MathDB
Problems
Contests
National and Regional Contests
China Contests
China Northern MO
2015 China Northern MO
2015 China Northern MO
Part of
China Northern MO
Subcontests
(8)
6
1
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2x1 tiles in 8x8 grid
The figure obtained by removing one small unit square from the
2
×
2
2\times 2
2
×
2
grid table is called an
L
L
L
''shape". .Put
k
k
k
L-shapes in an
8
×
8
8\times 8
8
×
8
grid table. Each
L
L
L
-shape can be rotated, but each
L
L
L
shape is required to cover exactly three small unit squares in the grid table, and the common area covered by any two
L
L
L
shapes is
0
0
0
, and except for these
k
k
k
L
L
L
shapes, no other
L
L
L
shapes can be placed. Find the minimum value of
k
k
k
.
8
2
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sum \frac{a_i}{ s-a_i}+\frac{ka_n}{s-a_n} >= \frac{n-1}{n-2
Given a positive integer
n
≥
3
n \ge 3
n
≥
3
. Find the smallest real number
k
k
k
such that for any positive real number except
a
1
,
a
2
,
.
.
,
a
n
a_1, a_2,..,a_n
a
1
,
a
2
,
..
,
a
n
,
∑
i
=
1
n
−
1
a
i
s
−
a
i
+
k
a
n
s
−
a
n
≥
n
−
1
n
−
2
\sum_{i=1}^{n-1}\frac{a_i}{ s-a_i}+\frac{ka_n}{s-a_n} \ge \frac{n-1}{n-2}
i
=
1
∑
n
−
1
s
−
a
i
a
i
+
s
−
a
n
k
a
n
≥
n
−
2
n
−
1
where,
s
=
a
1
+
a
2
+
.
.
+
a
n
s=a_1+a_2+..+a_n
s
=
a
1
+
a
2
+
..
+
a
n
a_{n+1} = (p_n^2+2015)(p_nq_n), bounded sequence?
The sequence
{
a
n
}
\{a_n\}
{
a
n
}
is defined as follows:
a
1
a_1
a
1
is a positive rational number,
a
n
=
p
n
q
n
a_n= \frac{p_n}{q_n}
a
n
=
q
n
p
n
, (
n
=
1
,
2
,
…
n= 1,2,…
n
=
1
,
2
,
…
) is a positive integer, where
p
n
p_n
p
n
and
q
n
q_n
q
n
are positive integers that are relatively prime, then
a
n
+
1
=
p
n
2
+
2015
p
n
q
n
a_{n+1} = \frac{p_n^2+2015}{p_nq_n}
a
n
+
1
=
p
n
q
n
p
n
2
+
2015
Is there a
1
>
2015
_1>2015
1
>
2015
, making the sequence
{
a
n
}
\{a_n\}
{
a
n
}
a bounded sequence? Justify your conclusion.
4
2
Hide problems
partition of 1-16 with a subset with elements a,b,c such that a+b=c
If the set
S
=
{
1
,
2
,
3
,
…
,
16
}
S = \{1,2,3,…,16\}
S
=
{
1
,
2
,
3
,
…
,
16
}
is partitioned into
n
n
n
subsets, there must be a subset in which elements
a
,
b
,
c
a, b, c
a
,
b
,
c
(can be the same) exist, satisfying
a
+
b
=
c
a+ b=c
a
+
b
=
c
. Find the maximum value of
n
n
n
.
combo NT with 108 pos. integers <= 2015
It is known that
a
1
,
a
2
,
.
.
.
a
108
a_1, a_2,...a_{108}
a
1
,
a
2
,
...
a
108
are
108
108
108
different positive integers not exceeding
2015
2015
2015
. Prove that there is a positive integer
k
k
k
such that there are at least four different pairs
(
i
,
j
)
(i, j)
(
i
,
j
)
satisfying
a
i
−
a
j
=
k
a_i-a_j =k
a
i
−
a
j
=
k
.
3
1
Hide problems
\phi (n)= 2^5 / 47 n , prime factorization
If
n
=
p
1
a
1
,
p
2
a
2
.
.
.
p
s
a
s
n=p_1^{a_1},p_2^{a_2}...p_s^{a_s}
n
=
p
1
a
1
,
p
2
a
2
...
p
s
a
s
then
ϕ
(
n
)
=
n
(
1
−
1
p
1
)
(
1
−
1
p
2
)
.
.
.
(
1
−
1
p
s
)
\phi (n)=n \left(1- \frac{1}{p_1}\right)\left(1 - \frac{1}{p_2}\right)...\left(1- \frac{1}{p_s}\right)
ϕ
(
n
)
=
n
(
1
−
p
1
1
)
(
1
−
p
2
1
)
...
(
1
−
p
s
1
)
. Find the smallest positive integer
n
n
n
such that
ϕ
(
n
)
=
2
5
47
n
.
\phi (n)=\frac{2^5}{47}n.
ϕ
(
n
)
=
47
2
5
n
.
