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Problems
Contests
National and Regional Contests
Moldova Contests
Moldova National Olympiad
2002 Moldova National Olympiad
2002 Moldova National Olympiad
Part of
Moldova National Olympiad
Subcontests
(7)
12.5
1
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Integral inequality
Let
0
≤
a
≤
b
≤
c
≤
3
0 \le a \le b \le c \le 3
0
≤
a
≤
b
≤
c
≤
3
Prove :
(
a
−
b
)
(
a
2
−
9
)
+
(
a
−
c
)
(
b
2
−
9
)
+
(
b
−
c
)
(
c
2
−
9
)
≤
36
(a-b)(a^2-9)+(a-c)(b^2-9)+(b-c)(c^2-9) \le 36
(
a
−
b
)
(
a
2
−
9
)
+
(
a
−
c
)
(
b
2
−
9
)
+
(
b
−
c
)
(
c
2
−
9
)
≤
36
12.8
1
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Series
∑
n
=
1
∞
3
n
.
s
i
n
3
(
π
3
n
)
=
?
\bf{\sum_{n=1}^{\infty}3^n.sin^3(\frac{\pi}{3^n})=?}
n
=
1
∑
∞
3
n
.si
n
3
(
3
n
π
)
=
?
4
10
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3
10
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2
10
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1
10
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12.6
1
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One interesting locus [collinear points and circle]
Let A,B,C be three collinear points and a circle T(A,r). If M and N are two diametrical opposite variable points on T, Find locus geometrical of the intersection BM and CN.