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### If $a+b+c+abc=4,$ prove $\frac{1}{\sqrt{a^2+4bc}}+\frac{1}{\sqrt{b^2+4ac}}+\frac{1}{\sqrt{c^2+4ba}}\ge \frac{5}{4}.$

Remark 1: This is my third proof (the idea) which is better than my two old proofs. Remark 2: Some years ago, I used a similar idea for the problem: Let $a, b, c > 0$ with $a + b + c = 3$. Prove ...
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### Monochromatic $4$-Cycle in Bipartite Complete Graph

Let one part be $\{\,v_1, v_2, \ldots, v_n\,\}$ and another part be $\{\,u_1, u_2, \ldots, u_n\,\}$. Let's show that for $n = 5$ the graph always has a monochromatic $4$-cycle. Suppose the opposite. ...
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So regrading the question you have in their solution : Let me explain in such a way : a 3 digit number in base 10 (what we use currently : the decimal representation) is expressed as $a 10^0 + b 10^1 +... • 537 1 vote Accepted ### German MO combinatorics problem 1995 Just to clarify user 10354138 answer beacause there might be a small mistake: $$\frac{k}{2n-k+2}+\frac{2n-k+1}{k+1}\ge 2\cdot \frac{n+1}{n+2}$$$\$\iff \frac{k}{2n-k+2}+\frac{2n-k+1}{k+1}-\frac{2(n+1)}{...
• 26

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