MathDB
Problems
Contests
National and Regional Contests
China Contests
XES Mathematics Olympiad
the 12th XMO
the 12th XMO
Part of
XES Mathematics Olympiad
Subcontests
(5)
Problem 1
1
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KS _|_ SR
As shown in the figure, it is known that the quadrilateral
A
B
C
D
ABCD
A
BC
D
satisfies
∠
A
D
B
=
∠
A
C
B
=
9
0
o
\angle ADB = \angle ACB = 90^o
∠
A
D
B
=
∠
A
CB
=
9
0
o
. Suppose
A
C
AC
A
C
and
B
D
BD
B
D
intersect at point
P
P
P
, point
R
R
R
lies on
C
D
CD
C
D
and
R
P
⊥
A
B
RP \perp AB
RP
⊥
A
B
.
M
M
M
and
N
N
N
are the midpoints of
A
B
AB
A
B
and
C
D
CD
C
D
respectively. Point
K
K
K
is a point on the extension line of
N
M
NM
NM
, the circumscribed circles of
△
D
K
C
\vartriangle DKC
△
DK
C
and
△
A
K
B
\vartriangle AKB
△
A
K
B
intersect at point
S
S
S
. Prove that
K
S
⊥
S
R
KS \perp SR
K
S
⊥
SR
. https://cdn.artofproblemsolving.com/attachments/5/d/fc0a391f8ebcdee792e9b226cbf55a058251a1.png
Problem 5
1
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Inequality
Let
a
,
b
,
c
∈
R
+
a,b,c\in\mathbb R_+
a
,
b
,
c
∈
R
+
satisfy that
(
1
+
a
)
(
1
+
b
)
(
1
+
c
)
=
(
a
b
−
a
−
b
+
1
)
(
1
+
c
)
+
(
b
c
−
b
−
c
+
1
)
(
1
+
a
)
+
(
c
a
−
c
−
a
+
1
)
(
1
+
b
)
.
\sqrt{(1+a)(1+b)(1+c)}=\sqrt{(ab-a-b+1)(1+c)}+\sqrt{(bc-b-c+1)(1+a)}+\sqrt{(ca-c-a+1)(1+b)}.
(
1
+
a
)
(
1
+
b
)
(
1
+
c
)
=
(
ab
−
a
−
b
+
1
)
(
1
+
c
)
+
(
b
c
−
b
−
c
+
1
)
(
1
+
a
)
+
(
c
a
−
c
−
a
+
1
)
(
1
+
b
)
.
Find the value range of
a
+
b
+
c
.
a+b+c.
a
+
b
+
c
.
Problem 4
1
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Graph theory
求最小的
n
,
n,
n
,
使得对任意有
1000
{1000}
1000
个顶点且每个顶点度均为
4
{4}
4
的简单图
G
,
G,
G
,
都一定可以从中取掉
n
{n}
n
条边
,
,
,
使
G
{G}
G
变为二部图
.
.
.
Problem 3
1
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number theory
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
.
Problem 2
1
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12th XMO Problem 2
Let
a
1
,
a
2
,
⋯
,
a
22
∈
[
1
,
2
]
,
a_1,a_2,\cdots,a_{22}\in [1,2],
a
1
,
a
2
,
⋯
,
a
22
∈
[
1
,
2
]
,
find the maximum value of
∑
i
=
1
22
a
i
a
i
+
1
(
∑
i
=
1
22
a
i
)
2
\dfrac{\sum\limits_{i=1}^{22}a_ia_{i+1}}{\left( \sum\limits_{i=1}^{22}a_i\right) ^2}
(
i
=
1
∑
22
a
i
)
2
i
=
1
∑
22
a
i
a
i
+
1
where
a
23
=
a
1
.
a_{23}=a_1.
a
23
=
a
1
.