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Romanian Masters of Mathematics Collection
2021 Romanian Master of Mathematics
2
2
Part of
2021 Romanian Master of Mathematics
Problems
(1)
Xenia and Sergey play an NT game
Source: RMM 2021/2
10/13/2021
Xenia and Sergey play the following game. Xenia thinks of a positive integer
N
N
N
not exceeding
5000
5000
5000
. Then she fixes
20
20
20
distinct positive integers
a
1
,
a
2
,
⋯
,
a
20
a_1, a_2, \cdots, a_{20}
a
1
,
a
2
,
⋯
,
a
20
such that, for each
k
=
1
,
2
,
⋯
,
20
k = 1,2,\cdots,20
k
=
1
,
2
,
⋯
,
20
, the numbers
N
N
N
and
a
k
a_k
a
k
are congruent modulo
k
k
k
. By a move, Sergey tells Xenia a set
S
S
S
of positive integers not exceeding
20
20
20
, and she tells him back the set
{
a
k
:
k
∈
S
}
\{a_k : k \in S\}
{
a
k
:
k
∈
S
}
without spelling out which number corresponds to which index. How many moves does Sergey need to determine for sure the number Xenia thought of?Sergey Kudrya, Russia
RMM
number theory