American Mathematical Monthly Problems
Solutions to selected problems from the American Mathematical Monthly. If you want to comment or share solutions, contact me at tommaso.mazzoli@tuwien.ac.at
Useful Resources
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Problem 12407 by an anonymous contributor (New Delhi, India) Let \(r\) be a positive real. Evaluate \[ \int_0^\infty \frac{x^{r-1}}{(1+x^2)(1+x^{2r})} \mathop{}\!\mathrm{d} x \] |
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Problem 12406 by R. Mortini and R. Rupp (Luxembourg/Germany) For fixed \(p \in \mathbb{R}\), find all functions \(f : [0, 1] \to \mathbb{R}\) that are continuous at 0 and 1 and satisfy \[ f(x^2) + 2p \cdot f(x) = (x + p)^2 \] for all \(x \in [0, 1]\). |
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Problem 12388 by A. Garcia (France) \[ \int_0^a \frac{\ln^2(x)\arctan(x)}{x^2-2\cos(\alpha)x+1} \mathop{}\!\mathrm{d} x \] |
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Problem 12243 by M. Lawrence Glasser (USA) \[ I_a = \int_0^a \frac{t}{\sinh t \, \sqrt{1-\frac{\sinh^2(t)}{\sinh^2(a)}}} \mathop{}\!\mathrm{d} t \] |
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Problem 12229 by Moubinool Omarjee (France) \[ 30240\left(\int_0^1 xf(x) \mathop{}\!\mathrm{d} x\right)^2 \le \int_0^1 \left(f''(x) \right)^2 \mathop{}\!\mathrm{d}x \] |
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Problem 12221 by N. Batir (Turkey) \[ \int_0^1 \frac{\ln(x^6+1)}{x^2+1} \mathop{}\!\mathrm{d} x = \frac{\pi\ln 6 }{2}- 3G \] where \(G\) is Catalan's constant. |
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Problem 12142 by J. A. Scott (UK) \[ \int_a^b ( f''(x))^2 \mathop{}\!\mathrm{d} x \ge \frac{980}{\big(8\sqrt 2 -1\big)^2} \frac{(f(a)+f(b))^2}{(b-a)^3} \] |