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Mathematics
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LHF
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42
Problem from the 2020 Latvian "Sophomore's Dream" competition
Mar 2, 2020
26
Prove that $e^\pi+\frac{1}{\pi} < \pi^e+1$
Feb 24, 2020
18
Proof of $(1-x)x^n \leq \frac{n^n}{\left(n+1\right)^{n+1}}$ without use of derivatives
Feb 16, 2020
12
Prove that $p(x)=x^4-x+\frac{1}{2}$ has no real roots.
Mar 6, 2020
12
$f(x)=\frac{\sin x}{x}$, prove that $|f^{(n)}(x)|\le \frac{1}{n+1}$
Mar 1, 2020
10
Prove that $\int_0^1\sqrt{f^4(x)+(\int_0^1f(t)\, dt)^4}\, dx\le \sqrt{2}\int_0^1f^2(x)\,dx$
Mar 11, 2020
10
Evaluate $\lim_{n\to\infty}\sum_{k=1}^n\frac{k^3}{\sqrt{n^8+k}}$ & $\lim_{n\to\infty}n\bigg(\sum_{k=1}^n \frac{k^3}{\sqrt{n^8+k}}-\frac{1}{4}\bigg)$.
Feb 28, 2020
8
Find $\lim_{n\to \infty}\int _0^{\frac{\pi}{2}} \sqrt{1+\sin^nx}$
Feb 23, 2020
8
Show that $\int_0^{\pi}\sqrt{9\sin^2t+16\cos^2t}\leq4\pi$
Mar 8, 2020
7
Find $\lim\limits_{n \to \infty} n \int_2^e (\ln x)^n \, dx$.
Mar 4, 2020