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Turkey Team Selection Test
1990 Turkey Team Selection Test
2
x_1 + x_2 + ... + x_n = 0
x_1 + x_2 + ... + x_n = 0
Source: Turkey TST 1990 - P2
September 11, 2013
algebra
polynomial
calculus
derivative
inequalities proposed
inequalities
Problem Statement
For real numbers
x
i
x_i
x
i
, the statement
x
1
+
x
2
+
x
3
=
0
⇒
x
1
x
2
+
x
2
x
3
+
x
3
x
1
≤
0
x_1 + x_2 + x_3 = 0 \Rightarrow x_1x_2 + x_2x_3 + x_3x_1 \leq 0
x
1
+
x
2
+
x
3
=
0
⇒
x
1
x
2
+
x
2
x
3
+
x
3
x
1
≤
0
is always true. (Prove!) For which
n
≥
4
n\geq 4
n
≥
4
integers, the statement
x
1
+
x
2
+
⋯
+
x
n
=
0
⇒
x
1
x
2
+
x
2
x
3
+
⋯
+
x
n
−
1
x
n
+
x
n
x
1
≤
0
x_1 + x_2 + \dots + x_n = 0 \Rightarrow x_1x_2 + x_2x_3 + \dots + x_{n-1}x_n + x_nx_1 \leq 0
x
1
+
x
2
+
⋯
+
x
n
=
0
⇒
x
1
x
2
+
x
2
x
3
+
⋯
+
x
n
−
1
x
n
+
x
n
x
1
≤
0
is always true. Justify your answer.
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