deutschland
by smileapple, Aug 28, 2024, 7:12 AM
mein Herz in Flammen
will dich lieben und verdammen
dein Atem kalt
so jung und doch so alt
deine Liebe
ist Fluch und Segen
meine Liebe
kann ich dir nicht geben
Problem (Germany 2017/5). Prove that for all non-negative numbers
, 
, 
satisfying 
, one has ![\[ 1 \le \frac{x}{1-yz}+\frac{y}{1-zx}+\frac{z}{1-xy} \le \frac{9}{8}. \]](//latex.artofproblemsolving.com/e/f/f/eff8ef2c56398c98f398bef5383294a075eb5ad8.png)
Solution. For clarity we shall use the dummy variables
, 
, 
instead of , , .
The lower bound is clear as
Now, set
. The line 
describes the line tangent to 
at 
, and we observe that
which clearly holds on ![$[0,1]$](//latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
. It then follows by AM-GM that ![\[\sum_{\text{cyc}}\frac{a}{1-bc}\le\sum_{\text{cyc}}\frac{a}{1-\frac{(1-a)^2}{4}}=f(a)+f(b)+f(c)\le\frac3{64}\sum_\text{cyc}(21a+1)=\frac98,\]](//latex.artofproblemsolving.com/9/5/f/95fe31ab0cd2e0603f21f11dfe6b5fd7ef4d52f9.png)
as desired. 
will dich lieben und verdammen
dein Atem kalt
so jung und doch so alt
deine Liebe
ist Fluch und Segen
meine Liebe
kann ich dir nicht geben
Problem (Germany 2017/5). Prove that for all non-negative numbers
, 
, 
satisfying 
, one has ![\[ 1 \le \frac{x}{1-yz}+\frac{y}{1-zx}+\frac{z}{1-xy} \le \frac{9}{8}. \]](http://latex.artofproblemsolving.com/e/f/f/eff8ef2c56398c98f398bef5383294a075eb5ad8.png)
Solution. For clarity we shall use the dummy variables
, 
, 
instead of , , .The lower bound is clear as
Now, set
. The line 
describes the line tangent to 
at 
, and we observe that
which clearly holds on ![$[0,1]$](http://latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
. It then follows by AM-GM that ![\[\sum_{\text{cyc}}\frac{a}{1-bc}\le\sum_{\text{cyc}}\frac{a}{1-\frac{(1-a)^2}{4}}=f(a)+f(b)+f(c)\le\frac3{64}\sum_\text{cyc}(21a+1)=\frac98,\]](http://latex.artofproblemsolving.com/9/5/f/95fe31ab0cd2e0603f21f11dfe6b5fd7ef4d52f9.png)
as desired. 
January 2025
November 2024