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Problem - S. Stewart (Australia) Prove [\sum_{n=1}^\infty\frac{\bar{H}_{2n}}{n^2}=\frac{3\zeta(3)}{4}] where [\bar{H}n=\sum{k=1}^n\frac{(-1)^{k-1}}{k}] is the nth skew-harmonic number. Soluti...
Problem - S. Sharma (India) Show that [\int_0^\infty\frac{x\sinh(x)}{3+4\sinh^2(x)}\,\mathrm dx=\frac{\pi^2}{24}.] Solution Call the integral in question $I$. Let $x\mapsto \ln(x)$, so that $\m...
Elliptic curves are a serious topic of study in pure mathematics, they are at the forefront of modern research in number theory, and they have been used to great effect in cryptographic systems. B...
Problem - M. Tetiva (Romania) Evaluate [\sum_{n=1}^\infty\left(H_n-\ln(n)-\gamma-\frac{1}{2n}\right)] where $H_n=\sum_{k=1}^n\frac{1}{k}$ and $\gamma$ is the Euler-Mascheroni constant. Solution...
Problem - P. Perfetti (Italy) Prove [\int_1^\infty\frac{\ln\left(x^4-2x^2+2\right)}{x\sqrt{x^2-1}}\,\mathrm dx=\pi\ln(2+\sqrt{2}).] Solution Call the stated integral $I$ and let $x=\sqrt{u+1}$,...
Problem - G. Apostolopoulos (Greece) Prove [\sum_{k=2}^\infty\frac{1}{k}\int_0^1\left{\frac{1}{\sqrt[k]{x}}\right}\,\mathrm dx=\gamma] where $\gamma$ is the Euler-Mascheroni constant. Solution ...
Problem - T. Amdberhan and V. Moll (USA) Show that [\int_0^\infty\frac{\cos(x)\sin\left(\sqrt{1+x^2}\right)}{\sqrt{1+x^2}}\,\mathrm dx=\frac{\pi}{4}.] Solution Call the integral in question $I$...
Problem - S. P. Adriopoulos (Greece) Let $x$ be a real number between $0$ and $1$. Prove [\prod_{n=1}^\infty(1-x^n)\geq\text{exp}\left(\frac{1}{2}-\frac{1}{2(1-x)^2}\right).] Solution Call thes...
Problem - C. I. Valean (Romania) Prove that [\int_0^1\frac{x\ln(1+x)}{1+x^2}\,\mathrm dx=\frac{\pi^2}{96}+\frac{\ln^2(2)}{8}.] Solution First we define a new function called $I(\alpha)$ as $\d...
Problem Find all $n\in\mathbb{N}$ such that [f(x)=\prod_{k=1}^nf^{(k)}(x)] has a real solution of the form $f(x)=ax^r$ with $a\neq0$. Solution We show that such an $f$ exists if $n\neq1$ or $n...
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