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Saudi Arabia Contests
Saudi Arabia Pre-TST + Training Tests
2014 Saudi Arabia Pre-TST
1.1
\sqrt{a^2_j-a^2_k < \frac{1}{\sqrt{n} +\sqrt{n - 1}}
\sqrt{a^2_j-a^2_k < \frac{1}{\sqrt{n} +\sqrt{n - 1}}
Source: 2014 Saudi Arabia Pre-TST 1.1
September 13, 2020
algebra
inequalities
Problem Statement
Let
a
1
,
a
2
,
.
.
.
,
a
2
n
a_1, a_2,...,a_{2n}
a
1
,
a
2
,
...
,
a
2
n
be positive real numbers such that
a
i
+
a
n
+
i
=
1
a_i + a_{n+i} = 1
a
i
+
a
n
+
i
=
1
, for all
i
=
1
,
.
.
.
,
n
i = 1,...,n
i
=
1
,
...
,
n
. Prove that there exist two different integers
1
≤
j
,
k
≤
2
n
1 \le j, k \le 2n
1
≤
j
,
k
≤
2
n
for which
a
j
2
−
a
k
2
<
1
n
+
n
−
1
\sqrt{a^2_j-a^2_k} < \frac{1}{\sqrt{n} +\sqrt{n - 1}}
a
j
2
−
a
k
2
<
n
+
n
−
1
1
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