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National and Regional Contests
Estonia Contests
Estonia Team Selection Test
2005 Estonia Team Selection Test
2005 Estonia Team Selection Test
Part of
Estonia Team Selection Test
Subcontests
(5)
5
1
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2005 points On a horizontal line, black or white
On a horizontal line,
2005
2005
2005
points are marked, each of which is either white or black. For every point, one finds the sum of the number of white points on the right of it and the number of black points on the left of it. Among the
2005
2005
2005
sums, exactly one number occurs an odd number of times. Find all possible values of this number.
4
1
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(6x^2 -24x -4a) and (x^3 + ax^2 + bx - 8) have non-negative real roots
Find all pairs
(
a
,
b
)
(a, b)
(
a
,
b
)
of real numbers such that the roots of polynomials
6
x
2
−
24
x
−
4
a
6x^2 -24x -4a
6
x
2
−
24
x
−
4
a
and
x
3
+
a
x
2
+
b
x
−
8
x^3 + ax^2 + bx - 8
x
3
+
a
x
2
+
b
x
−
8
are all non-negative real numbers.
3
1
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(x + y)^x = x^y diophantine
Find all pairs
(
x
,
y
)
(x, y)
(
x
,
y
)
of positive integers satisfying the equation
(
x
+
y
)
x
=
x
y
(x + y)^x = x^y
(
x
+
y
)
x
=
x
y
.
2
1
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on planet Automory, everyone loves exactly 1and honours exactly 1
On the planet Automory, there are infinitely many inhabitants. Every Automorian loves exactly one Automorian and honours exactly one Automorian. Additionally, the following can be noticed:
∙
\bullet
∙
each Automorian is loved by some Automorian;
∙
\bullet
∙
if Automorian
A
A
A
loves Automorian
B
B
B
, then also all Automorians honouring
A
A
A
love
B
B
B
,
∙
\bullet
∙
if Automorian
A
A
A
honours Automorian
B
B
B
, then also all Automorians loving
A
A
A
honour
B
B
B
. Is it correct to claim that every Automorian honours and loves the same Automorian?
1
1
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<PMQ_1 + < PMQ_2, common tangent of unequal circles
On a plane, a line
ℓ
\ell
ℓ
and two circles
c
1
c_1
c
1
and
c
2
c_2
c
2
of different radii are given such that
ℓ
\ell
ℓ
touches both circles at point
P
P
P
. Point
M
≠
P
M \ne P
M
=
P
on
ℓ
\ell
ℓ
is chosen so that the angle
Q
1
M
Q
2
Q_1MQ_2
Q
1
M
Q
2
is as large as possible where
Q
1
Q_1
Q
1
and
Q
2
Q_2
Q
2
are the tangency points of the tangent lines drawn from
M
M
M
to
c
i
c_i
c
i
and
c
2
c_2
c
2
, respectively, differing from
ℓ
\ell
ℓ
. Find
∠
P
M
Q
1
+
∠
P
M
Q
2
\angle PMQ_1 + \angle PMQ_2
∠
PM
Q
1
+
∠
PM
Q
2
·