Two algebra questions I have two questions that I need help with:
1) How many single digit even natural number solutions are there for the equation $A+B+C+D = 24$ such that $A+B > C+D$
A)20 B)11 C)16 D)24
2) Three positive real numbers $x,y,z$ are such that $x+y+Z = 1$. which of the following inequalities best discribe the relation between $XY,YZ,ZX$.
A) $xy+yz+zx > 1/3$
B) $xy+yz+zx <1/3$
C) $xy+yz+zx \le 1/3$
D) $xy+yz+zx \le 2/3$

For the inequality note that:
$$(x-y)^2 + (y-z)^2+(z-x)^2 = 2(x^2+y^2+z^2)-2(xy+yz+zx) \geq 0$$
so that $$x^2+y^2+z^2 \geq xy+yz+zx$$
Now
$$1=(x+y+z)^2=(x^2+y^2+z^2) + 2(xy+yz+zx)\geq 3(xy+yz+zx)$$
I think that is pre-calculus algebra, and not nearly as well known a trick as it should be.
