Category Archives: problem

2016-12 A series

Let \(k\) be a positive integer. Let \(a_n=1\) if \(n\) is not a multiple of \(k+1\), and \(a_n=-k\) if \(n\) is a multiple of \(k+1\). Compute \[\sum_{n=1}^\infty \frac{a_n}{n}.\]

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2016-11 Infinite series

For a positive integer \( n \), define \( f(n) \) by
\[
f(n) =
\begin{cases}
0 & \text{ if } n \equiv 0 \pmod{5} \\
1 & \text{ if } n \equiv \pm 1 \pmod{5} \\
-1 & \text{ if } n \equiv \pm 2 \pmod{5}
\end{cases}.
\]
Compute the infinite series
\[
\sum_{n=1}^{\infty} \frac{f(n)}{n} = 1 – \frac{1}{2} – \frac{1}{3} + \frac{1}{4} + \frac{1}{6} – \dots.
\]

(This is the last problem of this semester. Thank you.)

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2016-10 Factorization

Suppose that \( A \) is an \( n \times n \) matrix with integer entries and \( \det A = p_1^{e_1} p_2^{e_2} \dots p_k^{e_k} \) for primes \( p_1, p_2, \dots, p_k \) and positive integers \( e_1, e_2, \dots, e_k \). Prove that there exist \( n \times n \) matrices \( B_1, B_2, \dots, B_k \) with integer entries such that \( A = B_1 B_2 \dots B_k \) and \( \det B_1 = p_1^{e_1}, \det B_2 = p_2^{e_2}, \dots, \det B_k = p_k^{e_k} \).

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2016-9 Determinant of a matrix

Let \( A=(a_{ij})_{ij}\) be an \(n\times n\) matrix, where \[ a_{ij}=\begin{cases} 3 &\text{if }i=j,\\ (-1)^{\lvert i-j\rvert}&\text{otherwise.}\end{cases}\] Compute the determinant of \(A\).

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2016-7 Sum-free

For a set \( A \subset \mathbb{R} \), let \( f(A) \) be the size of the largest set \( B \subset A \) such that \( (B+B) \cap B = \emptyset \). For a positive integer \( n \), let \( f(n) = \min_{0 \notin A, |A|=n} f(A) \). Prove that \( f(n) \geq n/3 \).

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2016-6 Convex function

Suppose that \( f \) is a real-valued convex function on \( \mathbb{R} \). Prove that the function \( X \mapsto \mathrm{Tr } f(X) \) on the vector space of \( N \times N \) Hermitian matrices is convex.

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2016-5 Partition into 4 sets

Let \(A_1,A_2,\ldots,A_n\) be subsets of \(\{1,2,\ldots,n\}\) such that \(i\notin A_i\) for all \(i\). Prove that there exist four sets \(C_1,C_2,C_3,C_4\) such that \(C_1\cup C_2\cup C_3\cup C_4=\{1,2,\ldots,n\} \) and for all \(i\) and \(j\), if \(i\in C_j\), then \( \lvert A_i\cap C_j\rvert \le \frac12 \lvert A_i\rvert\).

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2016-4 Distances in a tree

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 \[ \det (D_T(x))=(1-x^2)^{n-1}.\]

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2016-3 Non-finitely generated subgroup

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.

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