MathDB
Problems
Contests
National and Regional Contests
Singapore Contests
Singapore Senior Math Olympiad
2011 Singapore Senior Math Olympiad
2011 Singapore Senior Math Olympiad
Part of
Singapore Senior Math Olympiad
Subcontests
(5)
5
1
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Inequality
Given
x
1
,
x
2
,
…
,
x
n
>
0
,
n
≥
5
x_1,x_2,\dots,x_n>0,n\geq 5
x
1
,
x
2
,
…
,
x
n
>
0
,
n
≥
5
, show that
x
1
x
2
x
1
2
+
x
2
2
+
2
x
3
x
4
+
x
2
x
3
x
2
2
+
x
3
2
+
2
x
4
x
5
+
⋯
+
x
n
x
1
x
n
2
+
x
1
2
+
2
x
2
x
3
≤
n
−
1
2
\frac{x_1x_2}{x_1^2+x_2^2+2x_3x_4}+\frac{x_2x_3}{x_2^2+x_3^2+2x_4x_5}+\cdots+\frac{x_nx_1}{x_n^2+x_1^2+2x_2x_3}\leq \frac{n-1}{2}
x
1
2
+
x
2
2
+
2
x
3
x
4
x
1
x
2
+
x
2
2
+
x
3
2
+
2
x
4
x
5
x
2
x
3
+
⋯
+
x
n
2
+
x
1
2
+
2
x
2
x
3
x
n
x
1
≤
2
n
−
1
4
1
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Sets of consecutive integers
Let
n
n
n
and
k
k
k
be positive integers with
n
≥
k
≥
2
n\geq k\geq 2
n
≥
k
≥
2
. For
i
=
1
,
…
,
n
i=1,\dots,n
i
=
1
,
…
,
n
, let
S
i
S_i
S
i
be a nonempty set of consecutive integers such that among any
k
k
k
of them, there are two with nonempty intersection. Prove that there is a set
X
X
X
of
k
−
1
k-1
k
−
1
integers such that each
S
i
S_i
S
i
,
i
=
1
,
…
,
n
i=1,\dots,n
i
=
1
,
…
,
n
contains at least one integer in
X
X
X
.
3
1
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Trigonometric equality
Find all positive integers
n
n
n
such that
cos
π
n
cos
2
π
n
cos
3
π
n
=
1
n
+
1
\cos\frac{\pi}{n}\cos\frac{2\pi}{n}\cos\frac{3\pi}{n}=\frac{1}{n+1}
cos
n
π
cos
n
2
π
cos
n
3
π
=
n
+
1
1
2
1
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Difference=gcd
Determine if there is a set
S
S
S
of 2011 positive integers so that for every pair
m
,
n
m,n
m
,
n
of distinct elements of
S
S
S
,
∣
m
−
n
∣
=
(
m
,
n
)
|m-n|=(m,n)
∣
m
−
n
∣
=
(
m
,
n
)
. Here
(
m
,
n
)
(m,n)
(
m
,
n
)
denotes the greatest common divisor of
m
m
m
and
n
n
n
.
1
1
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Prove equilateral
In the triangle
A
B
C
ABC
A
BC
, the altitude at
A
A
A
, the bisector of
∠
B
\angle B
∠
B
and the median at
C
C
C
meet at a common point. Prove (or disprove?) that the triangle
A
B
C
ABC
A
BC
is equilateral.