MathDB
Problems
Contests
National and Regional Contests
China Contests
China Team Selection Test
1993 China Team Selection Test
1993 China Team Selection Test
Part of
China Team Selection Test
Subcontests
(3)
2
2
Hide problems
a/b + c/d for fixed n
Let
n
≥
2
,
n
∈
N
n \geq 2, n \in \mathbb{N}
n
≥
2
,
n
∈
N
,
a
,
b
,
c
,
d
∈
N
a,b,c,d \in \mathbb{N}
a
,
b
,
c
,
d
∈
N
,
a
b
+
c
d
<
1
\frac{a}{b} + \frac{c}{d} < 1
b
a
+
d
c
<
1
and
a
+
c
≤
n
,
a + c \leq n,
a
+
c
≤
n
,
find the maximum value of
a
b
+
c
d
\frac{a}{b} + \frac{c}{d}
b
a
+
d
c
for fixed
n
.
n.
n
.
Subset without squares
Let
S
=
{
(
x
,
y
)
∣
x
=
1
,
2
,
…
,
1993
,
y
=
1
,
2
,
3
,
4
}
S = \{(x,y) | x = 1, 2, \ldots, 1993, y = 1, 2, 3, 4\}
S
=
{(
x
,
y
)
∣
x
=
1
,
2
,
…
,
1993
,
y
=
1
,
2
,
3
,
4
}
. If
T
⊂
S
T \subset S
T
⊂
S
and there aren't any squares in
T
.
T.
T
.
Find the maximum possible value of
∣
T
∣
.
|T|.
∣
T
∣.
The squares in T use points in S as vertices.
3
2
Hide problems
n-colored graph without triangles
A graph
G
=
(
V
,
E
)
G=(V,E)
G
=
(
V
,
E
)
is given. If at least
n
n
n
colors are required to paints its vertices so that between any two same colored vertices no edge is connected, then call this graph ''
n
−
n-
n
−
colored''. Prove that for any
n
∈
N
n \in \mathbb{N}
n
∈
N
, there is a
n
−
n-
n
−
colored graph without triangles.
Extend DP until it cuts the circumcircle again at N
Let
A
B
C
ABC
A
BC
be a triangle and its bisector at
A
A
A
cuts its circumcircle at
D
.
D.
D
.
Let
I
I
I
be the incenter of triangle
A
B
C
,
ABC,
A
BC
,
M
M
M
be the midpoint of
B
C
,
BC,
BC
,
P
P
P
is the symmetric to
I
I
I
with respect to
M
M
M
(Assuming
P
P
P
is in the circumcircle). Extend
D
P
DP
D
P
until it cuts the circumcircle again at
N
.
N.
N
.
Prove that among segments
A
N
,
B
N
,
C
N
AN, BN, CN
A
N
,
BN
,
CN
, there is a segment that is the sum of the other two.
1
2
Hide problems
find the value of f(p)
For all primes
p
≥
3
,
p \geq 3,
p
≥
3
,
define
F
(
p
)
=
∑
k
=
1
p
−
1
2
k
120
F(p) = \sum^{\frac{p-1}{2}}_{k=1}k^{120}
F
(
p
)
=
∑
k
=
1
2
p
−
1
k
120
and
f
(
p
)
=
1
2
−
{
F
(
p
)
p
}
f(p) = \frac{1}{2} - \left\{ \frac{F(p)}{p} \right\}
f
(
p
)
=
2
1
−
{
p
F
(
p
)
}
, where
{
x
}
=
x
−
[
x
]
,
\{x\} = x - [x],
{
x
}
=
x
−
[
x
]
,
find the value of
f
(
p
)
.
f(p).
f
(
p
)
.
2 * x^4 + 1 = y^2
Find all integer solutions to
2
x
4
+
1
=
y
2
.
2 x^4 + 1 = y^2.
2
x
4
+
1
=
y
2
.