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National and Regional Contests
Moldova Contests
Moldova Team Selection Test
2010 Moldova Team Selection Test
2010 Moldova Team Selection Test
Part of
Moldova Team Selection Test
Subcontests
(4)
3
2
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Find angle
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral. We have that \angle BAC\equal{}3\angle CAD, AB\equal{}CD, \angle ACD\equal{}\angle CBD. Find angle
∠
A
C
D
\angle ACD
∠
A
C
D
Isosceles triangle with height $HM$
Let
A
B
C
ABC
A
BC
be an acute triangle.
H
H
H
is the orthocenter and
M
M
M
is the middle of the side
B
C
BC
BC
. A line passing through
H
H
H
and perpendicular to
H
M
HM
H
M
intersect the segment
A
B
AB
A
B
and
A
C
AC
A
C
in
P
P
P
and
Q
Q
Q
. Prove that MP \equal{} MQ
2
2
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maximum of |sin(x)| and |sin(x+2010)|
Prove that for any real number
x
x
x
the following inequality is true: \max\{|\sin x|, |\sin(x\plus{}2010)|\}>\dfrac1{\sqrt{17}}
Integer part of an expression
Let
x
1
,
x
2
,
…
,
x
n
x_1, x_2, \ldots, x_n
x
1
,
x
2
,
…
,
x
n
be positive real numbers with sum
1
1
1
. Find the integer part of: E\equal{}x_1\plus{}\dfrac{x_2}{\sqrt{1\minus{}x_1^2}}\plus{}\dfrac{x_3}{\sqrt{1\minus{}(x_1\plus{}x_2)^2}}\plus{}\cdots\plus{}\dfrac{x_n}{\sqrt{1\minus{}(x_1\plus{}x_2\plus{}\cdots\plus{}x_{n\minus{}1})^2}}
1
2
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Diophantine equation
Find all
3
3
3
-digit numbers such that placing to the right side of the number its successor we get a
6
6
6
-digit number which is a perfect square.
A polynomial with negative roots
Let p\in\mathbb{R}_\plus{} and k\in\mathbb{R}_\plus{}. The polynomial F(x)\equal{}x^4\plus{}a_3x^3\plus{}a_2x^2\plus{}a_1x\plus{}k^4 with real coefficients has
4
4
4
negative roots. Prove that F(p)\geq(p\plus{}k)^4
4
2
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Dividing a cube
Let
n
≥
6
n\geq6
n
≥
6
be a even natural number. Prove that any cube can be divided in \dfrac{3n(n\minus{}2)}4\plus{}2 cubes.
Hardest in ARO 2008
In a chess tournament 2n\plus{}3 players take part. Every two play exactly one match. The schedule is such that no two matches are played at the same time, and each player, after taking part in a match, is free in at least
n
n
n
next (consecutive) matches. Prove that one of the players who play in the opening match will also play in the closing match.