What methods one could use to prove inequalities IMO problems?
Off the top of your head, you might want to shout out that AM-GM inequality, Cauchy Schwarz inequality. Jensen's inequality are among the "hot" and popular methods that you would consider using to effectively prove the inequality hard IMO problem.
Showing posts with label Cauchy Schwarz inequality. Show all posts
Showing posts with label Cauchy Schwarz inequality. Show all posts
2015 IMO contest problem: If$x,\,y,\,z$ are real numbers such that $x+2y+3z=6$ and $x^2+4y^2+9z^2=12$, evaluate $xyz$.
If$x,\,y,\,z$ are real numbers such that $x+2y+3z=6$ and $x^2+4y^2+9z^2=12$, evaluate $xyz$.
There is something that this Olympiad problem might trick us because it's obvious that $x^2,\,4y^2,\,9z^2$ are squares of $x,\,2y,\,3z$ and hence, one has reason to believe that the proper first step in solving this problem is to square the first given equation:
$x+2y+3z=6$
There is something that this Olympiad problem might trick us because it's obvious that $x^2,\,4y^2,\,9z^2$ are squares of $x,\,2y,\,3z$ and hence, one has reason to believe that the proper first step in solving this problem is to square the first given equation:
$x+2y+3z=6$
Compare M and N
Given that $p,\,q,\,r,\,s,\,a,\,b$ are positive real numbers with $M=\sqrt{ap+br}\cdot \sqrt{\dfrac{q}{a}+\dfrac{s}{b}}$ and $N=\sqrt{pq}+\sqrt{rs}$. Compare $M$ and $N$.
This question has the best form of quality because it allows us to approach it in at least 3 different ways, 3 of which that are enlighting and inspiring.
This question has the best form of quality because it allows us to approach it in at least 3 different ways, 3 of which that are enlighting and inspiring.
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