MathDB
Problems
Contests
National and Regional Contests
China Contests
China National Olympiad
1999 China National Olympiad
1999 China National Olympiad
Part of
China National Olympiad
Subcontests
(3)
2
2
Hide problems
Results about the sequence of polynomials f_n(x)
Let
a
a
a
be a real number. Let
(
f
n
(
x
)
)
n
≥
0
(f_n(x))_{n\ge 0}
(
f
n
(
x
)
)
n
≥
0
be a sequence of polynomials such that
f
0
(
x
)
=
1
f_0(x)=1
f
0
(
x
)
=
1
and
f
n
+
1
(
x
)
=
x
f
n
(
x
)
+
f
n
(
a
x
)
f_{n+1}(x)=xf_n(x)+f_n(ax)
f
n
+
1
(
x
)
=
x
f
n
(
x
)
+
f
n
(
a
x
)
for all non-negative integers
n
n
n
. a) Prove that
f
n
(
x
)
=
x
n
f
n
(
x
−
1
)
f_n(x)=x^nf_n\left(x^{-1}\right)
f
n
(
x
)
=
x
n
f
n
(
x
−
1
)
for all non-negative integers
n
n
n
. b) Find an explicit expression for
f
n
(
x
)
f_n(x)
f
n
(
x
)
.
Maximum value of lambda
Determine the maximum value of
λ
\lambda
λ
such that if
f
(
x
)
=
x
3
+
a
x
2
+
b
x
+
c
f(x) = x^3 +ax^2 +bx+c
f
(
x
)
=
x
3
+
a
x
2
+
b
x
+
c
is a cubic polynomial with all its roots nonnegative, then
f
(
x
)
≥
λ
(
x
−
a
)
3
f(x)\geq\lambda(x -a)^3
f
(
x
)
≥
λ
(
x
−
a
)
3
for all
x
≥
0
x\geq0
x
≥
0
. Find the equality condition.
3
2
Hide problems
Maximum number of connected groups among 99 space stations
There are
99
99
99
space stations. Each pair of space stations is connected by a tunnel. There are
99
99
99
two-way main tunnels, and all the other tunnels are strictly one-way tunnels. A group of
4
4
4
space stations is called connected if one can reach each station in the group from every other station in the group without using any tunnels other than the
6
6
6
tunnels which connect them. Determine the maximum number of connected groups.
Interesting colourings of a cube
A
4
×
4
×
4
4\times4\times4
4
×
4
×
4
cube is composed of
64
64
64
unit cubes. The faces of
16
16
16
unit cubes are to be coloured red. A colouring is called interesting if there is exactly
1
1
1
red unit cube in every
1
×
1
×
4
1\times1\times 4
1
×
1
×
4
rectangular box composed of
4
4
4
unit cubes. Determine the number of interesting colourings.
1
2
Hide problems
F is orthocentre of ABC iff HD||CF and H is on (ABC)
Let
A
B
C
ABC
A
BC
be an acute triangle with
∠
C
>
∠
B
\angle C>\angle B
∠
C
>
∠
B
. Let
D
D
D
be a point on
B
C
BC
BC
such that
∠
A
D
B
\angle ADB
∠
A
D
B
is obtuse, and let
H
H
H
be the orthocentre of triangle
A
B
D
ABD
A
B
D
. Suppose that
F
F
F
is a point inside triangle
A
B
C
ABC
A
BC
that is on the circumcircle of triangle
A
B
D
ABD
A
B
D
. Prove that
F
F
F
is the orthocenter of triangle
A
B
C
ABC
A
BC
if and only if
H
D
∣
∣
C
F
HD||CF
HD
∣∣
CF
and
H
H
H
is on the circumcircle of triangle
A
B
C
ABC
A
BC
.
Help me
Let
m
m
m
be a positive integer. Prove that there are integers
a
,
b
,
k
a, b, k
a
,
b
,
k
, such that both
a
a
a
and
b
b
b
are odd,
k
≥
0
k\geq0
k
≥
0
and
2
m
=
a
19
+
b
99
+
k
⋅
2
1999
2m=a^{19}+b^{99}+k\cdot2^{1999}
2
m
=
a
19
+
b
99
+
k
⋅
2
1999