MathDB
Problems
Contests
National and Regional Contests
India Contests
India National Olympiad
1995 India National Olympiad
5
Solve summation
Solve summation
Source: INMO 1995 Problem 5
October 6, 2005
LaTeX
inequalities unsolved
inequalities
Problem Statement
Let
n
≥
2
n \geq 2
n
≥
2
. Let
a
1
,
a
2
,
a
3
,
…
a
n
a_1 , a_2 , a_3 , \ldots a_n
a
1
,
a
2
,
a
3
,
…
a
n
be
n
n
n
real numbers all less than
1
1
1
and such that
∣
a
k
−
a
k
+
1
∣
<
1
|a_k - a_{k+1} | < 1
∣
a
k
−
a
k
+
1
∣
<
1
for
1
≤
k
≤
n
−
1
1 \leq k \leq n-1
1
≤
k
≤
n
−
1
. Show that
a
1
a
2
+
a
2
a
3
+
a
3
a
4
+
…
+
a
n
−
1
a
n
+
a
n
a
1
<
2
n
−
1.
\dfrac{a_1}{a_2} + \dfrac{a_2}{a_3} + \dfrac{a_3}{a_4} + \ldots + \dfrac{a_{n-1}}{a_n} + \dfrac{a_n}{a_1} < 2 n - 1 .
a
2
a
1
+
a
3
a
2
+
a
4
a
3
+
…
+
a
n
a
n
−
1
+
a
1
a
n
<
2
n
−
1.
Back to Problems
View on AoPS