MathDB
Problems
Contests
National and Regional Contests
Mongolia Contests
Mongolian Mathematical Olympiad
1999 Mongolian Mathematical Olympiad
Problem 5
inequality
inequality
Source:
February 23, 2017
Inequality
inequalities
Problem Statement
Given
a
;
b
;
c
a;b;c
a
;
b
;
c
satisfying
a
2
+
b
2
+
c
2
=
2
a^{2}+b^{2}+c^{2}=2
a
2
+
b
2
+
c
2
=
2
. Prove that: a)
∣
a
+
b
+
c
−
a
b
c
∣
⩽
2
\left | a+b+c-abc \right |\leqslant 2
∣
a
+
b
+
c
−
ab
c
∣
⩽
2
. b)
∣
a
3
+
b
3
+
c
3
−
3
a
b
c
∣
⩽
2
2
\left | a^{3}+b^{3}+c^{3}-3abc \right |\leqslant 2\sqrt{2}
a
3
+
b
3
+
c
3
−
3
ab
c
⩽
2
2
Back to Problems
View on AoPS