MathDB
Problems
Contests
National and Regional Contests
China Contests
China National Olympiad
1997 China National Olympiad
1997 China National Olympiad
Part of
China National Olympiad
Subcontests
(3)
3
2
Hide problems
Split 1,2,..,3n into 3 sequences of n terms
Prove that there are infinitely many natural numbers
n
n
n
such that we can divide
1
,
2
,
…
,
3
n
1,2,\ldots ,3n
1
,
2
,
…
,
3
n
into three sequences
(
a
n
)
,
(
b
n
)
(a_n),(b_n)
(
a
n
)
,
(
b
n
)
and
(
c
n
)
(c_n)
(
c
n
)
, with
n
n
n
terms in each, satisfying the following conditions: i)
a
1
+
b
1
+
c
1
=
a
2
+
b
2
+
c
2
=
…
=
a
n
+
b
n
+
c
n
a_1+b_1+c_1= a_2+b_2+c_2=\ldots =a_n+b_n+c_n
a
1
+
b
1
+
c
1
=
a
2
+
b
2
+
c
2
=
…
=
a
n
+
b
n
+
c
n
and
a
1
+
b
1
+
c
1
a_1+b_1+c_1
a
1
+
b
1
+
c
1
is divisible by
6
6
6
; ii)
a
1
+
a
2
+
…
+
a
n
=
b
1
+
b
2
+
…
+
b
n
=
c
1
+
c
2
+
…
+
c
n
,
a_1+a_2+\ldots +a_n= b_1+b_2+\ldots +b_n=c_1+c_2+\ldots +c_n,
a
1
+
a
2
+
…
+
a
n
=
b
1
+
b
2
+
…
+
b
n
=
c
1
+
c
2
+
…
+
c
n
,
and
a
1
+
a
2
+
…
+
a
n
a_1+a_2+\ldots +a_n
a
1
+
a
2
+
…
+
a
n
is divisible by
6
6
6
.
Sequence satisfying the condition a_(m+n) < a_n + a_m
Let
(
a
n
)
(a_n)
(
a
n
)
be a sequence of non-negative real numbers satisfying
a
n
+
m
≤
a
n
+
a
m
a_{n+m}\le a_n+a_m
a
n
+
m
≤
a
n
+
a
m
for all non-negative integers
m
,
n
m,n
m
,
n
. Prove that if
n
≥
m
n\ge m
n
≥
m
then
a
n
≤
m
a
1
+
(
n
m
−
1
)
a
m
a_n\le ma_1+\left(\dfrac{n}{m}-1\right)a_m
a
n
≤
m
a
1
+
(
m
n
−
1
)
a
m
holds.
2
2
Hide problems
BIjective map on {1,2,...,17}
Let
A
=
{
1
,
2
,
3
,
⋯
,
17
}
A=\{1,2,3,\cdots ,17\}
A
=
{
1
,
2
,
3
,
⋯
,
17
}
. A mapping
f
:
A
→
A
f:A\rightarrow A
f
:
A
→
A
is defined as follows:
f
[
1
]
(
x
)
=
f
(
x
)
,
f
[
k
+
1
]
(
x
)
=
f
(
f
[
k
]
(
x
)
)
f^{[1]}(x)=f(x), f^{[k+1]}(x)=f(f^{[k]}(x))
f
[
1
]
(
x
)
=
f
(
x
)
,
f
[
k
+
1
]
(
x
)
=
f
(
f
[
k
]
(
x
))
for
k
∈
N
k\in\mathbb{N}
k
∈
N
. Suppose that
f
f
f
is bijective and that there exists a natural number
M
M
M
such that: i) when
m
<
M
m<M
m
<
M
and
1
≤
i
≤
16
1\le i\le 16
1
≤
i
≤
16
, we have
f
[
m
]
(
i
+
1
)
−
f
[
m
]
(
i
)
≠
±
1
(
m
o
d
17
)
f^{[m]}(i+1)- f^{[m]}(i) \not=\pm 1\pmod{17}
f
[
m
]
(
i
+
1
)
−
f
[
m
]
(
i
)
=
±
1
(
mod
17
)
and
f
[
m
]
(
1
)
−
f
[
m
]
(
17
)
≠
±
1
(
m
o
d
17
)
f^{[m]}(1)- f^{[m]}(17) \not=\pm 1\pmod{17}
f
[
m
]
(
1
)
−
f
[
m
]
(
17
)
=
±
1
(
mod
17
)
; ii) when
1
≤
i
≤
16
1\le i\le 16
1
≤
i
≤
16
, we have
f
[
M
]
(
i
+
1
)
−
f
[
M
]
(
i
)
=
±
1
(
m
o
d
17
)
f^{[M]}(i+1)- f^{[M]}(i)=\pm 1 \pmod{17}
f
[
M
]
(
i
+
1
)
−
f
[
M
]
(
i
)
=
±
1
(
mod
17
)
and
f
[
M
]
(
1
)
−
f
[
M
]
(
17
)
=
±
1
(
m
o
d
17
)
f^{[M]}(1)- f^{[M]}(17)=\pm 1\pmod{17}
f
[
M
]
(
1
)
−
f
[
M
]
(
17
)
=
±
1
(
mod
17
)
. Find the maximal value of
M
M
M
.
