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Polish MO Finals
1981 Polish MO Finals
3
3
Part of
1981 Polish MO Finals
Problems
(1)
\prod (x-a^k )/(x+a^k ineq
Source: Polish MO Finals 1981 p3
8/24/2024
Prove that for any natural number
n
n
n
and real numbers
a
a
a
and
x
x
x
satisfying
a
n
+
1
≤
x
≤
1
a^{n+1} \le x \le 1
a
n
+
1
≤
x
≤
1
and
0
<
a
<
1
0 < a < 1
0
<
a
<
1
it holds that
∏
k
=
1
n
∣
x
−
a
k
x
+
a
k
∣
≤
∏
k
=
1
n
1
−
a
k
1
+
a
k
\prod_{k=1}^n \left|\frac{x-a^k}{x+a^k}\right| \le \prod_{k=1}^n \frac{1-a^k}{1+a^k}
k
=
1
∏
n
x
+
a
k
x
−
a
k
≤
k
=
1
∏
n
1
+
a
k
1
−
a
k
algebra
inequalities