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Moscow Mathematical Olympiad
2018 Moscow Mathematical Olympiad
3
Number of roots
Number of roots
Source: Moscow Olympiad 2018, Grade 9, P3
July 13, 2018
number theory
Problem Statement
a
1
,
a
2
,
.
.
.
,
a
k
a_1,a_2,...,a_k
a
1
,
a
2
,
...
,
a
k
are positive integers and
1
a
1
+
1
a
2
+
.
.
.
+
1
a
k
>
1
\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_k}>1
a
1
1
+
a
2
1
+
...
+
a
k
1
>
1
. Prove that equation
[
n
a
1
]
+
[
n
a
2
]
+
.
.
.
+
[
n
a
k
]
=
n
[\frac{n}{a_1}]+[\frac{n}{a_2}]+...+[\frac{n}{a_k}]=n
[
a
1
n
]
+
[
a
2
n
]
+
...
+
[
a
k
n
]
=
n
has no more than
a
1
∗
a
2
∗
.
.
.
∗
a
k
a_1*a_2*...*a_k
a
1
∗
a
2
∗
...
∗
a
k
postivie integer solutions in
n
n
n
.
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