MathDB
Problems
Contests
National and Regional Contests
China Contests
China National Olympiad
1998 China National Olympiad
1998 China National Olympiad
Part of
China National Olympiad
Subcontests
(3)
3
2
Hide problems
Least n such that any n element subset satisfies condition
Let
S
=
{
1
,
2
,
…
,
98
}
S=\{1,2,\ldots ,98\}
S
=
{
1
,
2
,
…
,
98
}
. Find the least natural number
n
n
n
such that we can pick out
10
10
10
numbers in any
n
n
n
-element subset of
S
S
S
satisfying the following condition: no matter how we equally divide the
10
10
10
numbers into two groups, there exists a number in one group such that it is coprime to the other numbers in that group, meanwhile there also exists a number in the other group such that it is not coprime to any of the other numbers in the same group.
Maximal value of |x_k|
Let
x
1
,
x
2
,
…
,
x
n
x_1,x_2,\ldots ,x_n
x
1
,
x
2
,
…
,
x
n
be real numbers, where
n
≥
2
n\ge 2
n
≥
2
, satisfying
∑
i
=
1
n
x
i
2
+
∑
i
=
1
n
−
1
x
i
x
i
+
1
=
1
\sum_{i=1}^{n}x^2_i+ \sum_{i=1}^{n-1}x_ix_{i+1}=1
∑
i
=
1
n
x
i
2
+
∑
i
=
1
n
−
1
x
i
x
i
+
1
=
1
. For each
k
k
k
, find the maximal value of
∣
x
k
∣
|x_k|
∣
x
k
∣
.
2
2
Hide problems
Do there exist such 2n positive integers?
Given a positive integer
n
>
1
n>1
n
>
1
, determine with proof if there exist
2
n
2n
2
n
pairwise different positive integers
a
1
,
…
,
a
n
,
b
1
,
…
b
n
a_1,\ldots ,a_n,b_1,\ldots b_n
a
1
,
…
,
a
n
,
b
1
,
…
b
n
such that
a
1
+
…
+
a
n
=
b
1
+
…
+
b
n
a_1+\ldots +a_n=b_1+\ldots +b_n
a
1
+
…
+
a
n
=
b
1
+
…
+
b
n
and
n
−
1
>
∑
i
=
1
n
a
i
−
b
i
a
i
+
b
i
>
n
−
1
−
1
1998
.
n-1>\sum_{i=1}^{n}\frac{a_i-b_i}{a_i+b_i}>n-1-\frac{1}{1998}.
n
−
1
>
i
=
1
∑
n
a
i
+
b
i
a
i
−
b
i
>
n
−
1
−
1998
1
.
Determine the position of D from condition
Let
D
D
D
be a point inside acute triangle
A
B
C
ABC
A
BC
satisfying the condition
D
A
⋅
D
B
⋅
A
B
+
D
B
⋅
D
C
⋅
B
C
+
D
C
⋅
D
A
⋅
C
A
=
A
B
⋅
B
C
⋅
C
A
.
DA\cdot DB\cdot AB+DB\cdot DC\cdot BC+DC\cdot DA\cdot CA=AB\cdot BC\cdot CA.
D
A
⋅
D
B
⋅
A
B
+
D
B
⋅
D
C
⋅
BC
+
D
C
⋅
D
A
⋅
C
A
=
A
B
⋅
BC
⋅
C
A
.
Determine (with proof) the geometric position of point
D
D
D
.
1
2
Hide problems
Find sin A if sqrt(2) OI = AB - AC
Let
A
B
C
ABC
A
BC
be a non-obtuse triangle satisfying
A
B
>
A
C
AB>AC
A
B
>
A
C
and
∠
B
=
4
5
∘
\angle B=45^{\circ}
∠
B
=
4
5
∘
. The circumcentre
O
O
O
and incentre
I
I
I
of triangle
A
B
C
ABC
A
BC
are such that
2
O
I
=
A
B
−
A
C
\sqrt{2}\ OI=AB-AC
2
O
I
=
A
B
−
A
C
. Find the value of
sin
A
\sin A
sin
A
.
Divisible by 2^2000
Find all natural numbers
n
>
3
n>3
n
>
3
, such that
2
2000
2^{2000}
2
2000
is divisible by
1
+
C
n
1
+
C
n
2
+
C
n
3
1+C^1_n+C^2_n+C^3_n
1
+
C
n
1
+
C
n
2
+
C
n
3
.