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Problems
Contests
National and Regional Contests
China Contests
South East Mathematical Olympiad
2010 South East Mathematical Olympiad
2010 South East Mathematical Olympiad
Part of
South East Mathematical Olympiad
Subcontests
(4)
2
2
Hide problems
2010 smo (2)
For any set
A
=
{
a
1
,
a
2
,
⋯
,
a
m
}
A=\{a_1,a_2,\cdots,a_m\}
A
=
{
a
1
,
a
2
,
⋯
,
a
m
}
, let
P
(
A
)
=
a
1
a
2
⋯
a
m
P(A)=a_1a_2\cdots a_m
P
(
A
)
=
a
1
a
2
⋯
a
m
. Let
n
=
(
2010
99
)
n={2010\choose99}
n
=
(
99
2010
)
, and let
A
1
,
A
2
,
⋯
,
A
n
A_1, A_2,\cdots,A_n
A
1
,
A
2
,
⋯
,
A
n
be all
99
99
99
-element subsets of
{
1
,
2
,
⋯
,
2010
}
\{1,2,\cdots,2010\}
{
1
,
2
,
⋯
,
2010
}
. Prove that
2010
∣
∑
i
=
1
n
P
(
A
i
)
2010|\sum^{n}_{i=1}P(A_i)
2010∣
∑
i
=
1
n
P
(
A
i
)
.
2010 smo (6)
Let
N
∗
\mathbb{N}^*
N
∗
be the set of positive integers. Define
a
1
=
2
a_1=2
a
1
=
2
, and for
n
=
1
,
2
,
…
,
n=1, 2, \ldots,
n
=
1
,
2
,
…
,
a
n
+
1
=
min
{
λ
∣
1
a
1
+
1
a
2
+
⋯
1
a
n
+
1
λ
<
1
,
λ
∈
N
∗
}
a_{n+1}=\min\{\lambda|\frac{1}{a_1}+\frac{1}{a_2}+\cdots\frac{1}{a_n}+\frac{1}{\lambda}<1,\lambda\in \mathbb{N}^*\}
a
n
+
1
=
min
{
λ
∣
a
1
1
+
a
2
1
+
⋯
a
n
1
+
λ
1
<
1
,
λ
∈
N
∗
}
Prove that
a
n
+
1
=
a
n
2
−
a
n
+
1
a_{n+1}=a_n^2-a_n+1
a
n
+
1
=
a
n
2
−
a
n
+
1
for
n
=
1
,
2
,
…
n=1,2,\ldots
n
=
1
,
2
,
…
.
3
2
Hide problems
2010 smo (3)
The incircle of triangle
A
B
C
ABC
A
BC
touches
B
C
BC
BC
at
D
D
D
and
A
B
AB
A
B
at
F
F
F
, intersects the line
A
D
AD
A
D
again at
H
H
H
and the line
C
F
CF
CF
again at
K
K
K
. Prove that
F
D
×
H
K
F
H
×
D
K
=
3
\frac{FD\times HK}{FH\times DK}=3
F
H
×
DK
F
D
×
HK
=
3
2010 SMO (7)
Let
n
n
n
be a positive integer. The real numbers
a
1
,
a
2
,
⋯
,
a
n
a_1,a_2,\cdots,a_n
a
1
,
a
2
,
⋯
,
a
n
and
r
1
,
r
2
,
⋯
,
r
n
r_1,r_2,\cdots,r_n
r
1
,
r
2
,
⋯
,
r
n
are such that
a
1
≤
a
2
≤
⋯
≤
a
n
a_1\leq a_2\leq \cdots \leq a_n
a
1
≤
a
2
≤
⋯
≤
a
n
and
0
≤
r
1
≤
r
2
≤
⋯
≤
r
n
0\leq r_1\leq r_2\leq \cdots \leq r_n
0
≤
r
1
≤
r
2
≤
⋯
≤
r
n
. Prove that
∑
i
=
1
n
∑
j
=
1
n
a
i
a
j
min
(
r
i
,
r
j
)
≥
0
\sum_{i=1}^n\sum_{j=1}^n a_i a_j \min (r_i,r_j)\geq 0
∑
i
=
1
n
∑
j
=
1
n
a
i
a
j
min
(
r
i
,
r
j
)
≥
0
4
2
Hide problems
2010 smo (4)
Let
a
a
a
and
b
b
b
be positive integers such that
1
≤
a
<
b
≤
100
1\leq a<b\leq 100
1
≤
a
<
b
≤
100
. If there exists a positive integer
k
k
k
such that
a
b
∣
a
k
+
b
k
ab|a^k+b^k
ab
∣
a
k
+
b
k
, we say that the pair
(
a
,
b
)
(a, b)
(
a
,
b
)
is good. Determine the number of good pairs.
2010 SMO (8)
A
1
,
A
2
,
⋯
,
A
8
A_1,A_2,\cdots,A_8
A
1
,
A
2
,
⋯
,
A
8
are fixed points on a circle. Determine the smallest positive integer
n
n
n
such that among any
n
n
n
triangles with these eight points as vertices, two of them will have a common side.
1
2
Hide problems
2010 SMO
Let
a
,
b
,
c
∈
{
0
,
1
,
2
,
⋯
,
9
}
a,b,c\in\{0,1,2,\cdots,9\}
a
,
b
,
c
∈
{
0
,
1
,
2
,
⋯
,
9
}
.The quadratic equation
a
x
2
+
b
x
+
c
=
0
ax^2+bx+c=0
a
x
2
+
b
x
+
c
=
0
has a rational root. Prove that the three-digit number
a
b
c
abc
ab
c
is not a prime number.
2010 smo (5)
A
B
C
ABC
A
BC
is a triangle with a right angle at
C
C
C
.
M
1
M_1
M
1
and
M
2
M_2
M
2
are two arbitrary points inside
A
B
C
ABC
A
BC
, and
M
M
M
is the midpoint of
M
1
M
2
M_1M_2
M
1
M
2
. The extensions of
B
M
1
,
B
M
BM_1,BM
B
M
1
,
BM
and
B
M
2
BM_2
B
M
2
intersect
A
C
AC
A
C
at
N
1
,
N
N_1,N
N
1
,
N
and
N
2
N_2
N
2
respectively. Prove that
M
1
N
1
B
M
1
+
M
2
N
2
B
M
2
≥
2
M
N
B
M
\frac{M_1N_1}{BM_1}+\frac{M_2N_2}{BM_2}\geq 2\frac{MN}{BM}
B
M
1
M
1
N
1
+
B
M
2
M
2
N
2
≥
2
BM
MN