AoPS Academy
such that 
are the lengths of a right triangle.
from the domain 
and put those numbers into the given function f(x)=1/x.It gives us 1=1/2.But it's not true.So ,it can't be one-one function.But in the answer,it is one-one function.Would anyone enlighten me where is my fault? Thank you!
and 
are enclosed in a cylindrical tin of diameter 
. Both balls hit the top surface and the shell of the cylinder. The largest ball also hits the bottom surface. Show that 
and are enclosed in a cylindrical tin of diameter . Both balls hit the top surface and the shell of the cylinder. The largest ball also hits the bottom surface. Show that 
![[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
import graph; size(10cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -15.36, xmax = 15.36, ymin = -7.04, ymax = 7.04; /* image dimensions */
pen ccqqqq = rgb(0.8,0,0); pen ffdxqq = rgb(1,0.8431372549019608,0);
draw((-7.2,3.88)--(-7.22,-1.84)--(0.44,-1.8)--(0.34,3.94)--cycle, linewidth(1.2));
draw(circle((-4.6,0.84), 2.616907150615757), linewidth(2) + ccqqqq);
draw(circle((-1,2.56), 1.3730906283528068), linewidth(2) + ffdxqq);
/* draw figures */
draw((-7.2,3.88)--(-7.22,-1.84), linewidth(1.2));
draw((-7.22,-1.84)--(0.44,-1.8), linewidth(1.2));
draw((0.44,-1.8)--(0.34,3.94), linewidth(1.2));
draw((0.34,3.94)--(-7.2,3.88), linewidth(1.2));
/* dots and labels */
dot((-7.2,3.88),dotstyle);
label("$A$", (-7.12,4.08), NE * labelscalefactor);
dot((-7.22,-1.84),dotstyle);
label("$B$", (-7.14,-1.64), NE * labelscalefactor);
dot((0.44,-1.8),dotstyle);
label("$C$", (0.52,-1.6), NE * labelscalefactor);
dot((0.34,3.94),dotstyle);
label("$D$", (0.42,4.14), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy]](http://latex.artofproblemsolving.com/c/5/5/c55e76828322d77b41ef4cf883ee48ea0c98d8d1.png)
Therefore, . Kinda went fast. Sorry about that.
![\[
\frac{(x+y+z)^5 - x^5 - y^5 - z^5}{(x+y)(y+z)(z+x)} = 0
\]](http://latex.artofproblemsolving.com/7/8/f/78f8e6c2570684631dc288144e13568d71e68ec8.png)
with 
such that
.
