LHF

 41 Problem from the 2020 Latvian “Sophomore's Dream” competition 18 Proof of $(1-x)x^n \leq \frac{n^n}{\left(n+1\right)^{n+1}}$ without use of derivatives 12 Prove that $p(x)=x^4-x+\frac{1}{2}$ has no real roots. 12 $f(x)=\frac{\sin x}{x}$, prove that $|f^{(n)}(x)|\le \frac{1}{n+1}$ 10 Prove that $\int_0^1\sqrt{f^4(x)+(\int_0^1f(t)\, dt)^4}\, dx\le \sqrt{2}\int_0^1f^2(x)\,dx$

### Reputation (8,178)

 +10 Proof of $(1-x)x^n \leq \frac{n^n}{\left(n+1\right)^{n+1}}$ without use of derivatives +10 Hard inequality for positive numbers +10 If $I_n=\int_0 ^1{\frac{x^{n+1}}{x+3}}dx$, prove $\lim_{n \to \infty} n I_n=\frac{1}{4}$ +10 Find all the functions $f\left( x+y\right) +xy=f\left( x\right) f\left( y\right)$

### Questions (11)

 24 Prove that $e^\pi+\frac{1}{\pi} < \pi^e+1$ 4 Evaluate $\lim\limits_{n\to\infty}\left(\sqrt [n+1]{\frac {a_{n+1}}{b_{n+1}}}-\sqrt[n]{\frac {a_n}{b_n}}\ \right)$ 3 Given a recurrence formula, evaluate $\lim\limits_{n\to \infty} n^2 x_n^3$ 3 Asymmetric inequality in three variables $\frac{3(a+b)^2(b+c)^2}{4ab^2c} \geq 7+\frac{5(a^2+2b^2+c^2)}{(a+b)(b+c)}$ 2 Maximize product $(a^3-a^2+2)(b^3-b^2+2)(2c^3+5c^2+9)$

### Tags (92)

 197 inequality × 76 95 definite-integrals × 15 165 calculus × 39 81 algebra-precalculus × 32 159 real-analysis × 32 70 sequences-and-series × 27 144 limits × 45 69 contest-math × 12 123 integration × 18 59 a.m.-g.m.-inequality × 15

### Bookmarks (15)

 24 Prove that $e^\pi+\frac{1}{\pi} < \pi^e+1$ 9 $f(x)=\frac{\sin x}{x}$, prove that $|f^{(n)}(x)|\le \frac{1}{n+1}$ [duplicate] 6 Prove that $\int_0^1\sqrt{f^4(x)+(\int_0^1f(t)\, dt)^4}\, dx\le \sqrt{2}\int_0^1f^2(x)\,dx$ 4 Minimum three variable expression $\frac{(a^2+b^2+c^2)^3}{(a+b+c)^3|(a-b)(b-c)(c-a)|}$ 3 Asymmetric inequality in three variables $\frac{3(a+b)^2(b+c)^2}{4ab^2c} \geq 7+\frac{5(a^2+2b^2+c^2)}{(a+b)(b+c)}$