MathLabs

Problem 1

Let x,y,zx,y,z be nonnegative real numbers with x+y+z=1x+y+z=1. Show that 0≤xy+yz+zx−2xyz≤7270\le xy+yz+zx-2xyz\le\frac{7}{27}.
Step 2 of 4: Reduce to two parameters
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In plain words

Once the sum of two variables is fixed, their product is largest when they are equal; this turns a three-variable problem into a one-variable boundary check.

s=x+y,p=xy,f=s−s2+p(2s−1),0≤p≤s2/4s=x+y,\quad p=xy,\quad f=s-s^2+p(2s-1),\quad 0\le p\le s^2/4
Detailed analysis

Substitute z=1−sz=1-s. Then the target expression is f=s−s2+p(2s−1)f=s-s^2+p(2s-1). For fixed ss, AM-GM gives p=xyp=xy and 0≤p≤s2/40\le p\le s^2/4.