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January 2nd, 2019, 12:50 PM  #1 
Newbie Joined: Jan 2019 From: gd Posts: 1 Thanks: 0  Inequality problem
Let triangle with side a,b,c such that a+b+c=10. Prove this in.png I need help with this problem by use of triangle inequality, AG or similar inequality that I know. So I ask for hint. Last edited by skipjack; January 3rd, 2019 at 06:42 AM. 
January 2nd, 2019, 01:15 PM  #2  
Math Team Joined: May 2013 From: The Astral plane Posts: 2,093 Thanks: 853 Math Focus: Wibbly wobbly timeywimey stuff.  Quote:
So: $\displaystyle a + b + c = 10$ Here's the inequality in the image: $\displaystyle a^2 + b^2 + c^2 + \frac{2}{5} abc < 50$ Can we take it that you have to derive the inequality from the first equation? Dan Last edited by skipjack; January 3rd, 2019 at 06:44 AM.  
January 2nd, 2019, 06:04 PM  #3 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,923 Thanks: 1123 Math Focus: Elementary mathematics and beyond 
I've posted the image inline.

January 2nd, 2019, 06:23 PM  #4 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,923 Thanks: 1123 Math Focus: Elementary mathematics and beyond 
Because the variables in the inequality can be interchanged with respect to value and position the equation has cyclic symmetry and an extremum is reached when all three sides are equal. After you come up with a value for that all you have to do is devise another triangle to show that the LHS of the inequality is less than the previously calculated value.

January 3rd, 2019, 08:03 AM  #5 
Global Moderator Joined: Dec 2006 Posts: 20,388 Thanks: 2015  

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