Find the value of
\[
\sum_{n=1}^{\infty} \frac{1+ \frac{1}{2} + \dots + \frac{1}{n}}{n(2n-1)}.
\]
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Find the value of
\[
\sum_{n=1}^{\infty} \frac{1+ \frac{1}{2} + \dots + \frac{1}{n}}{n(2n-1)}.
\]
Let \(A\), \(B\) be matrices over the reals with \(n\) rows. Let \(M=\begin{pmatrix}A &B\end{pmatrix}\). Prove that \[ \det(M^TM)\le \det(A^TA)\det(B^TB).\]
Find all positive integers \( a, b, c \) satisfying
\[
3^a + 5^b = 2^c.
\]
Does there exist a constant \(\varepsilon>0\) such that for each positive integer \(n\) and each subset \(A\) of \(\{1,2,\ldots,n\}\) with \(\lvert A\rvert<\varepsilon n\), there exists an artihmetic progression \(S\) in \(\{1,2,\ldots,n\}\) such that \( S\cap A=\emptyset\) and \(\lvert S\rvert >\varepsilon n\)?
For \( \theta>0 \), let
\[
f(\theta) = \sum_{n=1}^{\infty} \left( \frac{1}{n+ \theta} – \frac{1}{n+ 3\theta} \right).
\]
Find \( \sup_{\theta > 0} f(\theta) \).
Does there exist infinitely many positive integers \(n\) such that the first digit of \(2^n\) is \(9\)?
Suppose that \( f : (2, \infty) \to (-2, 2) \) is a continuous function and there exists a positive constant \( m \) such that \( | 1 + xf(x) + (f(x))^2 | \leq m \) for any \( x > 2 \). Prove that, for any \( x > 2 \),
\[
\left| f(x) – \frac{\sqrt{x^2 -4}-x}{2} \right| \leq 6 \sqrt{m}.
\]
Prove (or disprove) that exactly one of the following is true for every subset \(A\) of \(\{ (i,j): i,j\in\{1,2,\ldots,n\}, i\neq j\}\).
(i) There exists a sequence of distinct integers \(i_1,i_2,\ldots,i_k\in \{1,2,\ldots,n\}\) for some integer \(k>1\) such that \( (i_1,i_2), (i_2,i_3),\ldots,(i_{k-1},i_k), (i_k,i_1)\in A\).
(ii) There exists a collection of finite sets \( A_1,A_2,\ldots,A_n\) such that for all distinct \(i,j\in\{1,2,\ldots,n\}\), \((i,j)\in A\) if and only if \( \lvert A_i\cap A_j\rvert > \frac12 \lvert A_i\rvert \) and \( \lvert A_i\cap A_j\rvert \le \frac12 \lvert A_j\rvert \)
For an integer \( n \geq 4 \), find the solutions of the equation
\[
\sum_{k=1}^n \frac{\sin \frac{k\pi}{n+1}}{\sin (\frac{k\pi}{n+1} -x)} = 0.
\]
Let \(a_1,a_2,\ldots,a_n\) be distinct points in \(\mathbb R^4\). Does there exist a non-zero polynomial \(P(x_1,x_2,x_3,x_4)\) such that
(1) the degree of \(P\) is at most \(\lceil\sqrt{5} n^{1/4}\rceil\) and
(2) \(P(a_i)=0\) for all \(i=1,2,\ldots,n\)?