L>4 BOOK OF MATHEMATICAL PROBLEMS. 



120. If x, y, z be three positive quantities whose sum is 

 unity, then will 



121. Prove that 

 (a'+b* + c* + d t ) 



122. Prove that 



\n~ 



123. If a, 5, c be three positive quantities of which any two 

 are together greater than the third, then will 



1 1 11119- 



+ T + , > - + r + ~ 



b+ca c + a b a+bc a b c a + b+c 

 and, x, y, z being any real quantities, 



cannot be negative. 



If x + y + z = 0, cfyz + b*zx + c*xy cannot be positive. 



124. If a, 5, c be positive and not all equal, the expressions 

 a (a - b) (a - c) + b (b - c} (b - a) + c (c - a) (c - b), 

 a a (a-b)(a-c) + b 3 (b-c)(b-a)+b\c-a)(c-b), 



are positive. 



125. Prove that 



{ax (b + c) + by (c + a) + cz (a + 5)}* 



> 4a6c (x + y + z) (ax + by + cz) ; 

 a, 6, c, a, y, z being all positive, and a, b, c unequal. 



126. If 



the greatest value of either of these equals is ^ , x, y t z being each 

 positive and less than unity. 



