MathDB
Problems
Contests
National and Regional Contests
China Contests
China Team Selection Test
1994 China Team Selection Test
1994 China Team Selection Test
Part of
China Team Selection Test
Subcontests
(3)
3
2
Hide problems
line of symmetry and every red point is reflected
Find the smallest
n
∈
N
n \in \mathbb{N}
n
∈
N
such that if any 5 vertices of a regular
n
n
n
-gon are colored red, there exists a line of symmetry
l
l
l
of the
n
n
n
-gon such that every red point is reflected across
l
l
l
to a non-red point.
all the vertices of S are vertices of T
For any 2 convex polygons
S
S
S
and
T
T
T
, if all the vertices of
S
S
S
are vertices of
T
T
T
, call
S
S
S
a sub-polygon of
T
T
T
. I. Prove that for an odd number
n
≥
5
n \geq 5
n
≥
5
, there exists
m
m
m
sub-polygons of a convex
n
n
n
-gon such that they do not share any edges, and every edge and diagonal of the
n
n
n
-gon are edges of the
m
m
m
sub-polygons. II. Find the smallest possible value of
m
m
m
.
2
1
Hide problems
the numbers form an arithmetic progression
An
n
n
n
by
n
n
n
grid, where every square contains a number, is called an
n
n
n
-code if the numbers in every row and column form an arithmetic progression. If it is sufficient to know the numbers in certain squares of an
n
n
n
-code to obtain the numbers in the entire grid, call these squares a key. a.) Find the smallest
s
∈
N
s \in \mathbb{N}
s
∈
N
such that any
s
s
s
squares in an
n
−
n-
n
−
code
(
n
≥
4
)
(n \geq 4)
(
n
≥
4
)
form a key. b.) Find the smallest
t
∈
N
t \in \mathbb{N}
t
∈
N
such that any
t
t
t
squares along the diagonals of an
n
n
n
-code
(
n
≥
4
)
(n \geq 4)
(
n
≥
4
)
form a key.
1
2
Hide problems
5n numbers inequality
Given
5
n
5n
5
n
real numbers
r
i
,
s
i
,
t
i
,
u
i
,
v
i
≥
1
(
1
≤
i
≤
n
)
r_i, s_i, t_i, u_i, v_i \geq 1 (1 \leq i \leq n)
r
i
,
s
i
,
t
i
,
u
i
,
v
i
≥
1
(
1
≤
i
≤
n
)
, let
R
=
1
n
∑
i
=
1
n
r
i
R = \frac {1}{n} \sum_{i=1}^{n} r_i
R
=
n
1
∑
i
=
1
n
r
i
,
S
=
1
n
∑
i
=
1
n
s
i
S = \frac {1}{n} \sum_{i=1}^{n} s_i
S
=
n
1
∑
i
=
1
n
s
i
,
T
=
1
n
∑
i
=
1
n
t
i
T = \frac {1}{n} \sum_{i=1}^{n} t_i
T
=
n
1
∑
i
=
1
n
t
i
,
U
=
1
n
∑
i
=
1
n
u
i
U = \frac {1}{n} \sum_{i=1}^{n} u_i
U
=
n
1
∑
i
=
1
n
u
i
,
V
=
1
n
∑
i
=
1
n
v
i
V = \frac {1}{n} \sum_{i=1}^{n} v_i
V
=
n
1
∑
i
=
1
n
v
i
. Prove that
∏
i
=
1
n
r
i
s
i
t
i
u
i
v
i
+
1
r
i
s
i
t
i
u
i
v
i
−
1
≥
(
R
S
T
U
V
+
1
R
S
T
U
V
−
1
)
n
\prod_{i=1}^{n}\frac {r_i s_i t_i u_i v_i + 1}{r_i s_i t_i u_i v_i - 1} \geq \left(\frac {RSTUV +1}{RSTUV - 1}\right)^n
∏
i
=
1
n
r
i
s
i
t
i
u
i
v
i
−
1
r
i
s
i
t
i
u
i
v
i
+
1
≥
(
RST
U
V
−
1
RST
U
V
+
1
)
n
.
all sets comprising of 4 natural numbers
Find all sets comprising of 4 natural numbers such that the product of any 3 numbers in the set leaves a remainder of 1 when divided by the remaining number.