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National and Regional Contests
China Contests
(China) National High School Mathematics League
2013 China Second Round Olympiad
3
2013 China Second Round Olympiad (C) Test 2 Q3
2013 China Second Round Olympiad (C) Test 2 Q3
Source: 13 Oct 2013
October 15, 2013
inequalities proposed
inequalities
Problem Statement
The integers
n
>
1
n>1
n
>
1
is given . The positive integer
a
1
,
a
2
,
⋯
,
a
n
a_1,a_2,\cdots,a_n
a
1
,
a
2
,
⋯
,
a
n
satisfing condition : (1)
a
1
<
a
2
<
⋯
<
a
n
a_1<a_2<\cdots<a_n
a
1
<
a
2
<
⋯
<
a
n
; (2)
a
1
2
+
a
2
2
2
,
a
2
2
+
a
3
2
2
,
⋯
,
a
n
−
1
2
+
a
n
2
2
\frac{a^2_1+a^2_2}{2},\frac{a^2_2+a^2_3}{2},\cdots,\frac{a^2_{n-1}+a^2_n}{2}
2
a
1
2
+
a
2
2
,
2
a
2
2
+
a
3
2
,
⋯
,
2
a
n
−
1
2
+
a
n
2
are all perfect squares . Prove that :
a
n
≥
2
n
2
−
1.
a_n\ge 2n^2-1.
a
n
≥
2
n
2
−
1.
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