6. Let a, b, c, x, y, and z be real numbers with a² + b² + c² = 1 = x² + y² + z². Show that:

(a) 

(b) 

7. Let a, b, c, d, and e be real numbers. Show the following:

(a) 

(b) 

(c) 

(d) 

8. Show that  for all real numbers a and b.

9. Show that if x and y are positive real numbers with x + y = 1, then

10. Show that if x, y and z are positive real numbers with x + y + z = 1, then

11. Let a, b, and c be positive real numbers. Show that 

12. Let a, b, c, and d be positive. Show that

13. Let A_n, G_n, and H_n be the arithmetic, geometric, and harmonic mean, respectively, of positive numbers a₁, a₂, ..., a_n. Assuming  show that 
 
 

Monday, June 22, 1998