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Problems
Contests
National and Regional Contests
China Contests
XES Mathematics Olympiad
the 16th XMO
the 16th XMO
Part of
XES Mathematics Olympiad
Subcontests
(4)
4
1
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Springs are fired inXMO
Given an integer
n
n
n
,For a sequence of
X
X
X
with the number of
n
n
n
and
Y
Y
Y
with the number of
100
n
100n
100
n
, we call it a spring . We have two following rules
■
\blacksquare
■
Choose four adjacent character , if it is
Y
X
X
Y
YXXY
Y
XX
Y
, than it can be changed into
X
Y
Y
X
XYYX
X
YY
X
■
\blacksquare
■
Choose. four adjacent character , if it is
X
Y
Y
X
XYYX
X
YY
X
, than it can be changed into
Y
X
X
Y
YXXY
Y
XX
Y
If spring
A
A
A
can become
B
B
B
using the rules , than we call they are [color=#3D85C6]similar Thy to find the maximum of
C
C
C
such that there exists
C
C
C
distinct springs and they are [color=#3D85C6]similar
3
1
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Interesting number theory inXMO
m
m
m
is an integer satisfying
m
≥
2024
m \ge 2024
m
≥
2024
,
p
p
p
is the smallest prime factor of
m
m
m
, for an arithmetic sequence
{
a
n
}
\{a_n\}
{
a
n
}
of positive numbers with the common difference
m
m
m
satisfying : for any integer
1
≤
i
≤
p
2
1 \le i \le \frac{p}{2}
1
≤
i
≤
2
p
, there doesn’t exist an integer
x
,
y
≤
max
{
a
1
,
m
}
x , y \le \max \{a_1 , m\}
x
,
y
≤
max
{
a
1
,
m
}
such that
a
i
=
x
y
a_i=xy
a
i
=
x
y
Try to proof that there exists a positive real number
c
c
c
such that for any
1
≤
i
≤
j
≤
n
1\le i \le j \le n
1
≤
i
≤
j
≤
n
,
g
c
d
(
a
i
,
a
j
)
=
c
×
g
c
d
(
i
,
j
)
gcd(a_i , a_j ) = c \times gcd(i , j)
g
c
d
(
a
i
,
a
j
)
=
c
×
g
c
d
(
i
,
j
)
2
1
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Geometry inXMO need calculating
In a triangle
A
B
C
ABC
A
BC
, let
O
O
O
be the circumcenter ,
A
O
AO
A
O
meet
B
C
BC
BC
at
K
K
K
, A circle
Ω
\Omega
Ω
with the centre
T
T
T
and the center
K
K
K
and the radius
A
K
AK
A
K
meet
A
C
AC
A
C
again at
T
T
T
,
D
D
D
is a point on the plain satisfies that
B
C
BC
BC
is the bisector of the angle
∠
A
B
D
\angle ABD
∠
A
B
D
, let the orthocenter of the triangle
A
B
C
ABC
A
BC
and
B
C
D
BCD
BC
D
be
M
M
M
and
N
N
N
. If
M
N
/
/
A
C
MN//AC
MN
//
A
C
than
D
T
DT
D
T
is tangent to
Ω
\Omega
Ω
1
1
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Tricky Algebra
Let
a
1
,
a
2
,
…
,
a
n
≥
0.
a_1,a_2,\ldots ,a_n\ge 0.
a
1
,
a
2
,
…
,
a
n
≥
0.
For all
1
≤
k
≤
n
1\le k\le n
1
≤
k
≤
n
define
b
k
:
=
min
1
≤
i
<
j
≤
n
,
j
−
i
≤
2
∣
2
a
k
−
a
i
−
a
j
∣
.
b_k:=\min_{1\le i<j\le n,j-i\le 2}|2a_k-a_i-a_j|.
b
k
:=
1
≤
i
<
j
≤
n
,
j
−
i
≤
2
min
∣2
a
k
−
a
i
−
a
j
∣.
Here the index mod
n
.
n.
n
.
Find the maximum value of
b
1
+
b
2
+
⋯
+
b
n
a
1
+
a
2
+
⋯
+
a
n
.
\frac{b_1+b_2+\cdots +b_n}{a_1+a_2+\cdots +a_n}.
a
1
+
a
2
+
⋯
+
a
n
b
1
+
b
2
+
⋯
+
b
n
.
Proposed by Zheng Wang