1|>2} translates to

\displaystyle r^2+2r\cos \theta +1 >4

which is equivalent to

\displaystyle r^2+2r\cos \theta -3 >0.

Since {r} must be positive, we have as a consequence

\displaystyle r > \sqrt{\cos^2 \theta+3}-\cos \theta.

The condition {|z^3+1|>1} is equivalent to

\displaystyle r^6+2r^3 \cos (3\theta) >0

which translates to

\displaystyle r^3+2 \cos (3\theta)>0.

Thus it is enough to prove that

\displaystyle (\sqrt{\cos^2 \theta+3}-\cos \theta)^3 \geq -2 \cos (3\theta)=-2\cos \theta (4 \cos^2 \theta -3).

Note that the result we need to prove is not trivial if and only if {\theta \in (\pi/6,\pi/2)} and on that interval {\cos \theta} is positive.

Denote {t=\cos \theta} then {t \in (0,\sqrt{3}/2)} and we want to prove that

\displaystyle (\sqrt{t^2+3}-t)^3 +2t(4t^2-3)\geq 0.

If we denote {f(t)=(\sqrt{t^2+3}-t)^3 +2t(4t^2-3)} then {f(\sqrt{3/8})=0=f'(\sqrt{3/8})=0}.

If we evaluate then we obtain

\displaystyle f(t)=(a^2+3)^{3/2}+3a^2\sqrt{a^2+3}+4a^3-15a

which is a sum of two convex functions (first two and last two terms) among which one is strictly convex, and therefore is strictly convex (on {[0,1]}).

This means that {\sqrt{3/8}} is the global minimum of {f} on {[0,1]} and therefore {f} is positive on that interval which finishes the proof.

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Categories: Complex analysis, Olympiad Tags: ,

IMC 2013 Problem 5

August 8, 2013 Leave a comment

Problem 5. Does there exist a sequence {(a_n)} of complex numbers such that for every positive integer {p} we have that {\sum_{n=1}^\infty a_n^p} converges if and only if {p} is not a prime?

Geometric mean for n complex numbers

January 29, 2011 Leave a comment

Let D be a closed disc in the complex plane. Prove that for all positive integers n, and for all complex numbers z_1,z_2,\ldots,z_n\in D  there exists a z\in D such that z^n = z_1\cdot z_2\cdots z_n.

Romanian TST 2004

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Every expansion has a 0

October 13, 2009 Leave a comment

Suppose f is entire such that for each x_0 \in \mathbb{C} at least one of the coefficients of the expansion f=\sum\limits_{n=0}^\infty c_n (z-z_0)^n is equal to 0. Prove that f is a polynomial.
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Power series & number theory

October 13, 2009 Leave a comment

Let F(z)=\sum\limits_{n=1}^\infty d(n) z^n where d(n) denotes the number of divisors of n. Calculate the radius of convergence of this series and prove that F(z)=\sum\limits_{n=1}^\infty \frac{z^n}{1-z^n}. Furthermore, prove that F(r)\geq \frac{1}{1-r}\log \frac{1}{1-r} for r \in (0,1).

Amazing property of entire functions

October 12, 2009 Leave a comment

Prove that if p,q,r are non-constant, non-vanishing entire functions with p+q+r=0 then there exists an entire function h such that p,q,r are constant multiples of h.
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Blaschke factors

October 12, 2009 Leave a comment

Take w,z \in \mathbb{C} such that \bar{z}w \neq 1. Prove that \left| \frac{w-z}{1-\bar{w}z} \right|<1 if |z|<1 and |w|<1 and also \left| \frac{w-z}{1-\bar{w}z} \right|=1 if |z|=1 or |w|=1.

Prove that for any w fixed in the unit disk \mathbb{D}, the mapping F_w(z)=\frac{w-z}{1-\bar{w}z} has the following properties:
1) F_w maps \mathbb{D} to \mathbb{D} and is holomorphic.
2) F_w interchanges 0 and w.
3) |z|=1 \Rightarrow |F_w(z)|=1.
4) F_w is an involution on \mathbb{D} and thus is bijective.
5) Construct a bijective, holomorphic function, which maps the unit disk to itself, takes the unit circle into itself and interchanges two given complex numbers z,w with |z|<1,|w|<1.

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