MathDB
Problems
Contests
National and Regional Contests
China Contests
China Northern MO
2014 China Northern MO
2014 China Northern MO
Part of
China Northern MO
Subcontests
(8)
8
1
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2player game blowing up a balloon
Two people,
A
A
A
and
B
B
B
, play the game of blowing up a balloon. The balloon will explode only when the volume of the balloon
V
>
2014
V>2014
V
>
2014
mL.
A
A
A
blows in
1
1
1
mL first, and then they takes turns blowing. It is agreed that the gas blown by each person must not be less than the gas blown by the other party last time and should not be more than twice the amount of gas the other party blew last time. The agreement is that the person who blows up the balloon loses. Who has a winning strategy ? Briefly explain it. (Do not consider the change in volume caused by the change in tension when the balloon is inflated).
4
1
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12 candidates in election, 6 members voting at an election committee
In an election, there are a total of
12
12
12
candidates. An election committee has
6
6
6
members voting. It is known that at most two candidates voted by any two committee members are the same. Find the maximum number of committee members.
5
1
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<HAB=<HAD if ABCD #, I incenter of BCD, H orthocenter of IBD
As shown in the figure, in the parallelogram
A
B
C
D
ABCD
A
BC
D
,
I
I
I
is the incenter of
△
B
C
D
\vartriangle BCD
△
BC
D
, and
H
H
H
is the orthocenter of
△
I
B
D
\vartriangle IBD
△
I
B
D
. Prove that
∠
H
A
B
=
∠
H
A
D
\angle HAB=\angle HAD
∠
H
A
B
=
∠
H
A
D
. https://cdn.artofproblemsolving.com/attachments/4/3/5fa16c208ef3940443854756ae7bdb9c4272ed.png
1
1
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converse of perpendicularity in isosceles triangle (ASU 1962)
As shown in the figure, given
△
A
B
C
\vartriangle ABC
△
A
BC
with
∠
B
\angle B
∠
B
,
∠
C
\angle C
∠
C
acute angles,
A
D
⊥
B
C
AD \perp BC
A
D
⊥
BC
,
D
E
⊥
A
C
DE \perp AC
D
E
⊥
A
C
,
M
M
M
midpoint of
D
E
DE
D
E
,
A
M
⊥
B
E
AM \perp BE
A
M
⊥
BE
. Prove that
△
A
B
C
\vartriangle ABC
△
A
BC
is isosceles. https://cdn.artofproblemsolving.com/attachments/a/8/f553c33557979f6f7b799935c3bde743edcc3c.png
2
1
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China Northern Mathematical Olympiad 2014 , Problem 2
Define a positive number sequence sequence
{
a
n
}
\{a_n\}
{
a
n
}
by
a
1
=
1
,
(
n
2
+
1
)
a
n
−
1
2
=
(
n
−
1
)
2
a
n
2
.
a_{1}=1,(n^2+1)a^2_{n-1}=(n-1)^2a^2_{n}.
a
1
=
1
,
(
n
2
+
1
)
a
n
−
1
2
=
(
n
−
1
)
2
a
n
2
.
Prove that
1
a
1
2
+
1
a
2
2
+
⋯
+
1
a
n
2
≤
1
+
1
−
1
a
n
2
.
\frac{1}{a^2_1}+\frac{1}{a^2_2}+\cdots +\frac{1}{a^2_n}\le 1+\sqrt{1-\frac{1}{a^2_n}} .
a
1
2
1
+
a
2
2
1
+
⋯
+
a
n
2
1
≤
1
+
1
−
a
n
2
1
.
3
1
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China Northern Mathematical Olympiad 2014 , Problem 3
Determine whether there exist an infinite number of positive integers
x
,
y
x,y
x
,
y
satisfying the condition:
x
2
+
y
∣
x
+
y
2
.
x^2+y \mid x+y^2.
x
2
+
y
∣
x
+
y
2
.
Please prove it.
7
1
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China Northern Mathematical Olympiad 2014 , Problem 7
Prove that there exist infinitely many positive integers
n
n
n
such that
3
n
+
2
3^n+2
3
n
+
2
and
5
n
+
2
5^n+2
5
n
+
2
are all composite numbers.
6
1
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China Northern Mathematical Olympiad 2014 , Problem 6
Let
x
,
y
,
z
,
w
x,y,z,w
x
,
y
,
z
,
w
be real numbers such that
x
+
2
y
+
3
z
+
4
w
=
1
x+2y+3z+4w=1
x
+
2
y
+
3
z
+
4
w
=
1
. Find the minimum of
x
2
+
y
2
+
z
2
+
w
2
+
(
x
+
y
+
z
+
w
)
2
x^2+y^2+z^2+w^2+(x+y+z+w)^2
x
2
+
y
2
+
z
2
+
w
2
+
(
x
+
y
+
z
+
w
)
2
.