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Problems
Contests
National and Regional Contests
China Contests
XES Mathematics Olympiad
the 12th XMO
Problem 3
Problem 3
Part of
the 12th XMO
Problems
(1)
number theory
Source: 12th XMO
4/13/2023
Let
a
0
=
0
,
a
1
∈
Z
+
.
a_0=0,a_1\in\mathbb Z_+.
a
0
=
0
,
a
1
∈
Z
+
.
For integer
n
≥
2
,
a
n
n\geq 2,a_n
n
≥
2
,
a
n
is the smallest positive integer satisfy that for
∀
0
≤
i
≠
j
≤
n
−
1
,
a
n
∤
(
a
i
−
a
j
)
.
\forall 0\leq i\neq j\leq n-1,a_n\nmid (a_i-a_j).
∀0
≤
i
=
j
≤
n
−
1
,
a
n
∤
(
a
i
−
a
j
)
.
(1) If
a
1
=
2023
,
a_1=2023,
a
1
=
2023
,
calculate
a
10000
.
a_{10000}.
a
10000
.
(2) If
a
t
≤
a
1
2
,
a_t\leq\frac{a_1}2,
a
t
≤
2
a
1
,
find the maximum value of
t
a
1
.
\frac t{a_1}.
a
1
t
.
number theory
xmo