MathDB
Problems
Contests
National and Regional Contests
China Contests
XES Mathematics Olympiad
the 15th XMO
the 15th XMO
Part of
XES Mathematics Olympiad
Subcontests
(4)
4
1
Hide problems
Classical NT in XMO
Let
p
p
p
be a prime number and
k
k
k
is a integer with
p
∣
2
k
−
1
p|2^k-1
p
∣
2
k
−
1
for
a
∈
{
1
,
2
,
…
,
p
−
1
}
a\in \{ 1,2,\ldots,p-1\}
a
∈
{
1
,
2
,
…
,
p
−
1
}
, let
m
a
m_{a}
m
a
be the only element that satisfies
p
∣
a
m
a
−
1
p|am_{a}-1
p
∣
a
m
a
−
1
and define
T
a
=
{
x
∈
{
1
,
2
,
…
p
−
1
}
∣
{
m
a
x
p
−
x
a
p
}
<
1
2
T_{a}= \{ x \in \{1,2,\ldots p-1\} | \{ \frac {m_{a}x}{p} - \frac {x}{ap} \} < \frac{1}{2}
T
a
=
{
x
∈
{
1
,
2
,
…
p
−
1
}
∣
{
p
m
a
x
−
a
p
x
}
<
2
1
and there exists integer y satisfying p | x-y^
k
+
1
k+1
k
+
1
}
\}
}
Try to proof that there exists an integer
m
m
m
and integers
1
≤
a
1
<
a
2
<
…
<
a
m
≤
p
−
1
1 \le a_1 <a_2< \ldots < a_{m} \le p-1
1
≤
a
1
<
a
2
<
…
<
a
m
≤
p
−
1
satisfying
∣
T
a
1
∣
=
∣
T
a
2
∣
=
…
=
∣
T
a
m
∣
=
m
|T_{a_1}| = |T_{a_2}| = \ldots = |T_{a_{m}}| = m
∣
T
a
1
∣
=
∣
T
a
2
∣
=
…
=
∣
T
a
m
∣
=
m
3
1
Hide problems
Combination in XMO
k
k
k
is an integer, there exists a triangulation for a regular polygon with
2024
2024
2024
sides and
2024
2024
2024
colored dots with
k
k
k
different colors meeting
(
1
)
(1)
(
1
)
each color will be used at least once
(
2
)
(2)
(
2
)
every small triangle will have at least
2
2
2
dots that will be in the same color. Try to find the maximum value of
k
k
k
2
1
Hide problems
Inequality in XMO
n
n
n
is a integer and
a
1
,
a
2
,
…
,
a
n
∈
[
−
1
,
1
]
a_1, a_2, \ldots, a_n\in[-1,1]
a
1
,
a
2
,
…
,
a
n
∈
[
−
1
,
1
]
are real numbers with
∑
i
=
1
n
a
i
=
0
\sum_{i=1}^{n}a_{i}=0
∑
i
=
1
n
a
i
=
0
,try to find the maximum value of
∑
1
≤
i
,
j
≤
n
,
i
≠
j
∣
a
i
−
a
j
2
∣
\sum_{1\leq i , j \leq n , i\ne j}|a_{i}-a^2_j|
1
≤
i
,
j
≤
n
,
i
=
j
∑
∣
a
i
−
a
j
2
∣
1
1
Hide problems
Boring geometry proving by angle chasing in XMO
A quadrilateral
A
B
C
D
ABCD
A
BC
D
with
A
B
⊥
B
C
AB \perp BC
A
B
⊥
BC
,
A
D
⊥
D
C
AD \perp DC
A
D
⊥
D
C
,
E
E
E
is a point that is on the line
B
D
BD
B
D
with
E
C
=
C
A
EC=CA
EC
=
C
A
,
F
F
F
,
G
G
G
is on the line
A
B
AB
A
B
A
D
AD
A
D
such that
E
F
⊥
A
C
EF\perp AC
EF
⊥
A
C
and
E
G
⊥
A
C
EG\perp AC
EG
⊥
A
C
,let
X
Y
X Y
X
Y
be the midpoint of segment
A
F
A
G
AF AG
A
F
A
G
, let
Z
W
Z W
Z
W
be the midpoint of segment
B
E
D
E
BE DE
BE
D
E
, try to proof that
(
W
B
X
)
(WBX)
(
W
BX
)
is tangent to
(
Z
D
Y
)
(ZDY)
(
Z
D
Y
)