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Problems
Contests
National and Regional Contests
China Contests
China Western Mathematical Olympiad
2003 China Western Mathematical Olympiad
2003 China Western Mathematical Olympiad
Part of
China Western Mathematical Olympiad
Subcontests
(4)
4
2
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Interior of a convex quadrilateral ABCD
Given that the sum of the distances from point
P
P
P
in the interior of a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
to the sides
A
B
,
B
C
,
C
D
,
D
A
AB, BC, CD, DA
A
B
,
BC
,
C
D
,
D
A
is a constant, prove that
A
B
C
D
ABCD
A
BC
D
is a parallelogram.
Pairs of students in same row and of same sex not greater 11
1650
1650
1650
students are arranged in
22
22
22
rows and
75
75
75
columns. It is known that in any two columns, the number of pairs of students in the same row and of the same sex is not greater than
11
11
11
. Prove that the number of boys is not greater than
928
928
928
.
3
2
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Odd integers among 1,..., 2n - 1 can be divided by d
Let
n
n
n
be a given positive integer. Find the smallest positive integer
u
n
u_n
u
n
such that for any positive integer
d
d
d
, in any
u
n
u_n
u
n
consecutive odd positive integers, the number of them that can be divided by
d
d
d
is not smaller than the number of odd integers among 1, 3, 5, \ldots, 2n \minus{} 1 that can be divided by
d
d
d
.
5-term sum of xi / [4 + xi^2] equal or less than one
The non-negative numbers
x
1
,
x
2
,
…
,
x
5
x_1, x_2, \ldots, x_5
x
1
,
x
2
,
…
,
x
5
satisfy \sum_{i \equal{} 1}^5 \frac {1}{1 \plus{} x_i} \equal{} 1. Prove that \sum_{i \equal{} 1}^5 \frac {x_i}{4 \plus{} x_i^2} \leq 1.
2
2
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Sum of differences over 2n-1 consecutive terms is one
Let
a
1
,
a
2
,
…
,
a
2
n
a_1, a_2, \ldots, a_{2n}
a
1
,
a
2
,
…
,
a
2
n
be
2
n
2n
2
n
real numbers satisfying the condition \sum_{i \equal{} 1}^{2n \minus{} 1} (a_{i \plus{} 1} \minus{} a_i)^2 \equal{} 1. Find the greatest possible value of (a_{n \plus{} 1} \plus{} a_{n \plus{} 2} \plus{} \ldots \plus{} a_{2n}) \minus{} (a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n).
Prove quadrilateral EFGH is a rectangle iff ABCD concyclic
A circle can be inscribed in the convex quadrilateral
A
B
C
D
ABCD
A
BC
D
. The incircle touches the sides
A
B
,
B
C
,
C
D
,
D
A
AB, BC, CD, DA
A
B
,
BC
,
C
D
,
D
A
at
A
1
,
B
1
,
C
1
,
D
1
A_1, B_1, C_1, D_1
A
1
,
B
1
,
C
1
,
D
1
respectively. The points
E
,
F
,
G
,
H
E, F, G, H
E
,
F
,
G
,
H
are the midpoints of
A
1
B
1
,
B
1
C
1
,
C
1
D
1
,
D
1
A
1
A_1B_1, B_1C_1, C_1D_1, D_1A_1
A
1
B
1
,
B
1
C
1
,
C
1
D
1
,
D
1
A
1
respectively. Prove that the quadrilateral
E
F
G
H
EFGH
EFG
H
is a rectangle if and only if
A
,
B
,
C
,
D
A, B, C, D
A
,
B
,
C
,
D
are concyclic.
1
2
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Find the smallest possible sum of the 4 numbers on a side
Place the numbers
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
1, 2, 3, 4, 5, 6, 7, 8
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
at the vertices of a cuboid such that the sum of any
3
3
3
numbers on a side is not less than
10
10
10
. Find the smallest possible sum of the 4 numbers on a side.
Prove that all the terms of the sequence are integral
The sequence
{
a
n
}
\{a_n\}
{
a
n
}
satisfies a_0 \equal{} 0, a_{n \plus{} 1} \equal{} ka_n \plus{} \sqrt {(k^2 \minus{} 1)a_n^2 \plus{} 1}, n \equal{} 0, 1, 2, \ldots, where
k
k
k
is a fixed positive integer. Prove that all the terms of the sequence are integral and that
2
k
2k
2
k
divides a_{2n}, n \equal{} 0, 1, 2, \ldots.