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Contests
National and Regional Contests
Moldova Contests
Moldova National Olympiad
2005 Moldova National Olympiad
2005 Moldova National Olympiad
Part of
Moldova National Olympiad
Subcontests
(4)
11.2
1
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Moldova National Olympiad 2005
Let
a
a
a
and
b
b
b
be two real numbers. Find these numbers given that the graphs of
f
:
R
→
R
,
f
(
x
)
=
2
x
4
−
a
2
x
2
+
b
−
1
f:\mathbb{R} \to \mathbb{R} , f(x)=2x^4-a^2x^2+b-1
f
:
R
→
R
,
f
(
x
)
=
2
x
4
−
a
2
x
2
+
b
−
1
and
g
:
R
→
R
,
g
(
x
)
=
2
a
x
3
−
1
g:\mathbb{R} \to \mathbb{R} ,g(x)=2ax^3-1
g
:
R
→
R
,
g
(
x
)
=
2
a
x
3
−
1
have exactly two points of intersection.
10.2
1
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System of equation
Find all positive solution of system of equation: \frac{xy}{2005y\plus{}2004x}\plus{}\frac{yz}{2004z\plus{}2003y}\plus{}\frac{zx}{2003x\plus{}2005z}\equal{}\frac{x^{2}\plus{}y^{2}\plus{}z^{2}}{2005^{2}\plus{}2004^{2}\plus{}2003^{2}}
10.7
1
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Find all functions...
Determine all strictly increasing functions
f
:
R
→
R
f: R\rightarrow R
f
:
R
→
R
satisfying relationship f(x\plus{}f(y))\equal{}f(x\plus{}y)\plus{}2005 for any real values of x and y.
10.4
1
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Inequality for positive numbers
Real numbers
x
1
,
x
2
,
.
.
,
x
n
x_{1},x_{2},..,x_{n}
x
1
,
x
2
,
..
,
x
n
are positive. Prove the inequality: \frac{x_{1}}{x_{2}\plus{}x_{3}}\plus{}\frac{x_{2}}{x_{3}\plus{}x_{4}}\plus{}...\plus{} \frac{x_{n\minus{}1}}{x_{n}\plus{}x_{1}}\plus{}\frac{x_{n}}{x_{1}\plus{}x_{2}}>(\sqrt{2}\minus{}1)n