MathDB
Problems
Contests
National and Regional Contests
China Contests
(China) National High School Mathematics League
2013 China Second Round Olympiad
2013 China Second Round Olympiad
Part of
(China) National High School Mathematics League
Subcontests
(4)
4
1
Hide problems
Exist subset with sum divisible by n
Let
n
,
k
n,k
n
,
k
be integers greater than
1
1
1
,
n
<
2
k
n<2^k
n
<
2
k
. Prove that there exist
2
k
2k
2
k
integers none of which are divisible by
n
n
n
, such that no matter how they are separated into two groups there exist some numbers all from the same group whose sum is divisible by
n
n
n
.
2
1
Hide problems
Sequence with infinitely many squares
Let
u
,
v
u,v
u
,
v
be positive integers. Define sequence
{
a
n
}
\{a_n\}
{
a
n
}
as follows:
a
1
=
u
+
v
a_1=u+v
a
1
=
u
+
v
, and for integers
m
≥
1
m\ge 1
m
≥
1
,
{
a
2
m
=
a
m
+
u
,
a
2
m
+
1
=
a
m
+
v
,
\begin{array}{lll} \begin{cases} a_{2m}=a_m+u, \\ a_{2m+1}=a_m+v, \end{cases} \end{array}
{
a
2
m
=
a
m
+
u
,
a
2
m
+
1
=
a
m
+
v
,
Let
S
m
=
a
1
+
a
2
+
…
+
a
m
(
m
=
1
,
2
,
…
)
S_m=a_1+a_2+\ldots +a_m(m=1,2,\ldots )
S
m
=
a
1
+
a
2
+
…
+
a
m
(
m
=
1
,
2
,
…
)
. Prove that there are infinitely many perfect squares in the sequence
{
S
n
}
\{S_n\}
{
S
n
}
.
3
2
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Maximum of smallest and largest scores in test
n
n
n
students take a test with
m
m
m
questions, where
m
,
n
≥
2
m,n\ge 2
m
,
n
≥
2
are integers. The score given to every question is as such: for a certain question, if
x
x
x
students fails to answer it correctly, then those who answer it correctly scores
x
x
x
points, while those who answer it wrongly scores
0
0
0
. The score of a student is the sum of his scores for the
m
m
m
questions. Arrange the scores in descending order
p
1
≥
p
2
≥
…
≥
p
n
p_1\ge p_2\ge \ldots \ge p_n
p
1
≥
p
2
≥
…
≥
p
n
. Find the maximum value of
p
1
+
p
n
p_1+p_n
p
1
+
p
n
.
2013 China Second Round Olympiad (C) Test 2 Q3
The integers
n
>
1
n>1
n
>
1
is given . The positive integer
a
1
,
a
2
,
⋯
,
a
n
a_1,a_2,\cdots,a_n
a
1
,
a
2
,
⋯
,
a
n
satisfing condition : (1)
a
1
<
a
2
<
⋯
<
a
n
a_1<a_2<\cdots<a_n
a
1
<
a
2
<
⋯
<
a
n
; (2)
a
1
2
+
a
2
2
2
,
a
2
2
+
a
3
2
2
,
⋯
,
a
n
−
1
2
+
a
n
2
2
\frac{a^2_1+a^2_2}{2},\frac{a^2_2+a^2_3}{2},\cdots,\frac{a^2_{n-1}+a^2_n}{2}
2
a
1
2
+
a
2
2
,
2
a
2
2
+
a
3
2
,
⋯
,
2
a
n
−
1
2
+
a
n
2
are all perfect squares . Prove that :
a
n
≥
2
n
2
−
1.
a_n\ge 2n^2-1.
a
n
≥
2
n
2
−
1.
1
3
Hide problems
Trisecting a chord
A
B
AB
A
B
is a chord of circle
ω
\omega
ω
,
P
P
P
is a point on minor arc
A
B
AB
A
B
,
E
,
F
E,F
E
,
F
are on segment
A
B
AB
A
B
such that
A
E
=
E
F
=
F
B
AE=EF=FB
A
E
=
EF
=
FB
.
P
E
,
P
F
PE,PF
PE
,
PF
meets
ω
\omega
ω
at
C
,
D
C,D
C
,
D
respectively. Prove that
E
F
⋅
C
D
=
A
C
⋅
B
D
EF\cdot CD=AC\cdot BD
EF
⋅
C
D
=
A
C
⋅
B
D
.
2013 China Second Round Olympiad (B) Test 2 Q1
For any positive integer
n
n
n
, Prove that there is not exist three odd integer
x
,
y
,
z
x,y,z
x
,
y
,
z
satisfing the equation
(
x
+
y
)
n
+
(
y
+
z
)
n
=
(
x
+
z
)
n
(x+y)^n+(y+z)^n=(x+z)^n
(
x
+
y
)
n
+
(
y
+
z
)
n
=
(
x
+
z
)
n
.
2013 China Second Round Olympiad (C) Test 2 Q1
Let
n
n
n
be a positive odd integer ,
a
1
,
a
2
,
⋯
,
a
n
a_1,a_2,\cdots,a_n
a
1
,
a
2
,
⋯
,
a
n
be any permutation of the positive integers
1
,
2
,
⋯
,
n
1,2,\cdots,n
1
,
2
,
⋯
,
n
. Prove that :
(
a
1
−
1
)
(
a
2
2
−
2
)
(
a
3
3
−
3
)
⋯
(
a
n
n
−
n
)
(a_1-1)(a^2_2-2)(a^3_3-3)\cdots (a^n_n-n)
(
a
1
−
1
)
(
a
2
2
−
2
)
(
a
3
3
−
3
)
⋯
(
a
n
n
−
n
)
is an even number.