Show that
$$\displaystyle \tan(3\pi/11) + 4\sin(2\pi/11) =\sqrt{11}$$
[Solution (math.stackexchange link).]
$$\displaystyle \tan(3\pi/11) + 4\sin(2\pi/11) =\sqrt{11}$$
[Solution (math.stackexchange link).]
bridge hands and math and computer science and programming and puzzles and etc and etc.
Sunday, October 14, 2018
Wednesday, October 3, 2018
Yet another inequality
$x_1, x_2, \dots, x_n$ are $n$ positive real numbers with sum $S$, $n \gt 1$.
Show that
$$ \sum_{i=1}^{n} \frac{x_i}{S - x_i} \ge \frac{n}{n-1}$$
Solution [Click here to expand/collapse]
Using AM >= GM we have that
$\sum y_i \sum \frac{1}{y_i} \ge n^2$
Applying this to (wlog, assuming $S = 1$) $$\left(\sum_{i=1}^{n} \frac{1}{1-x_i}\right) \left(\sum_{i=1}^{n} (1 - x_i)\right) \ge n^2$$ i.e $$ \sum_{i=1}^{n} \frac{1}{1-x_i} \ge \frac{n^2}{n-1}$$ And so $$ \sum_{i=1}^{n} \left(\frac{1}{1-x_i} - 1\right) \ge \frac{n^2}{n-1} -n$$ Which gives us what we want to prove.
Show that
$$ \sum_{i=1}^{n} \frac{x_i}{S - x_i} \ge \frac{n}{n-1}$$
Solution [Click here to expand/collapse]
Using AM >= GM we have that
$\sum y_i \sum \frac{1}{y_i} \ge n^2$
Applying this to (wlog, assuming $S = 1$) $$\left(\sum_{i=1}^{n} \frac{1}{1-x_i}\right) \left(\sum_{i=1}^{n} (1 - x_i)\right) \ge n^2$$ i.e $$ \sum_{i=1}^{n} \frac{1}{1-x_i} \ge \frac{n^2}{n-1}$$ And so $$ \sum_{i=1}^{n} \left(\frac{1}{1-x_i} - 1\right) \ge \frac{n^2}{n-1} -n$$ Which gives us what we want to prove.