Evaluate the following integral for \( z \in \mathbb{C}^+ \).
\[
\frac{1}{2\pi} \int_{-2}^2 \log (z-x) \sqrt{4-x^2} dx.
\]
Category Archives: problem
2015-15 A sequence periodic modulo m for all m
Does there exist an infinite sequence such that (i) every integer appears infinitely many times and (ii) the sequence is periodic modulo \(m\) for every positive integer \(m\)?
2015-14 Local and absolute maximum
Find all positive integers \(n\) such that the following statement holds:
Let \(f:\mathbb{R}^n\to \mathbb {R}\) be a differentiable function that has a unique critical point \(c\). If \(f\) has a local maximum at \(c\), then \(f(c)\) is an absolute maximum of \(f\).
2015-13 Minimum
Find the minimum value of
\[
\int_{\mathbb{R}} f(x) \log f(x) dx
\]
among functions \(f: \mathbb{R} \rightarrow \mathbb{R}^+ \cup \{ 0 \}\) that satisfy the condition
\[
\int_{\mathbb{R}} f(x) dx = \int_{\mathbb{R}} x^2 f(x) dx = 1.
\]
2015-12 Rank
Let \(A\) be an \(n\times n\) matrix with complex entries. Prove that if \(A^2=A^*\), then \[\operatorname{rank}(A+A^*)=\operatorname{rank}(A).\] (Here, \(A^*\) is the conjugate transpose of \(A\).)
(This is the last problem of this semester. Thank you for participating KAIST Math Problem of the Week.)
2015-11 Limit
Does \(\frac{1}{n \sin n}\) converge as \(n\) goes to infinity?
2015-10 Product of sine functions
Let \(w_1,w_2,\ldots,w_n\) be positive real numbers such that \( \sum_{i=1}^n w_i=1\). Prove that if \(x_1,x_2,\ldots,x_n\in [0,\pi]\), then \[ \sin \left(\prod_{i=1}^n x_i^{w_i} \right) \ge \prod_{i=1}^n (\sin x_i)^{w_i}.\]
2015-9 Sum of squares
Let \(n\ge 1\) and \(a_0,a_1,a_2,\ldots,a_{n}\) be non-negative integers. Prove that if \[ N=\frac{a_0^2+a_1^2+a_2^2+\cdots+a_{n}^2}{1+a_0a_1a_2\cdots a_{n}}\] is an integer, then \(N\) is the sum of \(n\) squares of integers.
2015-8 all lines
Does there exist a subset \(A\) of \(\mathbb{R}^2\) such that \( \lvert A\cap L\rvert=2\) for every straight line \(L\)?
2015-7 Binomial Identity
Prove or disprove that \[ \sum_{i=0}^r (-1)^i \binom{i+k}{k} \binom{n}{r-i} = \binom{n-k-1}{r}\] if \(k, r\) are non-negative integers and \(0\le r\le n-k-1\).