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 Nov14 comment Finding an angle between side and a segment from specified point inside an equilateral triangle A proof of $d_3+d_5=d_9$: it suffices to show $\sin 54^{\circ}-\sin 18^{\circ}=\sin 30^{\circ}$, that is $2\cos 36^{\circ}\sin 18^{\circ}=1/2$. Using the fact that $\alpha=\sin 18^{\circ}$ satisfies $4(1-\alpha^2)-3=2\alpha$ (from $\cos 54^{\circ}=\sin 36^{\circ}$), we can show that $2(1-2\alpha^2)\alpha=1/2$. Nov14 comment Finding an angle between side and a segment from specified point inside an equilateral triangle This lemma is stated in www-math.mit.edu/~poonen/papers/ngon.pdf (pages 4-5). I'll check if it is really true. May13 comment Prove $|\vec{a_1}-\vec{b}|+ \cdots +|\vec{a_n}-\vec{b}| > n$ The condition given is $|\vec{b}|<1$, not $|\vec{b}|>1$. May1 comment Variance over two periods with known variances? A clear explanation, easily generalizable to cases with more than two sets. Apr23 comment Geometric interpretation for sum of fourth powers 30 comes from the fact that the Bernoulli number $B_4=-1/30$. See "Faulhaber's formula" in Wikipedia. Jan17 comment The volume of the solid obtained by revolving the region bounded by $y^2=x$ and $x=2{y}$ about the $y$-axis. The interval of integration is [0, 2], not [0, 1]; the result seems correct. Jan17 comment Let $a, b$ and $c$ be the lengths of the sides of an arbitrary triangle. Pick out the true statements. What will the value of $x$ be if (i) $a=b=c$, or (ii) $a=b\gg c$? Nov20 comment Negative real parts and of the solution of a polynomial and stable matrices This is called "the Routh-Hurwitz criterion". For a proof, see p.78 (Theorem 11) of this book. Nov18 comment Proof of Convergence: Babylonian Method $x_{n+1}=\frac{1}{2}(x_n + \frac{a}{x_n})$ @Clash en.wikipedia.org/wiki/… Nov15 comment Elegant way to prove this inequality You're welcome. Nov12 comment What is the integral of $x(1-x)^8$? Why $x=(x-1)+1$? Because I want to treat $x-1$ as sort of a unit, which makes the calculation easier. Nov12 comment What is the integral of $x(1-x)^8$? $(x-1+1)(x-1)^8=(x-1)\cdot (x-1)^8+1\cdot (x-1)^8$ follows from the distributive law. Nov11 comment Finding pairs of integers such that $x^2+3y$ and $y^2+3x$ are both perfect squares Symmetry ($x^2+3y$ and $y^2+3x$) suggests that without loss of generality we can assume $y\ge x$. Also notice that if $x\sim y$, $y^2+3x$ is not much larger than $y^2$, so it may be equal to $(y+1)^2$ or $(y+2)^2$... To discuss this rigorously you need to assume $y\ge x$. Nov10 comment Show that the value of a definite integral is unity As Dinesh points out, $g(x)=\frac{f(x)}{f(x)+f(6-x)}$ satisfies $g(x)+g(6-x)=1$ (more generally, $g(x)+g(a-x)=b$ where $a$ and $b$ are constants); this is the condition where you can use this integration trick. Nov10 comment Is $x^2+ax+a$ irreducible over ring $\mathbb{Z}$ of integers? @pedja I think in that case, the same method as posted by Bruno Joyal works (i.e. think in $\mathbb{F}_2[x]$). Nov10 comment Must every event have a probability? The probability that you can find life on Mars (or a Nessie in Loch Ness). I don't think it is "impossible to define any probability", but I think it is mathematically ill-defined (is it?). Nov10 comment Show that the value of a definite integral is unity Because $y$ is a dummy variable; please see my edited comment. Nov10 comment Show that the value of a definite integral is unity I posted an answer below. Nov10 comment Show that the value of a definite integral is unity In general: $\int_2^4 \frac{f(x)}{f(x)+f(6-x)}\,dx=1$. Nov9 comment Equivalent conditions to $0\leq x+\frac{1}{2}x(1-x)a\leq 1$? @JavaMan Not necessarily "opening downwards" -- if $a<0$.