7
2
Hide problems
min xyz if x|(y^5+1),y|(z^5+1),z|(x^5+1)
It is known that odd prime numbers
x
,
y
z
x, y z
x
,
yz
satisfy
x
∣
(
y
5
+
1
)
,
y
∣
(
z
5
+
1
)
,
z
∣
(
x
5
+
1
)
.
x|(y^5+1),y|(z^5+1),z|(x^5+1).
x
∣
(
y
5
+
1
)
,
y
∣
(
z
5
+
1
)
,
z
∣
(
x
5
+
1
)
.
Find the minimum value of the product
x
y
z
xyz
x
yz
.
infinite n such that C_n^{S_n} is even, where S_n =[1+1/2+..+1/n]
Use
[
x
]
[x]
[
x
]
to represent the greatest integer no more than a real number
x
x
x
. Let
S
n
=
[
1
+
1
2
+
1
3
+
.
.
.
+
1
n
]
,
(
n
=
1
,
2
,
.
.
,
)
S_n=\left[1+\frac12 +\frac13+...+\frac{1}{n}\right], \,\, (n =1,2,..,)
S
n
=
[
1
+
2
1
+
3
1
+
...
+
n
1
]
,
(
n
=
1
,
2
,
..
,
)
Prove that there are infinitely many
n
n
n
such that
C
n
S
n
C_n^{S_n}
C
n
S
n
is an even number. PS. Attached is the original wording which forgets left [ . I hope it is ok where I put it.
1
1
Hide problems
xyz/w + yzw/x + zwx/y + wxy/z = 4, diophantine
Find all integer solutions to the equation
x
y
z
w
+
y
z
w
x
+
z
w
x
y
+
w
x
y
z
=
4
\frac{xyz}{w}+\frac{yzw}{x}+\frac{zwx}{y}+\frac{wxy}{z}=4
w
x
yz
+
x
yz
w
+
y
z
w
x
+
z
w
x
y
=
4
5
1
Hide problems
converse of orthocenter property, given 3 pairs of equal angles
As shown in figure , points
D
,
E
,
F
D,E,F
D
,
E
,
F
lies the sides
A
B
AB
A
B
,
B
C
BC
BC
,
C
A
CA
C
A
of the acute angle
△
A
B
C
\vartriangle ABC
△
A
BC
respectively. If
∠
E
D
C
=
∠
C
D
F
\angle EDC = \angle CDF
∠
E
D
C
=
∠
C
D
F
,
∠
F
E
A
=
∠
A
E
D
\angle FEA=\angle AED
∠
FE
A
=
∠
A
E
D
,
∠
D
F
B
=
∠
B
F
E
\angle DFB =\angle BFE
∠
D
FB
=
∠
BFE
, prove that the
C
D
CD
C
D
,
A
E
AE
A
E
,
B
F
BF
BF
are the altitudes of
△
A
B
C
\vartriangle ABC
△
A
BC
. https://cdn.artofproblemsolving.com/attachments/3/d/5ddf48e298ad1b75691c13935102b26abe73c1.png
2
2
Hide problems
angle chasing, tangents at circumcircle of a right triangle
It is known that
⊙
O
\odot O
⊙
O
is the circumcircle of
△
A
B
C
\vartriangle ABC
△
A
BC
wwith diameter
A
B
AB
A
B
. The tangents of
⊙
O
\odot O
⊙
O
at points
B
B
B
and
C
C
C
intersect at
P
P
P
. The line perpendicular to
P
A
PA
P
A
at point
A
A
A
intersects the extension of
B
C
BC
BC
at point
D
D
D
. Extend
D
P
DP
D
P
at length
P
E
=
P
B
PE = PB
PE
=
PB
. If
∠
A
D
P
=
4
0
o
\angle ADP = 40^o
∠
A
D
P
=
4
0
o
, find the measure of
∠
E
\angle E
∠
E
.
is (AB+BC+CA)/AD^2 irrational?
As shown in figure , a circle of radius
1
1
1
passes through vertex
A
A
A
of
△
A
B
C
\vartriangle ABC
△
A
BC
and is tangent to the side
B
C
BC
BC
at the point
D
D
D
, intersect sides
A
B
AB
A
B
and
A
C
AC
A
C
at points
E
E
E
and
F
F
F
respectively . Also
E
F
EF
EF
bisects
∠
A
F
D
\angle AFD
∠
A
F
D
, and
∠
A
D
C
=
8
0
o
\angle ADC = 80^o
∠
A
D
C
=
8
0
o
, Is there a triangle that satisfies the condition, so that
A
B
+
B
C
+
C
A
A
D
2
\frac{AB+BC+CA}{AD^2}
A
D
2
A
B
+
BC
+
C
A
is an irrational number, and the irrational number is the root of a quadratic equation with integral coefficients? If it does not exist, please prove it; if it exists, find the quadratic equation that satisfies the condition. https://cdn.artofproblemsolving.com/attachments/b/9/9e3b955b6d6df35832dd0c0a2d1d2a1e1cce94.png