Let \(T\) be a tree on \(n\) vertices \(V=\{1,2,\ldots,n\}\). For two vertices \(i\) and \(j\), let \(d_{ij}\) be the distance between \(i\) and \(j\), that is the number of edges in the unique path from \(i\) to \(j\). Let \(D_T(x)=(x^{d_{ij}})_{i,j\in V}\) be the \(n\times n\) matrix. Prove that \[ \det (D_T(x))=(1-x^2)^{n-1}.\]
Let \( G \) be a subgroup of \( GL_2 (\mathbb{R}) \) generated by \( \begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix} \) and \( \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \). Let \( H \) be a subset of \( G \) that consists of all matrices in \( G \) whose diagonal entries are \( 1 \). Prove that \( H \) is a subgroup of \( G \) but not finitely generated.
For \( a \geq 0 \), find
\[
\lim_{n \to \infty} n \int_{-1}^0 \left( x + \frac{x^2}{2} + e^{ax} \right)^n dx.
\]
Prove that for every \( x_1, x_2,\ldots,x_n\in [0,1]\), there exist \(\varepsilon_1,\varepsilon_2,\ldots,\varepsilon_n\in\{1/2,-1/2\}\) such that for all \(k=1,2,\ldots,n-1\), \[ \left\lvert \sum_{i=1}^k \varepsilon_i x_i-\sum_{i=k+1}^n \varepsilon_i x_i \right\rvert\le 1.\]
Let \( A, B \) are \( n \times n \) Hermitian matrices and \( p, q \in [1, \infty] \) with \( \frac{1}{p} + \frac{1}{q} = 1 \). Prove that
\[
| Tr (AB) | \leq \| A \|_{S^p} \| B \|_{S^q}.
\]
(Here, \(\| A \|_{S^p} \) is the \(p\)-Schatten norm of \( A \), defined by
\[
\| A \|_{S^p} = \left( \sum_{i=1}^n |\lambda_i|^p \right)^{1/p},
\]
where \( \lambda_1, \lambda_2, \dots, \lambda_n \) are the eigenvalues of \( A \).)
Let \(f:[0,1)\to[0,1)\) be a function such that \[ f(x)=\begin{cases} 2x,&\text{if }0\le 2x\lt 1,\\ 2x-1, & \text{if } 1\le 2x\lt 2.\end{cases}\] Find all \(x\) such that \[ f(f(f(f(f(f(f(x)))))))=x.\]
Evaluate the following integral \[ \int_0^\infty \frac{\cos cx}{(\cosh x)^2} \, dx\] for a real constant \(c\).
Assume that a function \( f : (0, 1) \to [0, \infty) \) satisfies \( f(x) = 0 \) at all but countably many points \( x_1, x_2, \cdots \). Let \( y_n = f(x_n) \). Prove that, if \( \sum_{n=1}^{\infty} y_n < \infty \), then \( f \) is differentiable at some point.
Prove or disprove the following statement:
There exists a function \( f : \mathbb{R} \to \mathbb{R} \) such that
(1) \( f \equiv 0 \) almost everywhere, and
(2) for any nonempty open interval \(I\), \( f(I) = \mathbb{R} \).
Evaluate \[ \sum_{k=0}^{n} 2^k \tan \frac{\pi}{4\cdot 2^{n-k}}.\]