Understand the problem

Let $a, b, c$ be the lengths of sides of a (possibly degenerate) triangle. Prove the inequality
$$(a+b)\sqrt{ab}+(a+c)\sqrt{ac}+(b+c)\sqrt{bc} \geq (a+b+c)^2/2.$$

Let $a, b, c$ be the lengths of sides of a triangle. Prove the inequality
$$(a+b)\sqrt{ab}+(a+c)\sqrt{ac}+(b+c)\sqrt{bc} \geq (a+b+c)^2/2.$$

Source of the problem

Caucasus Mathematical Olympiad

Topic
Inequalities
Difficulty Level
Easy
Suggested Book
An Excursion in Mathematics

Start with hints

Do you really need a hint? Try it first!

The inequality is homogeneous, hence it might be a good idea to make the substitutions a=2R\sin A etc. The triangle inequality might also come in handy.
As the triangle is allowed to be degenerate, we have a+b\ge c. This implies ac+bc\ge c^2.
Summing, we get 2(ab+bc+ca)\ge a^2+b^2+c^2. It is possible to use the AM-GM inequality here to get the LHS of the inequality. Also, show that this implies 2(ab+bc+ca)\ge\frac{(a+b+c)^2}{2}.
Note that (a+b)\sqrt{ab}\ge 2ab by AM-GM. Hence, we get LHS\ge 2(ab+bc+ca)\ge a^2+b^2+c^2\ge\frac{(a+b+c)^2}{2}

Watch video

Connected Program at Cheenta

Math Olympiad Program

Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year. Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.

Similar Problems

Squares and Triangles | AIME I, 2008 | Question 2

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2008 based on Squares and Triangles.

Percentage Problem | AIME I, 2008 | Question 1

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2008 based on Percentage. you may use sequential hints.

Smallest Positive Integer | PRMO 2019 | Question 14

Try this beautiful problem from the Pre-RMO, 2019 based on Smallest Positive Integer. You may use sequential hints to solve the problem.

Circles and Triangles | AIME I, 2012 | Question 13

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2012 based on Circles and triangles.

Complex Numbers and Triangles | AIME I, 2012 | Question 14

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2012 based on Complex Numbers and Triangles.

Triangles and Internal bisectors | PRMO 2019 | Question 10

Try this beautiful problem from the Pre-RMO, 2019 based on Triangles and Internal bisectors. You may use sequential hints to solve the problem.

Angles in a circle | PRMO-2018 | Problem 80

Try this beautiful problem from PRMO, 2018 based on Angles in a circle. You may use sequential hints to solve the problem.

Linear Equations | AMC 8, 2007 | Problem 20

Try this beautiful problem from Algebra based on Linear equations from AMC-8, 2007. You may use sequential hints to solve the problem.

Digit Problem from SMO, 2012 | Problem 14

Try this beautiful problem from Singapore Mathematics Olympiad, SMO, 2012 based on digit. You may use sequential hints to solve the problem.

Problem on Semicircle | AMC 8, 2013 | Problem 20

Try this beautiful problem from AMC-8, 2013, (Problem-20) based on area of semi circle.You may use sequential hints to solve the problem.