MathDB
Problems
Contests
National and Regional Contests
China Contests
China Western Mathematical Olympiad
2009 China Western Mathematical Olympiad
2009 China Western Mathematical Olympiad
Part of
China Western Mathematical Olympiad
Subcontests
(4)
4
2
Hide problems
k consecutive integers are composite
Prove that for every given positive integer
k
k
k
, there exist infinitely many
n
n
n
, such that
2
n
+
3
n
−
1
,
2
n
+
3
n
−
2
,
…
,
2
n
+
3
n
−
k
2^{n}+3^{n}-1, 2^{n}+3^{n}-2,\ldots, 2^{n}+3^{n}-k
2
n
+
3
n
−
1
,
2
n
+
3
n
−
2
,
…
,
2
n
+
3
n
−
k
are all composite numbers.
convex sequence
The real numbers
a
1
,
a
2
,
…
,
a
n
a_{1},a_{2},\ldots ,a_{n}
a
1
,
a
2
,
…
,
a
n
where
n
≥
3
n\ge 3
n
≥
3
are such that
∑
i
=
1
n
a
i
=
0
\sum_{i=1}^{n}a_{i}=0
∑
i
=
1
n
a
i
=
0
and
2
a
k
≤
a
k
−
1
+
a
k
+
1
2a_{k}\le\ a_{k-1}+a_{k+1}
2
a
k
≤
a
k
−
1
+
a
k
+
1
for all
k
=
2
,
3
,
…
,
n
−
1
k=2,3,\ldots ,n-1
k
=
2
,
3
,
…
,
n
−
1
. Find the least
f
(
n
)
f(n)
f
(
n
)
such that, for all
k
∈
{
1
,
2
,
…
,
n
}
k\in\left\{1,2,\ldots ,n\right\}
k
∈
{
1
,
2
,
…
,
n
}
, we have
∣
a
k
∣
≤
f
(
n
)
max
{
∣
a
1
∣
,
∣
a
n
∣
}
|a_{k}|\le f(n)\max\left\{|a_{1}|,|a_{n}|\right\}
∣
a
k
∣
≤
f
(
n
)
max
{
∣
a
1
∣
,
∣
a
n
∣
}
.
3
2
Hide problems
four points are cyclic
Let
H
H
H
be the orthocenter of acute triangle
A
B
C
ABC
A
BC
and
D
D
D
the midpoint of
B
C
BC
BC
. A line through
H
H
H
intersects
A
B
,
A
C
AB,AC
A
B
,
A
C
at
F
,
E
F,E
F
,
E
respectively, such that
A
E
=
A
F
AE=AF
A
E
=
A
F
. The ray
D
H
DH
DH
intersects the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
at
P
P
P
. Prove that
P
,
A
,
E
,
F
P,A,E,F
P
,
A
,
E
,
F
are concyclic.
find the least n
A total of
n
n
n
people compete in a mathematical match which contains
15
15
15
problems where
n
>
12
n>12
n
>
12
. For each problem,
1
1
1
point is given for a right answer and
0
0
0
is given for a wrong answer. Analysing each possible situation, we find that if the sum of points every group of
12
12
12
people get is no less than
36
36
36
, then there are at least
3
3
3
people that got the right answer of a certain problem, among the
n
n
n
people. Find the least possible
n
n
n
.
2
2
Hide problems
AD passes through the circumcenter
Given an acute triangle
A
B
C
ABC
A
BC
,
D
D
D
is a point on
B
C
BC
BC
. A circle with diameter
B
D
BD
B
D
intersects line
A
B
,
A
D
AB,AD
A
B
,
A
D
at
X
,
P
X,P
X
,
P
respectively (different from
B
,
D
B,D
B
,
D
).The circle with diameter
C
D
CD
C
D
intersects
A
C
,
A
D
AC,AD
A
C
,
A
D
at
Y
,
Q
Y,Q
Y
,
Q
respectively (different from
C
,
D
C,D
C
,
D
). Draw two lines through
A
A
A
perpendicular to
P
X
,
Q
Y
PX,QY
PX
,
Q
Y
, the feet are
M
,
N
M,N
M
,
N
respectively.Prove that
△
A
M
N
\triangle AMN
△
A
MN
is similar to
△
A
B
C
\triangle ABC
△
A
BC
if and only if
A
D
AD
A
D
passes through the circumcenter of
△
A
B
C
\triangle ABC
△
A
BC
.
find the least k
Given an integer
n
≥
3
n\ge\ 3
n
≥
3
, find the least positive integer
k
k
k
, such that there exists a set
A
A
A
with
k
k
k
elements, and
n
n
n
distinct reals
x
1
,
x
2
,
…
,
x
n
x_{1},x_{2},\ldots,x_{n}
x
1
,
x
2
,
…
,
x
n
such that
x
1
+
x
2
,
x
2
+
x
3
,
…
,
x
n
−
1
+
x
n
,
x
n
+
x
1
x_{1}+x_{2}, x_{2}+x_{3},\ldots, x_{n-1}+x_{n}, x_{n}+x_{1}
x
1
+
x
2
,
x
2
+
x
3
,
…
,
x
n
−
1
+
x
n
,
x
n
+
x
1
all belong to
A
A
A
.
1
2
Hide problems
exists a polynomial
Let
M
M
M
be the set of the real numbers except for finitely many elements. Prove that for every positive integer
n
n
n
there exists a polynomial
f
(
x
)
f(x)
f
(
x
)
with
deg
f
=
n
\deg f = n
de
g
f
=
n
, such that all the coefficients and the
n
n
n
real roots of
f
f
f
are all in
M
M
M
.
find the last two digits
Define a sequence
(
x
n
)
n
≥
1
(x_{n})_{n\geq 1}
(
x
n
)
n
≥
1
by taking
x
1
∈
{
5
,
7
}
x_{1}\in\left\{5,7\right\}
x
1
∈
{
5
,
7
}
; when
k
≥
1
k\ge 1
k
≥
1
,
x
k
+
1
∈
{
5
x
k
,
7
x
k
}
x_{k+1}\in\left\{5^{x_{k}},7^{x_{k}}\right\}
x
k
+
1
∈
{
5
x
k
,
7
x
k
}
. Determine all possible last two digits of
x
2009
x_{2009}
x
2009
.