中四程度pure math 5題(大陸)

1. WLOG, let a>b>c
By Rearrangement Inequality
a3 + b3 > a2b + b2a = ab(a+b) --- (i)
Similarly, b3 + c3 > bc(b+c) --- (ii)
a3 + c3 > ac(a+c) --- (iii)
(i)+(ii)+(iii), the result follows

2. By AM-GM Inequality
1/a + 1/b = (a+b)/ab > 2(sqrt ab) / ab = 2/(sqrt ab) --- (i)
Similarly, 1/b + 1/c > 2/(sqrt bc) --- (ii)
1/a + 1/c > 2/(sqrt ac) --- (iii)
(i)+(ii)+(iii)
2(1/a + 1/b + 1/c) > 2/(sqrt ab) + 2/(sqrt bc) + 2/(sqrt ac)
The result follows

3. By AM-GM Inequality
(a+b+c)(a2+b2+c2)
> [3 (abc)^(1/3)] x [3 (a2b2a2)^(1/3)]
= 9abc

4. By AM-HM Inequality
2/(a+b) + 2/(b+c) + 2/(c+a)
> 3 x {3 / [ (a+b)/2 + (b+c)/2 + (a+c)/2 ]}
= 3 x [3 / (a+b+c)]
= 9/(a+b+c)

5. By AM-GM Inequality
(ab)2+ (bc)2 > 2sqrt[(ab)2+(bc)2] = 2ab2c --- (i)
Similarly, (bc)2+ (ca)2 > 2bc2a --- (ii)
(ca)2+ (ab)2 > 2ca2b --- (iii)
(i)+(ii)+(iii)
(ab)2+ (bc)2 + (ca)2 > abc(a+b+c)

Hope the above information helps =) By 小儒