MathDB
Problems
Contests
National and Regional Contests
Moldova Contests
Moldova Team Selection Test
2017 Moldova Team Selection Test
1
Inequality in Moldova TST 2017, D1, P1
Inequality in Moldova TST 2017, D1, P1
Source: Moldova TST 2017, Day 1, Problem 1
March 6, 2017
algebra
Problem Statement
Let the sequence
(
a
n
)
n
⩾
1
(a_{n})_{n\geqslant 1}
(
a
n
)
n
⩾
1
be defined as:
a
n
=
A
n
+
2
1
A
n
+
3
2
A
n
+
4
3
A
n
+
5
4
5
4
3
,
a_{n}=\sqrt{A_{n+2}^{1}\sqrt[3]{A_{n+3}^{2}\sqrt[4]{A_{n+4}^{3}\sqrt[5]{A_{n+5}^{4}}}}},
a
n
=
A
n
+
2
1
3
A
n
+
3
2
4
A
n
+
4
3
5
A
n
+
5
4
,
where
A
m
k
A_{m}^{k}
A
m
k
are defined by
A
m
k
=
(
m
k
)
⋅
k
!
.
A_{m}^{k}=\binom{m}{k}\cdot k!.
A
m
k
=
(
k
m
)
⋅
k
!
.
Prove that
a
n
<
119
120
⋅
n
+
7
3
.
a_{n}<\frac{119}{120}\cdot n+\frac{7}{3}.
a
n
<
120
119
⋅
n
+
3
7
.
Back to Problems
View on AoPS