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Iran Team Selection Test
2024 Iran Team Selection Test
3
Absoulute value in inequality
Absoulute value in inequality
Source: Iran Team selection test 2024 - P3
May 19, 2024
algebra
inequalities
Problem Statement
For any real numbers
x
,
y
,
z
x , y ,z
x
,
y
,
z
prove that :
(
x
+
y
+
z
)
2
+
∑
c
y
c
(
x
+
y
)
(
y
+
z
)
1
+
∣
x
−
z
∣
≥
x
y
+
y
z
+
z
x
(x+y+z)^2 + \sum_{cyc}{\frac{(x+y)(y+z)}{1+|x-z|}} \ge xy+yz+zx
(
x
+
y
+
z
)
2
+
cyc
∑
1
+
∣
x
−
z
∣
(
x
+
y
)
(
y
+
z
)
≥
x
y
+
yz
+
z
x
Proposed by Navid Safaei
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