Difference between revisions of "1971 Canadian MO Problems/Problem 2"
Airplanes1 (talk | contribs) (Created page with "== Problem == Let <math>x</math> and <math>y</math> be positive real numbers such that <math>x+y=1</math>. Show that <math>\left(1+\frac{1}{x}\right)\left(1+\frac{1}{y}\right)\ge...") |
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which is true since by AM-GM, we get: <cmath>x^2 + y^2 \ge 2xy</cmath> <cmath>x^2 + 2xy + y^2 \ge 4xy</cmath> <cmath> (x+y)^2 \ge 4xy</cmath> and we are given that <math> x + y = 1</math>, so <cmath> (x+y)^2 \ge 4xy \Rightarrow 1 \ge 4xy </cmath> | which is true since by AM-GM, we get: <cmath>x^2 + y^2 \ge 2xy</cmath> <cmath>x^2 + 2xy + y^2 \ge 4xy</cmath> <cmath> (x+y)^2 \ge 4xy</cmath> and we are given that <math> x + y = 1</math>, so <cmath> (x+y)^2 \ge 4xy \Rightarrow 1 \ge 4xy </cmath> | ||
| − | {{Old CanadaMO box|num-b= | + | {{Old CanadaMO box|num-b=1|num-a=3|year=1971}} |
[[Category:Intermediate Algebra Problems]] | [[Category:Intermediate Algebra Problems]] | ||
and 
be positive real numbers such that 
. Show that 
.
![\[\left(1+\frac{1}{x}\right)\left(1+\frac{1}{y}\right) \ge 9\]](http://latex.artofproblemsolving.com/9/9/4/994f610534f32e11542c22ce7d62c666e71f10bb.png)
![\[1 + \frac{1}{x} + \frac{1}{y} + \frac{1}{xy} \ge 9\]](http://latex.artofproblemsolving.com/d/7/2/d72332f85b7d65c7d98de71dce88dcc3e5dc2cbb.png)
![\[\frac{xy+x+y+1}{xy}\ge 9\]](http://latex.artofproblemsolving.com/4/b/4/4b4e1b576f9d2caa35a0c558fe529fb1c5f94ab4.png)
![\[\frac{xy+2}{xy}\ge 9\]](http://latex.artofproblemsolving.com/c/6/4/c64b7ff59727c015cc983ec01401024f9e483e22.png)
![\[1 + \frac{2}{xy}\ge 9\]](http://latex.artofproblemsolving.com/a/f/3/af3ba42a7e8a263d2db8346afbef22f0af47a170.png)
![\[\frac{2}{xy} \ge 8\]](http://latex.artofproblemsolving.com/d/3/1/d316fd1b6cb7599afe728a2f3a755807f9b63f1e.png)
![\[1\ge 4xy\]](http://latex.artofproblemsolving.com/f/c/4/fc47ad8e142635e5d5a3639efe13bbc9de5074ca.png)
which is true since by AM-GM, we get: ![\[x^2 + y^2 \ge 2xy\]](http://latex.artofproblemsolving.com/7/8/6/786ced427c89df85dba5ab567e21d964d5f39f76.png)
![\[x^2 + 2xy + y^2 \ge 4xy\]](http://latex.artofproblemsolving.com/c/d/7/cd7164f867df4d279d4d9fb8db7fabc174644048.png)
![\[(x+y)^2 \ge 4xy\]](http://latex.artofproblemsolving.com/4/1/3/4136e6b02c70c2d58c684db6b4d5e65301adc95f.png)
and we are given that 
, so ![\[(x+y)^2 \ge 4xy \Rightarrow 1 \ge 4xy\]](http://latex.artofproblemsolving.com/4/4/5/445121f06d5c4056f8f92f8e28f050e6b06f34c4.png)
