MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Hanoi Open Mathematics Competition
2011 Hanoi Open Mathematics Competitions
12
12
Part of
2011 Hanoi Open Mathematics Competitions
Problems
(2)
Determine the minimum value
Source:
2/17/2016
Suppose that
a
>
0
;
b
>
0
a > 0; b > 0
a
>
0
;
b
>
0
and
a
+
b
≤
1
a + b \leq 1
a
+
b
≤
1
. Determine the minimum value of
M
=
1
a
b
+
1
a
2
+
a
b
+
1
a
b
+
b
2
+
1
a
2
+
b
2
M=\frac{1}{ab} +\frac{1}{a^2+ab}+\frac{1}{ab+b^2}+\frac{1}{a^2+b^2}
M
=
ab
1
+
a
2
+
ab
1
+
ab
+
b
2
1
+
a
2
+
b
2
1
.
inequalities
Help me[homc2011]
Source:
2/16/2016
Suppose that
∣
a
x
2
+
b
x
+
c
∣
≥
∣
x
2
−
1
∣
|ax^2+bx+c| \geq |x^2-1|
∣
a
x
2
+
b
x
+
c
∣
≥
∣
x
2
−
1∣
for all real numbers x. Prove that
∣
b
2
−
4
a
c
∣
≥
4
|b^2-4ac|\geq 4
∣
b
2
−
4
a
c
∣
≥
4
.
algebra