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Problems
Contests
National and Regional Contests
China Contests
South East Mathematical Olympiad
2012 South East Mathematical Olympiad
2012 South East Mathematical Olympiad
Part of
South East Mathematical Olympiad
Subcontests
(4)
4
2
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find maximal value of a+b-c+d
Let
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
be real numbers satisfying inequality
a
cos
x
+
b
cos
2
x
+
c
cos
3
x
+
d
cos
4
x
≤
1
a\cos x+b\cos 2x+c\cos 3x+d\cos 4x\le 1
a
cos
x
+
b
cos
2
x
+
c
cos
3
x
+
d
cos
4
x
≤
1
holds for arbitrary real number
x
x
x
. Find the maximal value of
a
+
b
−
c
+
d
a+b-c+d
a
+
b
−
c
+
d
and determine the values of
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
when that maximum is attained.
problem of grasshopper leaping
Let positive integers
m
,
n
m,n
m
,
n
satisfy
n
=
2
m
−
1
n=2^m-1
n
=
2
m
−
1
.
P
n
=
{
1
,
2
,
⋯
,
n
}
P_n =\{1,2,\cdots ,n\}
P
n
=
{
1
,
2
,
⋯
,
n
}
is a set that contains
n
n
n
points on an axis. A grasshopper on the axis can leap from one point to another adjacent point. Find the maximal value of
m
m
m
satisfying following conditions: (a)
x
,
y
x, y
x
,
y
are two arbitrary points in
P
n
P_n
P
n
; (b) starting at point
x
x
x
, the grasshopper leaps
2012
2012
2012
times and finishes at point
y
y
y
; (the grasshopper is allowed to travel
x
x
x
and
y
y
y
more than once) (c) there are even number ways for the grasshopper to do (b).
3
2
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composite number
For composite number
n
n
n
, let
f
(
n
)
f(n)
f
(
n
)
denote the sum of the least three divisors of
n
n
n
, and
g
(
n
)
g(n)
g
(
n
)
the sum of the greatest two divisors of
n
n
n
. Find all composite numbers
n
n
n
, such that
g
(
n
)
=
(
f
(
n
)
)
m
g(n)=(f(n))^m
g
(
n
)
=
(
f
(
n
)
)
m
(
m
∈
N
∗
m\in N^*
m
∈
N
∗
).
find the size of angle AME
In
△
A
B
C
\triangle ABC
△
A
BC
, point
D
D
D
lies on side
A
C
AC
A
C
such that
∠
A
B
D
=
∠
C
\angle ABD=\angle C
∠
A
B
D
=
∠
C
. Point
E
E
E
lies on side
A
B
AB
A
B
such that
B
E
=
D
E
BE=DE
BE
=
D
E
.
M
M
M
is the midpoint of segment
C
D
CD
C
D
. Point
H
H
H
is the foot of the perpendicular from
A
A
A
to
D
E
DE
D
E
. Given
A
H
=
2
−
3
AH=2-\sqrt{3}
A
H
=
2
−
3
and
A
B
=
1
AB=1
A
B
=
1
, find the size of
∠
A
M
E
\angle AME
∠
A
ME
.
2
2
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inscribed circle
The incircle
I
I
I
of
△
A
B
C
\triangle ABC
△
A
BC
is tangent to sides
A
B
,
B
C
,
C
A
AB,BC,CA
A
B
,
BC
,
C
A
at
D
,
E
,
F
D,E,F
D
,
E
,
F
respectively. Line
E
F
EF
EF
intersects lines
A
I
,
B
I
,
D
I
AI,BI,DI
A
I
,
B
I
,
D
I
at
M
,
N
,
K
M,N,K
M
,
N
,
K
respectively. Prove that
D
M
⋅
K
E
=
D
N
⋅
K
F
DM\cdot KE=DN\cdot KF
D
M
⋅
K
E
=
D
N
⋅
K
F
.
find the least natural number
Find the least natural number
n
n
n
, such that the following inequality holds:
n
−
2011
2012
−
n
−
2012
2011
<
n
−
2013
2011
3
−
n
−
2011
2013
3
\sqrt{\dfrac{n-2011}{2012}}-\sqrt{\dfrac{n-2012}{2011}}<\sqrt[3]{\dfrac{n-2013}{2011}}-\sqrt[3]{\dfrac{n-2011}{2013}}
2012
n
−
2011
−
2011
n
−
2012
<
3
2011
n
−
2013
−
3
2013
n
−
2011
.
1
2
Hide problems
find (l,m,n) to form a geometric sequence
Find a triple
(
l
,
m
,
n
)
(l, m, n)
(
l
,
m
,
n
)
of positive integers
(
1
<
l
<
m
<
n
)
(1<l<m<n)
(
1
<
l
<
m
<
n
)
, such that
∑
k
=
1
l
k
,
∑
k
=
l
+
1
m
k
,
∑
k
=
m
+
1
n
k
\sum_{k=1}^{l}k, \sum_{k=l+1}^{m}k, \sum_{k=m+1}^{n}k
∑
k
=
1
l
k
,
∑
k
=
l
+
1
m
k
,
∑
k
=
m
+
1
n
k
form a geometric sequence in order.
find six-composited numbers
A nonnegative integer
m
m
m
is called a “six-composited number” if
m
m
m
and the sum of its digits are both multiples of
6
6
6
. How many “six-composited numbers” that are less than
2012
2012
2012
are there?