MathDB
Problems
Contests
National and Regional Contests
China Contests
(China) National High School Mathematics League
2000 China Second Round Olympiad
2000 China Second Round Olympiad
Part of
(China) National High School Mathematics League
Subcontests
(3)
1
1
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A problem about equal areas
In acute-angled triangle
A
B
C
,
ABC,
A
BC
,
E
,
F
E,F
E
,
F
are on the side
B
C
,
BC,
BC
,
such that
∠
B
A
E
=
∠
C
A
F
,
\angle BAE=\angle CAF,
∠
B
A
E
=
∠
C
A
F
,
and let
M
,
N
M,N
M
,
N
be the projections of
F
F
F
onto
A
B
,
A
C
,
AB,AC,
A
B
,
A
C
,
respectively. The line
A
E
AE
A
E
intersects
⊙
(
A
B
C
)
\odot (ABC)
⊙
(
A
BC
)
at
D
D
D
(different from point
A
A
A
). Prove that
S
A
M
D
N
=
S
△
A
B
C
.
S_{AMDN}=S_{\triangle ABC}.
S
A
M
D
N
=
S
△
A
BC
.
3
1
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How many times do people contact with others
There are
n
n
n
people, and given that any
2
2
2
of them have contacted with each other at most once. In any group of
n
−
2
n-2
n
−
2
of them, any one person of the group has contacted with other people in this group for
3
k
3^k
3
k
times, where
k
k
k
is a non-negative integer. Determine all the possible value of
n
.
n.
n
.
2
1
Hide problems
a_n is always a perfect square.
Define the sequence
a
1
,
a
2
,
…
a_1, a_2, \ldots
a
1
,
a
2
,
…
and
b
1
,
b
2
,
…
b_1, b_2, \ldots
b
1
,
b
2
,
…
as
a
0
=
1
,
a
1
=
4
,
a
2
=
49
a_0=1,a_1=4,a_2=49
a
0
=
1
,
a
1
=
4
,
a
2
=
49
and for
n
≥
0
n \geq 0
n
≥
0
{
a
n
+
1
=
7
a
n
+
6
b
n
−
3
,
b
n
+
1
=
8
a
n
+
7
b
n
−
4.
\begin{cases} a_{n+1}=7a_n+6b_n-3, \\ b_{n+1}=8a_n+7b_n-4. \end{cases}
{
a
n
+
1
=
7
a
n
+
6
b
n
−
3
,
b
n
+
1
=
8
a
n
+
7
b
n
−
4.
Prove that for any non-negative integer
n
,
n,
n
,
a
n
a_n
a
n
is a perfect square.