MathDB
Problems
Contests
National and Regional Contests
China Contests
China Western Mathematical Olympiad
2010 China Western Mathematical Olympiad
2010 China Western Mathematical Olympiad
Part of
China Western Mathematical Olympiad
Subcontests
(8)
8
1
Hide problems
CWMO 2010, Day 2, Problem 8
Determine all possible values of integer
k
k
k
for which there exist positive integers
a
a
a
and
b
b
b
such that
b
+
1
a
+
a
+
1
b
=
k
\dfrac{b+1}{a} + \dfrac{a+1}{b} = k
a
b
+
1
+
b
a
+
1
=
k
.
1
1
Hide problems
CWMO 2010, Day 1, Problem 1
Suppose that
m
m
m
and
k
k
k
are non-negative integers, and
p
=
2
2
m
+
1
p = 2^{2^m}+1
p
=
2
2
m
+
1
is a prime number. Prove that (a)
2
2
m
+
1
p
k
≡
1
2^{2^{m+1}p^k} \equiv 1
2
2
m
+
1
p
k
≡
1
(
mod
p
k
+
1
)
(\text{mod } p^{k+1})
(
mod
p
k
+
1
)
; (b)
2
m
+
1
p
k
2^{m+1}p^k
2
m
+
1
p
k
is the smallest positive integer
n
n
n
satisfying the congruence equation
2
n
≡
1
2^n \equiv 1
2
n
≡
1
(
mod
p
k
+
1
)
(\text{mod } p^{k+1})
(
mod
p
k
+
1
)
.
6
1
Hide problems
CWMO 2010, Day 2, Problem 6
Δ
A
B
C
\Delta ABC
Δ
A
BC
is a right-angled triangle,
∠
C
=
9
0
∘
\angle C = 90^{\circ}
∠
C
=
9
0
∘
. Draw a circle centered at
B
B
B
with radius
B
C
BC
BC
. Let
D
D
D
be a point on the side
A
C
AC
A
C
, and
D
E
DE
D
E
is tangent to the circle at
E
E
E
. The line through
C
C
C
perpendicular to
A
B
AB
A
B
meets line
B
E
BE
BE
at
F
F
F
. Line
A
F
AF
A
F
meets
D
E
DE
D
E
at point
G
G
G
. The line through
A
A
A
parallel to
B
G
BG
BG
meets
D
E
DE
D
E
at
H
H
H
. Prove that
G
E
=
G
H
GE = GH
GE
=
G
H
.
2
1
Hide problems
CWMO 2010, Day 1, Problem 2
A
B
AB
A
B
is a diameter of a circle with center
O
O
O
. Let
C
C
C
and
D
D
D
be two different points on the circle on the same side of
A
B
AB
A
B
, and the lines tangent to the circle at points
C
C
C
and
D
D
D
meet at
E
E
E
. Segments
A
D
AD
A
D
and
B
C
BC
BC
meet at
F
F
F
. Lines
E
F
EF
EF
and
A
B
AB
A
B
meet at
M
M
M
. Prove that
E
,
C
,
M
E,C,M
E
,
C
,
M
and
D
D
D
are concyclic.
7
1
Hide problems
CWMO 2010, Day 2, Problem 7
There are
n
n
n
(
n
≥
3
)
(n \ge 3)
(
n
≥
3
)
players in a table tennis tournament, in which any two players have a match. Player
A
A
A
is called not out-performed by player
B
B
B
, if at least one of player
A
A
A
's losers is not a
B
B
B
's loser.Determine, with proof, all possible values of
n
n
n
, such that the following case could happen: after finishing all the matches, every player is not out-performed by any other player.
3
1
Hide problems
CWMO 2010, Day 1, Problem 3
Determine all possible values of positive integer
n
n
n
, such that there are
n
n
n
different 3-element subsets
A
1
,
A
2
,
.
.
.
,
A
n
A_1,A_2,...,A_n
A
1
,
A
2
,
...
,
A
n
of the set
{
1
,
2
,
.
.
.
,
n
}
\{1,2,...,n\}
{
1
,
2
,
...
,
n
}
, with
∣
A
i
∩
A
j
∣
≠
1
|A_i \cap A_j| \not= 1
∣
A
i
∩
A
j
∣
=
1
for all
i
≠
j
i \not= j
i
=
j
.
5
1
Hide problems
CWMO 2010, Day 2, Problem 5
Let
k
k
k
be an integer and
k
>
1
k > 1
k
>
1
. Define a sequence
{
a
n
}
\{a_n\}
{
a
n
}
as follows:
a
0
=
0
a_0 = 0
a
0
=
0
,
a
1
=
1
a_1 = 1
a
1
=
1
, and
a
n
+
1
=
k
a
n
+
a
n
−
1
a_{n+1} = ka_n + a_{n-1}
a
n
+
1
=
k
a
n
+
a
n
−
1
for
n
=
1
,
2
,
.
.
.
n = 1,2,...
n
=
1
,
2
,
...
.Determine, with proof, all possible
k
k
k
for which there exist non-negative integers
l
,
m
(
l
≠
m
)
l,m (l \not= m)
l
,
m
(
l
=
m
)
and positive integers
p
,
q
p,q
p
,
q
such that
a
l
+
k
a
p
=
a
m
+
k
a
q
a_l + ka_p = a_m + ka_q
a
l
+
k
a
p
=
a
m
+
k
a
q
.
4
1
Hide problems
CWMO 2010, Day 1, Problem 4
Let
a
1
,
a
2
,
.
.
,
a
n
,
b
1
,
b
2
,
.
.
.
,
b
n
a_1,a_2,..,a_n,b_1,b_2,...,b_n
a
1
,
a
2
,
..
,
a
n
,
b
1
,
b
2
,
...
,
b
n
be non-negative numbers satisfying the following conditions simultaneously:(1)
∑
i
=
1
n
(
a
i
+
b
i
)
=
1
\displaystyle\sum_{i=1}^{n} (a_i + b_i) = 1
i
=
1
∑
n
(
a
i
+
b
i
)
=
1
;(2)
∑
i
=
1
n
i
(
a
i
−
b
i
)
=
0
\displaystyle\sum_{i=1}^{n} i(a_i - b_i) = 0
i
=
1
∑
n
i
(
a
i
−
b
i
)
=
0
;(3)
∑
i
=
1
n
i
2
(
a
i
+
b
i
)
=
10
\displaystyle\sum_{i=1}^{n} i^2(a_i + b_i) = 10
i
=
1
∑
n
i
2
(
a
i
+
b
i
)
=
10
.Prove that
max
{
a
k
,
b
k
}
≤
10
10
+
k
2
\text{max}\{a_k,b_k\} \le \dfrac{10}{10+k^2}
max
{
a
k
,
b
k
}
≤
10
+
k
2
10
for all
1
≤
k
≤
n
1 \le k \le n
1
≤
k
≤
n
.