## Top new questions this week:

### How prove this inequality $\sin{\sin{\sin{\sin{x}}}}\le\frac{4}{5}\cos{\cos{\cos{\cos{x}}}}$

Nice Question: let $x\in [0,2\pi]$, show that: $$\sin{\sin{\sin{\sin{x}}}}\le\dfrac{4}{5}\cos{\cos{\cos{\cos{x}}}}?$$ I know this follow famous problem(1995 Russia Mathematical olympiad) …

(inequality)

### Conjecture $_2F_1\left(\frac14,\frac34;\,\frac23;\,\frac13\right)=\frac1{\sqrt{\sqrt{\frac4{\sqrt{2-\sqrt[3]4}}+\sqrt[3]{4}+4}-\sqrt{2-\sqrt[3]4}-2}}$

Using a numerical search on my computer I discovered the following inequality: $$\left|\,{_2F_1}\left(\frac14,\frac34;\,\frac23;\,\frac13\right)-\rho\,\right|<10^{-20000},\tag1$$ where $\rho$ is …

(calculus) (special-functions) (closed-form) (conjectures) (hypergeometric-function)

### How to prove there exists $c$ such $f(c)f'(c)+f''(c)=0$

Nice Question: let $f(x)$ have two derivative on $[0,1]$,and such $$f(0)=2,f'(0)=-2,f(1)=1$$ show that: there exist $c\in(0,1)$,such $$f(c)f'(c)+f''(c)=0$$ my try: since …

(analysis)

### $\sqrt{7\sqrt{7\sqrt{7\sqrt{7\sqrt{7\cdots}}}}}$ approximation

Is there any trick to evaluate this or this is an approximation, I mean I am not allowed to use calculator. $$\sqrt{7\sqrt{7\sqrt{7\sqrt{7\sqrt{7\cdots}}}}}$$

### How to prove that either $2^{500} + 15$ or $2^{500} + 16$ isn't a perfect square?

How would I prove that either $2^{500} + 15$ or $2^{500} + 16$ isn't a perfect square?

(elementary-number-theory) (proof-strategy)

### Integral $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{x\sin{x}}{1+\cos^4{x}}dx$

Question: Find the integral $$I=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\dfrac{x\sin{x}}{1+\cos^4{x}}dx$$ my try: since $$I=2\int_{0}^{\frac{\pi}{2}}\dfrac{x\sin{x}}{1+\cos^4{x}}dx$$ then I can't. I …

(integration) (definite-integral)

### Can $x^3+3x^2+1=0$ be solved using high school methods?

I encountered the following problem in a high-school math text, which I wasn't able to solve using factorization/factor theorem: Solve $x^3+3x^2+1=0$ Am I missing something here, or is indeed a more …

(calculus) (algebra-precalculus) (roots)

## Greatest hits from previous weeks:

### What should I learn first, Mathematica or MatLab?

I have a non-professional interest in math and I would like to be able to be curious and play around with some math that I'm learning about (both here and from calculus-and-higher level math classes). …

(math-software) (mathematica) (matlab)

### Probability of 3 Heads in 10 Coin Flips

What's the probability of getting 3 heads and 7 tails is one flips a fair coin 10 times. I just can't figure out how to model this correctly.

(probability)

### Approximating $1/z$ by polynomials

Let $C=\{\mathrm e^{\mathrm it}, 0\le t\le 3\pi/2\}$ and $f(z)=1/z$. By Runge's theorem, there is a sequence of polynomials $p_n(z)$ such that $$\lim_n p_n(z)=f(z)$$ uniformly on $C$. Does anyone …

(complex-analysis) (approximation)

### Corestriction map in lie algebra cohomology

Given a lie algebra $\mathfrak{g}$ over a field $k$, we can define the cohomology groups of $\mathfrak{g}$ as follows: $$H^n(\mathfrak{g},k):=\mathrm{Ext}_{U(\mathfrak{g})}^n(k,k)$$ where …

(abstract-algebra) (lie-algebras) (cohomology) (group-cohomology)
Given a surface $f(x,y,z)=0$, how could you determine that it's symmetric about some plane, and, if so, how would you find this plane. The special case where $f$ is a polynomial is of some interest. …