MathDB
Problems
Contests
National and Regional Contests
China Contests
China National Olympiad
2000 China National Olympiad
2000 China National Olympiad
Part of
China National Olympiad
Subcontests
(3)
3
2
Hide problems
Tennis Club hosts a series of doubles matches
A table tennis club hosts a series of doubles matches following several rules: (i) each player belongs to two pairs at most; (ii) every two distinct pairs play one game against each other at most; (iii) players in the same pair do not play against each other when they pair with others respectively. Every player plays a certain number of games in this series. All these distinct numbers make up a set called the “set of games”. Consider a set
A
=
{
a
1
,
a
2
,
…
,
a
k
}
A=\{a_1,a_2,\ldots ,a_k\}
A
=
{
a
1
,
a
2
,
…
,
a
k
}
of positive integers such that every element in
A
A
A
is divisible by
6
6
6
. Determine the minimum number of players needed to participate in this series so that a schedule for which the corresponding set of games is equal to set
A
A
A
exists.
There are 4 sheets in any n among 2000
A test contains
5
5
5
multiple choice questions which have
4
4
4
options in each. Suppose each examinee chose one option for each question. There exists a number
n
n
n
, such that for any
n
n
n
sheets among
2000
2000
2000
sheets of answer papers, there are
4
4
4
sheets of answer papers such that any two of them have at most
3
3
3
questions with the same answers. Find the minimum value of
n
n
n
.
2
2
Hide problems
Closed form expression for f_n for sequence a_n
A sequence
(
a
n
)
(a_n)
(
a
n
)
is defined recursively by
a
1
=
0
,
a
2
=
1
a_1=0, a_2=1
a
1
=
0
,
a
2
=
1
and for
n
≥
3
n\ge 3
n
≥
3
,
a
n
=
1
2
n
a
n
−
1
+
1
2
n
(
n
−
1
)
a
n
−
2
+
(
−
1
)
n
(
1
−
n
2
)
.
a_n=\frac12na_{n-1}+\frac12n(n-1)a_{n-2}+(-1)^n\left(1-\frac{n}{2}\right).
a
n
=
2
1
n
a
n
−
1
+
2
1
n
(
n
−
1
)
a
n
−
2
+
(
−
1
)
n
(
1
−
2
n
)
.
Find a closed-form expression for
f
n
=
a
n
+
2
(
n
1
)
a
n
−
1
+
3
(
n
2
)
a
n
−
2
+
…
+
(
n
−
1
)
(
n
n
−
2
)
a
2
+
n
(
n
n
−
1
)
a
1
f_n=a_n+2\binom{n}{1}a_{n-1}+3\binom{n}{2}a_{n-2}+\ldots +(n-1)\binom{n}{n-2}a_2+n\binom{n}{n-1}a_1
f
n
=
a
n
+
2
(
1
n
)
a
n
−
1
+
3
(
2
n
)
a
n
−
2
+
…
+
(
n
−
1
)
(
n
−
2
n
)
a
2
+
n
(
n
−
1
n
)
a
1
.
Find all n that can be expressed in such a way
Find all positive integers
n
n
n
such that there exists integers
n
1
,
…
,
n
k
≥
3
n_1,\ldots,n_k\ge 3
n
1
,
…
,
n
k
≥
3
, for some integer
k
k
k
, satisfying
n
=
n
1
n
2
⋯
n
k
=
2
1
2
k
(
n
1
−
1
)
⋯
(
n
k
−
1
)
−
1.
n=n_1n_2\cdots n_k=2^{\frac{1}{2^k}(n_1-1)\cdots (n_k-1)}-1.
n
=
n
1
n
2
⋯
n
k
=
2
2
k
1
(
n
1
−
1
)
⋯
(
n
k
−
1
)
−
1.
1
2
Hide problems
Sign of a+b-2r-2r
The sides
a
,
b
,
c
a,b,c
a
,
b
,
c
of triangle
A
B
C
ABC
A
BC
satisfy
a
≤
b
≤
c
a\le b\le c
a
≤
b
≤
c
. The circumradius and inradius of triangle
A
B
C
ABC
A
BC
are
R
R
R
and
r
r
r
respectively. Let
f
=
a
+
b
−
2
R
−
2
r
f=a+b-2R-2r
f
=
a
+
b
−
2
R
−
2
r
. Determine the sign of
f
f
f
by the measure of angle
C
C
C
.
'Innovated tuple' and 'innovated degree'
Given an ordered
n
n
n
-tuple
A
=
(
a
1
,
a
2
,
⋯
,
a
n
)
A=(a_1,a_2,\cdots ,a_n)
A
=
(
a
1
,
a
2
,
⋯
,
a
n
)
of real numbers, where
n
≥
2
n\ge 2
n
≥
2
, we define
b
k
=
max
a
1
,
…
a
k
b_k=\max{a_1,\ldots a_k}
b
k
=
max
a
1
,
…
a
k
for each k. We define
B
=
(
b
1
,
b
2
,
⋯
,
b
n
)
B=(b_1,b_2,\cdots ,b_n)
B
=
(
b
1
,
b
2
,
⋯
,
b
n
)
to be the “innovated tuple” of
A
A
A
. The number of distinct elements in
B
B
B
is called the “innovated degree” of
A
A
A
. Consider all permutations of
1
,
2
,
…
,
n
1,2,\ldots ,n
1
,
2
,
…
,
n
as an ordered
n
n
n
-tuple. Find the arithmetic mean of the first term of the permutations whose innovated degrees are all equal to
2
2
2