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National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
1968 Dutch Mathematical Olympiad
2
2
Part of
1968 Dutch Mathematical Olympiad
Problems
(1)
1/2 ( N / a +a ) >= \sqrt{N}
Source: Netherlands - Dutch NMO 1968 p2
1/31/2023
It holds:
N
,
a
>
0
N,a > 0
N
,
a
>
0
. Prove that
1
2
(
N
a
+
a
)
≥
N
\frac12 \left(\frac{N}{a}+a \right) \ge \sqrt{N}
2
1
(
a
N
+
a
)
≥
N
, and if
N
≥
1
N \ge 1
N
≥
1
and
a
=
[
N
]
a = [\sqrt{N}]
a
=
[
N
]
. Prove that if
a
≠
N
:
1
2
(
N
a
+
a
)
a \ne \sqrt{N}: \frac12 \left(\frac{N}{a}+a \right)
a
=
N
:
2
1
(
a
N
+
a
)
is a better approximation for
N
\sqrt{N}
N
than
a
a
a
.
algebra
inequalities