A goat tied to a corner of a rectangle

Why is ${2n \choose n}/2 $ odd if and only if $n=2^k$? I conjecture this inequality $\sqrt[4]{\frac{(xy+yz+xz)(x^2+y^2+z^2)}{9}}\ge\sqrt[3]{\frac{(x+y)(y+z)(z+x)}{8}}$ Derivative of the binomial $\binom x n$ with respect to $x$ Calculate $\int_0^\infty\frac{\sin(x)\log(x)}{x}\mathrm dx$. Are my calculations of a recursive prime-generating function based on logarithms correct? Series of binomial coefficients: $\sum\limits_{n=k}^{\infty}{\binom nk}x^n=\frac{x^k}{(1-x)^{k+1}}$ How does Cramer's rule work? Do two empty sets have any elements in common? Summation of reciprocals Elementary solution of exponential Diophantine equation $2^x - 3^y = 7$. A ping-pong ball lying inside a wine glass Function with derivative-like property: $f(ab) = af(b) + bf(a)$