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### Prove $\frac{a+b}{c} + \frac{b+c}{a}+\frac{a+c}{b} + 6 \ge 2\sqrt2(\sqrt\frac{1-a}{a} + \sqrt\frac{1-b}{b} + \sqrt\frac{1-c}{c})$ when $a + b + c = 1$

From a+b+c=1 we get that a+b=1-c etc, so the LHS is equal to $\frac{1-a}{a}+\frac{1-b}{b}+\frac{1-c}{c}+6$. We can use the notation $x=\frac{1-a}{a}, y=\frac{1-b}{b}, z=\frac{1-c}{c}$ and now the ...
Accepted

### Combinatorics board olympiad problem Rioplatenes P3

Let's assume that $302$ colors have been used on the board. Since, no rows can admit $5$ (or more) colors, at least $2$ rows must contain exactly $4$ colors so that all the ($4 \times 2$) colors are ...
Accepted

1 vote
Accepted

### Prove $\sum\limits_{\mathrm{cyc}} \sqrt{5a+5b+8ab}\ge 3\sqrt{2}+2\sqrt{5}$ for $ab+bc+ca=1$

A proof using Holder inequality. By Holder inequality, we have \begin{align*} &\left(\sum_{\mathrm{cyc}} \sqrt{5a + 5b + 8ab}\right)^2 \sum_{\mathrm{cyc}} (5a + 5b + 8ab)^2\Big(5a + 5b + (6\sqrt{...
1 vote

### Prove that $\sum\limits_{cyc}\sqrt{\frac{8ab+8ac+9bc}{(2b+c)(b+2c)}}\geq5$

My second proof. In Michael Rozenberg's answer, Nguyenhuyen_AG's idea is the so-called isolated fudging. I got the same form using the isolated fudging technique. We need to prove that \sqrt{\frac{...

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