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 Mar13 comment Evaluating the integral $\int \frac{1}{x+ \text{ ln }(x)}dx$ wolframalpha.com/input/… Mar13 comment The formula of Eclidean distance to a hyperplane. If you consider $H$ as a hyperplane in $\mathbb{R}^{mn}$ and $X$ and $Y$ are two coordinates (i.e. $X,\,Y\in\mathbb{R}$), then $mn$ real numbers, not just $X$ and $Y$, are necessary to specify a point in $\mathbb{R}^{mn}$. Mar12 comment Weaker version of the A.M.-G.M. inequality Mar12 comment Positive numbers inequality The above answer can be rescued if you show $\sum iy_i\ge \sum i\cdot\frac1n$ when $0< y_1\le y_2\le \cdots \le y_n$ and $y_1+\cdots+y_n=1$. Mar11 comment Positive numbers inequality Where did you use $y_1\le y_2\le \cdots \le y_n$? and why is $\left(\frac2{n(n+1)}\sum y_i\right)^{\log_2 3}\ge \frac1{n^2}$ (the last inequality)? Feb17 comment prove that $f_n = 37111111…111$ is never prime Try Factor[371], Factor[3711], Factor[37111] etc. (up to $n=10$) with WolframAlpha. Feb17 comment common probability Was your sampling not repeated? The above data only suggest that the data from the lab and that from the field are different. Are there any reasons for believing that they are "same things"? Feb14 comment Solving a system of equations with an inequality constraint What sort of functions are $f_1$, $f_2$ and $f_3$? Feb13 comment Model a chemical phenomenom Yes, periodic boundary conditions are often used in mathematical / physical modeling and are a good starting point. Feb12 comment Model a chemical phenomenom I think the lattice should reflect the actual shape of the receptor, but you could start with a rectangle (if it's 2D) or a cuboid (if it's 3D). Feb10 comment Express $\binom{n+2}{k}$ according to $\binom{n}{k}$ Similar question: math.stackexchange.com/questions/649567/… Feb5 comment Hard inequality with conditions Then you can construct a counterexample easily, following the above argument. (such as in the case $b_2/g_2=r_2>1$ and $b_3/g_3=r_3<1$) Feb5 comment Find one solution for system of inequalities (if exits) Thank you, I corrected the note. Feb4 comment Find one solution for system of inequalities (if exits) $A(1)=0.020302010304655=20302010304655/10^{15}$ and not equal to $326/160723$ etc. I don't see why you want to convert a finite decimal to a different rational number (circulating decimal). Feb4 comment Find one solution for system of inequalities (if exits) Really? Even though $0.112843485160579$ was changed to $0.112843485160578$? You might need to refresh the page. Feb4 comment Find one solution for system of inequalities (if exits) Thank you, I changed $A(8)$ to satisfy $A(1)+\cdots+A(8)=1$ and updated the results accordingly. Feb4 comment Find one solution for system of inequalities (if exits) @user64494 I'll check, but I think that can be dealt with without affecting inequality signs. Nov18 comment Finding an angle between side and a segment from specified point inside an equilateral triangle It's simple: note that arc length = angle (in radians) when the radius of the circle is 1 (i.e. unit circle). So when the arc length is $u$, the chord length is $2\sin(u/2)$. Nov15 comment Find the line that intercepts the lines $r$ and $s$ and forms congruent angles to the coordinate axes You're almost done. From the first part of your argument, if you set $\overrightarrow{AB}=(x,\,y,\,z)$, you get $|x|=|y|=|z|$. (The angle is not 45 degrees by the way, but the argument still stands.) Then you have four cases to consider: $x=y=z$, $x=-y=z$, $x=y=-z$, $x=-y=-z$. Nov15 comment Finding an angle between side and a segment from specified point inside an equilateral triangle I used $\cos 3\theta=4\cos^3\theta-3\cos\theta$ and $\sin 2\theta=2\sin\theta\cos\theta$. So $\cos 3\theta=\sin 2\theta$ becomes $\cos\theta(4\cos^2\theta-3)=\cos\theta(2\sin\theta)$, and dividing by $\cos\theta$ (which is not zero) we get $4(1-\sin^2\theta)-3=2\sin\theta$.