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Contests
National and Regional Contests
Moldova Contests
Moldova National Olympiad
2010 Moldova National Olympiad
2010 Moldova National Olympiad
Part of
Moldova National Olympiad
Subcontests
(5)
9.1
1
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Weird algebra pls help
a
a
a
,
b
b
b
,
c
c
c
are real. What is the highest value of
a
+
b
+
c
a+b+c
a
+
b
+
c
if
a
2
+
4
b
2
+
9
c
2
−
2
a
−
12
b
+
6
c
+
2
=
0
a^2+4b^2+9c^2-2a-12b+6c+2=0
a
2
+
4
b
2
+
9
c
2
−
2
a
−
12
b
+
6
c
+
2
=
0
12.4
1
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A special triangle
The perimeter of a triangle is a natural number, its circumradius is equal to
65
8
\frac{65}{8}
8
65
, and the inradius is equal to
4
4
4
. Find the sides of the triangle.
12.8
1
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Number to satisfy the definite integral
Find all
t
∈
R
t\in \mathbb R
t
∈
R
, such that
∫
0
π
2
∣
sin
x
+
t
cos
x
∣
d
x
=
1
\int_{0}^{\frac{\pi}{2}}\mid \sin x+t\cos x\mid dx=1
∫
0
2
π
∣
sin
x
+
t
cos
x
∣
d
x
=
1
.
12.5
1
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a infinity of triplets satisfying the relation
Prove that exists a infinity of triplets
a
,
b
,
c
∈
R
a, b, c\in\mathbb{R}
a
,
b
,
c
∈
R
satisfying simultaneously the relations
a
+
b
+
c
=
0
a+b+c=0
a
+
b
+
c
=
0
and
a
4
+
b
4
+
c
4
=
50
a^4+b^4+c^4=50
a
4
+
b
4
+
c
4
=
50
.Moldova National Math Olympiad 2010, 12th grade
11.4
1
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Limit of sum of inverse of squres
Let a_n\equal{}1\plus{}\dfrac1{2^2}\plus{}\dfrac1{3^2}\plus{}\cdots\plus{}\dfrac1{n^2} Find
lim
n
→
∞
a
n
\lim_{n\to\infty}a_n
lim
n
→
∞
a
n