Difference between revisions of "1961 IMO Problems/Problem 2"
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This can be proven because <math>a^2b^2\le\frac{a^4+b^4}{2}</math>. The equality holds when <math>a=b=c</math>, or when the triangle is equilateral. | This can be proven because <math>a^2b^2\le\frac{a^4+b^4}{2}</math>. The equality holds when <math>a=b=c</math>, or when the triangle is equilateral. | ||
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==Solution 2 By PEKKA== | ==Solution 2 By PEKKA== | ||
We firstly use the duality principle. | We firstly use the duality principle. |
Revision as of 19:03, 1 August 2022
Problem
Let ,
, and
be the lengths of a triangle whose area is S. Prove that
In what case does equality hold?
Solution
Substitute , where
This shows that the inequality is equivalent to .
This can be proven because . The equality holds when
, or when the triangle is equilateral.
Solution 2 By PEKKA
We firstly use the duality principle.
The LHS becomes
and the RHS becomes
If we use Heron's formula.
By AM-GM
Making this substitution
becomes
and once we take the square root of the area then our RHS becomes
Multiplying the RHS and the LHS by 3 we get the LHS to be
Our RHS becomes
Subtracting
we have the LHS equal to
and the RHS being
If LHS
RHS then LHS-RHS
LHS-RHS=
by the trivial inequality so therefore,
and we're done.
Video Solution
https://www.youtube.com/watch?v=ZYOB-KSEF3k&list=PLa8j0YHOYQQJGzkvK2Sm00zrh0aIQnof8&index=4 - AMBRIGGS