MathDB
Problems
Contests
National and Regional Contests
India Contests
India Pre-Regional Mathematical Olympiad
2016 India PRMO
12
12
Part of
2016 India PRMO
Problems
(1)
2016 preRMO p12, 1 + 1/\sqrt2+1/\sqrt3+..+ 1/\sqrt{99}+ 1/\sqrt{100}
Source:
8/9/2019
Let
S
=
1
+
1
2
+
1
3
+
1
4
+
.
.
.
+
1
99
+
1
100
S = 1 + \frac{1}{\sqrt2}+ \frac{1}{\sqrt3}+\frac{1}{\sqrt4}+...+ \frac{1}{\sqrt{99}}+ \frac{1}{\sqrt{100}}
S
=
1
+
2
1
+
3
1
+
4
1
+
...
+
99
1
+
100
1
. Find
[
S
]
[S]
[
S
]
.You may use the fact that
n
<
1
2
(
n
+
n
+
1
)
<
n
+
1
\sqrt{n} < \frac12 (\sqrt{n} +\sqrt{n+1}) <\sqrt{n+1}
n
<
2
1
(
n
+
n
+
1
)
<
n
+
1
for all integers
n
≥
1
n \ge 1
n
≥
1
.
Sum
radical
floor function
inequalities
algebra