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2018 ITAMO
3
ITAMO 2018 problem 3
ITAMO 2018 problem 3
Source:
May 9, 2018
Inequality
Problem Statement
Let
x
1
,
x
2
,
.
.
.
,
x
n
x_1,x_2, ... , x_n
x
1
,
x
2
,
...
,
x
n
be positive integers,Asumme that in their decimal representations no
x
i
x_i
x
i
"prolongs"
x
j
x_j
x
j
.For instance ,
123
123
123
prolongs
12
12
12
,
459
459
459
prolongs
4
4
4
, but
124
124
124
does not prolog
123
123
123
. Prove that :
1
x
1
+
1
x
2
+
.
.
.
+
1
x
n
<
3
\frac {1}{x_1}+\frac {1}{x_2}+...+\frac {1}{x_n} < 3
x
1
1
+
x
2
1
+
...
+
x
n
1
<
3
.
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