Cyclic quadrilaterals among the first 12 in a sequence
Let
A
1
B
1
C
1
D
1
A_1B_1C_1D_1
A
1
B
1
C
1
D
1
be an arbitrary convex quadrilateral.
P
P
P
is a point inside the quadrilateral such that each angle enclosed by one edge and one ray which starts at one vertex on that edge and passes through point
P
P
P
is acute. We recursively define points
A
k
,
B
k
,
C
k
,
D
k
A_k,B_k,C_k,D_k
A
k
,
B
k
,
C
k
,
D
k
symmetric to
P
P
P
with respect to lines
A
k
−
1
B
k
−
1
,
B
k
−
1
C
k
−
1
,
C
k
−
1
D
k
−
1
,
D
k
−
1
A
k
−
1
A_{k-1}B_{k-1}, B_{k-1}C_{k-1}, C_{k-1}D_{k-1},D_{k-1}A_{k-1}
A
k
−
1
B
k
−
1
,
B
k
−
1
C
k
−
1
,
C
k
−
1
D
k
−
1
,
D
k
−
1
A
k
−
1
respectively for
k
≥
2
k\ge 2
k
≥
2
. Consider the sequence of quadrilaterals
A
i
B
i
C
i
D
i
A_iB_iC_iD_i
A
i
B
i
C
i
D
i
. i) Among the first 12 quadrilaterals, which are similar to the 1997th quadrilateral and which are not? ii) Suppose the 1997th quadrilateral is cyclic. Among the first 12 quadrilaterals, which are cyclic and which are not?
1
2
Hide problems
Maximum value of sum of 12th powers of x_i
Let
x
1
,
x
2
,
…
,
x
1997
x_1,x_2,\ldots ,x_{1997}
x
1
,
x
2
,
…
,
x
1997
be real numbers satisfying the following conditions: i)
−
1
3
≤
x
i
≤
3
-\dfrac{1}{\sqrt{3}}\le x_i\le \sqrt{3}
−
3
1
≤
x
i
≤
3
for
i
=
1
,
2
,
…
,
1997
i=1,2,\ldots ,1997
i
=
1
,
2
,
…
,
1997
; ii)
x
1
+
x
2
+
⋯
+
x
1997
=
−
318
3
x_1+x_2+\cdots +x_{1997}=-318 \sqrt{3}
x
1
+
x
2
+
⋯
+
x
1997
=
−
318
3
. Determine (with proof) the maximum value of
x
1
12
+
x
2
12
+
…
+
x
1997
12
x^{12}_1+x^{12}_2+\ldots +x^{12}_{1997}
x
1
12
+
x
2
12
+
…
+
x
1997
12
.
Show that P, E an F are collinear
Consider a cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
. The extensions of its sides
A
B
,
D
C
AB,DC
A
B
,
D
C
meet at the point
P
P
P
and the extensions of its sides
A
D
,
B
C
AD,BC
A
D
,
BC
meet at the point
Q
Q
Q
. Suppose QE,QF are tangents to the circumcircle of quadrilateral
A
B
C
D
ABCD
A
BC
D
at
E
,
F
E,F
E
,
F
respectively. Show that
P
,
E
,
F
P,E,F
P
,
E
,
F
are collinear.