MathDB
Problems
Contests
National and Regional Contests
China Contests
China Team Selection Test
2000 China Team Selection Test
2000 China Team Selection Test
Part of
China Team Selection Test
Subcontests
(3)
3
2
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Sum of squares of digits of k in base a representation
For positive integer
a
≥
2
a \geq 2
a
≥
2
, denote
N
a
N_a
N
a
as the number of positive integer
k
k
k
with the following property: the sum of squares of digits of
k
k
k
in base a representation equals
k
k
k
. Prove that: a.)
N
a
N_a
N
a
is odd; b.) For every positive integer
M
M
M
, there exist a positive integer
a
≥
2
a \geq 2
a
≥
2
such that
N
a
≥
M
N_a \geq M
N
a
≥
M
.
Explicit value of N(4)
Let
n
n
n
be a positive integer. Denote
M
=
{
(
x
,
y
)
∣
x
,
y
are integers
,
1
≤
x
,
y
≤
n
}
M = \{(x, y)|x, y \text{ are integers }, 1 \leq x, y \leq n\}
M
=
{(
x
,
y
)
∣
x
,
y
are integers
,
1
≤
x
,
y
≤
n
}
. Define function
f
f
f
on
M
M
M
with the following properties: a.)
f
(
x
,
y
)
f(x, y)
f
(
x
,
y
)
takes non-negative integer value; b.)
∑
y
=
1
n
f
(
x
,
y
)
=
n
−
1
\sum^n_{y=1} f(x, y) = n - 1
∑
y
=
1
n
f
(
x
,
y
)
=
n
−
1
for 1 \eq x \leq n; c.) If
f
(
x
1
,
y
1
)
f
(
x
2
,
y
2
)
>
0
f(x_1, y_1)f(x2, y2) > 0
f
(
x
1
,
y
1
)
f
(
x
2
,
y
2
)
>
0
, then
(
x
1
−
x
2
)
(
y
1
−
y
2
)
≥
0.
(x_1 - x_2)(y_1 - y_2) \geq 0.
(
x
1
−
x
2
)
(
y
1
−
y
2
)
≥
0.
Find
N
(
n
)
N(n)
N
(
n
)
, the number of functions
f
f
f
that satisfy all the conditions. Give the explicit value of
N
(
4
)
N(4)
N
(
4
)
.
2
2
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Faculty fraction sum
Given positive integers
k
,
m
,
n
k, m, n
k
,
m
,
n
such that
1
≤
k
≤
m
≤
n
1 \leq k \leq m \leq n
1
≤
k
≤
m
≤
n
. Evaluate
∑
i
=
0
n
(
−
1
)
i
n
+
k
+
i
⋅
(
m
+
n
+
i
)
!
i
!
(
n
−
i
)
!
(
m
+
i
)
!
.
\sum^{n}_{i=0} \frac{(-1)^i}{n+k+i} \cdot \frac{(m+n+i)!}{i!(n-i)!(m+i)!}.
i
=
0
∑
n
n
+
k
+
i
(
−
1
)
i
⋅
i
!
(
n
−
i
)!
(
m
+
i
)!
(
m
+
n
+
i
)!
.
Period functiona and smallest period
a.) Let
a
,
b
a,b
a
,
b
be real numbers. Define sequence
x
k
x_k
x
k
and
y
k
y_k
y
k
such that x_0 = 1, y_0 = 0, x_{k+1} = a \cdot x_k - b \cdot y_l, y_{k+1} = x_k - a \cdot y_k \text{ for } k = 0,1,2, \ldots Prove that
x
k
=
∑
l
=
0
[
k
/
2
]
(
−
1
)
l
⋅
a
k
−
2
⋅
l
⋅
(
a
2
+
b
)
l
⋅
λ
k
,
l
x_k = \sum^{[k/2]}_{l=0} (-1)^l \cdot a^{k - 2 \cdot l} \cdot \left(a^2 + b \right)^l \cdot \lambda_{k,l}
x
k
=
l
=
0
∑
[
k
/2
]
(
−
1
)
l
⋅
a
k
−
2
⋅
l
⋅
(
a
2
+
b
)
l
⋅
λ
k
,
l
where
λ
k
,
l
=
∑
m
=
l
[
k
/
2
]
(
k
2
⋅
m
)
⋅
(
m
l
)
\lambda_{k,l} = \sum^{[k/2]}_{m=l} \binom{k}{2 \cdot m} \cdot \binom{m}{l}
λ
k
,
l
=
∑
m
=
l
[
k
/2
]
(
2
⋅
m
k
)
⋅
(
l
m
)
b.) Let
u
k
=
∑
l
=
0
[
k
/
2
]
λ
k
,
l
u_k = \sum^{[k/2]}_{l=0} \lambda_{k,l}
u
k
=
∑
l
=
0
[
k
/2
]
λ
k
,
l
. For positive integer
m
,
m,
m
,
denote the remainder of
u
k
u_k
u
k
divided by
2
m
2^m
2
m
as
z
m
,
k
z_{m,k}
z
m
,
k
. Prove that
z
m
,
k
,
z_{m,k},
z
m
,
k
,
k
=
0
,
1
,
2
,
…
k = 0,1,2, \ldots
k
=
0
,
1
,
2
,
…
is a periodic function, and find the smallest period.
1
2
Hide problems
China TST 2000 circumcircle of triangle ADE
Let
A
B
C
ABC
A
BC
be a triangle such that
A
B
=
A
C
AB = AC
A
B
=
A
C
. Let
D
,
E
D,E
D
,
E
be points on
A
B
,
A
C
AB,AC
A
B
,
A
C
respectively such that
D
E
=
A
C
DE = AC
D
E
=
A
C
. Let
D
E
DE
D
E
meet the circumcircle of triangle
A
B
C
ABC
A
BC
at point
T
T
T
. Let
P
P
P
be a point on
A
T
AT
A
T
. Prove that
P
D
+
P
E
=
A
T
PD + PE = AT
P
D
+
PE
=
A
T
if and only if
P
P
P
lies on the circumcircle of triangle
A
D
E
ADE
A
D
E
.
Coefficients of Gamma function
Let
F
F
F
be the set of all polynomials
Γ
\Gamma
Γ
such that all the coefficients of
Γ
(
x
)
\Gamma (x)
Γ
(
x
)
are integers and
Γ
(
x
)
=
1
\Gamma (x) = 1
Γ
(
x
)
=
1
has integer roots. Given a positive intger
k
k
k
, find the smallest integer
m
(
k
)
>
1
m(k) > 1
m
(
k
)
>
1
such that there exist
Γ
∈
F
\Gamma \in F
Γ
∈
F
for which
Γ
(
x
)
=
m
(
k
)
\Gamma (x) = m(k)
Γ
(
x
)
=
m
(
k
)
has exactly
k
k
k
distinct integer roots.