Given real numbers $a_1, a_2, a_3, a_4$, if $a_1 > (a_1 + a_2 + a_3 + a_4)/4$ , then there exists $i \in \{2, 3, 4\}$ such that $a_i < (a_1 + a_2 + a_3 + a_4)/4$.

I could solve the problem in my mind, but I can't put the solution on the paper. I just have difficulty in writing the solution of the question.


closed as off-topic by Namaste, José Carlos Santos, Ken Duna, Brian Borchers, Math Lover Nov 21 '17 at 16:53

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  • 3
    $\begingroup$ Are you asking for a proof of this? All I see is a sentence. Also, you should move the actual question to the body of the post. And if you can "solve the problem in [your] mind", you should at least try to explain how you did so in the post. $\endgroup$ – Kevin Long Nov 20 '17 at 17:15
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    $\begingroup$ Just to be clear, is it $a_1 + a_2 + a_3 + a_4/4$ or $\frac{a_1 + a_2 + a_3 + a_4}{4}$? $\endgroup$ – Math Lover Nov 20 '17 at 17:25
  • $\begingroup$ Are you sure you're not missing parentheses around some parts? $\endgroup$ – user499203 Nov 20 '17 at 17:25
  • $\begingroup$ Yes, I'm asking for a proof. I forgot to use parentheses where it should be like (a1+a2+a3+a4)/4 at both of the inequality. Sorry for asking lack of information. $\endgroup$ – Hasan Kayhan Nov 20 '17 at 17:30
  • $\begingroup$ suppose it is not true. Quickly you will run into a contradiction. $\endgroup$ – Doug M Nov 20 '17 at 17:31

Hint: Suppose not. Then we have $$a_2 \ge \frac{a_1+a_2+a_3+a_4}4$$ and the same for $a_3$ and $a_4$. Now add up $a_1+a_2+a_3+a_4$ and see what happens.

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    $\begingroup$ Thank you very much, it helped me a lot. $\endgroup$ – Hasan Kayhan Nov 20 '17 at 17:41

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