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National and Regional Contests
Slovenia Contests
Slovenia Team Selection Tests
2019 Slovenia Team Selection Test
2019 Slovenia Team Selection Test
Part of
Slovenia Team Selection Tests
Subcontests
(5)
5
1
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Slovenia 2019 TST1 P5
Let
A
B
C
ABC
A
BC
be a triangle and
D
,
E
D, E
D
,
E
and
F
F
F
the foots of heights from
A
,
B
A, B
A
,
B
and
C
C
C
respectively. Let
D
1
D_1
D
1
be such a point on
E
F
EF
EF
, that
D
F
=
D
1
E
DF = D_1 E
D
F
=
D
1
E
where
E
E
E
is between
D
1
D_1
D
1
and
F
F
F
. Similarly, let
D
2
D_2
D
2
be such a point on
E
F
EF
EF
, that
D
E
=
D
2
F
DE = D_2 F
D
E
=
D
2
F
where
F
F
F
is between
E
E
E
and
D
2
D_2
D
2
. Let the bisector of
D
D
1
DD_1
D
D
1
intersect
A
B
AB
A
B
at
P
P
P
and let the bisector of
D
D
2
DD_2
D
D
2
intersect
A
C
AC
A
C
at
Q
Q
Q
. Prove that,
P
Q
PQ
PQ
bisects
B
C
BC
BC
.
4
1
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Slovenia 2019 TST1 P4
Let
P
P
P
be the set of all prime numbers. Let
A
A
A
be some subset of
P
P
P
that has at least two elements. Let's say that for every positive integer
n
n
n
the following statement holds: If we take
n
n
n
different elements
p
1
,
p
2
.
.
.
p
n
∈
A
p_1,p_2...p_n \in A
p
1
,
p
2
...
p
n
∈
A
, every prime number that divides
p
1
p
2
⋯
p
n
−
1
p_1 p_2 \cdots p_n-1
p
1
p
2
⋯
p
n
−
1
is also an element of
A
A
A
. Prove, that
A
A
A
contains all prime numbers.
1
1
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Slovenia 2019 TST1 P1
Let
A
B
C
ABC
A
BC
be a non-right isosceles triangle such that
A
C
=
B
C
AC = BC
A
C
=
BC
. Let
D
D
D
be such a point on the perpendicular bisector of
A
B
AB
A
B
, that
A
D
AD
A
D
is tangent on the
A
B
C
ABC
A
BC
circumcircle. Let
E
E
E
be such a point on
A
B
AB
A
B
, that
C
E
CE
CE
and
A
D
AD
A
D
are perpendicular and let
F
F
F
be the second intersection of line
A
C
AC
A
C
and the circle
C
D
E
CDE
C
D
E
. Prove that
D
F
DF
D
F
and
A
B
AB
A
B
are parallel.
2
2
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Slovenia 2019 TST1 P2
Prove, that for any positive real numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
who satisfy
a
2
+
b
2
+
c
2
=
1
a^2+b^2+c^2=1
a
2
+
b
2
+
c
2
=
1
the following inequality holds.
1
a
−
a
+
1
b
−
b
+
1
c
−
c
≥
2
a
+
2
b
+
2
c
\sqrt{\frac{1}{a}-a}+\sqrt{\frac{1}{b}-b}+\sqrt{\frac{1}{c}-c} \geq \sqrt{2a}+\sqrt{2b}+\sqrt{2c}
a
1
−
a
+
b
1
−
b
+
c
1
−
c
≥
2
a
+
2
b
+
2
c
Slovenia 2019 TST2 P2
Determine all non-negative real numbers
a
a
a
, for which
f
(
a
)
=
0
f(a)=0
f
(
a
)
=
0
for all functions
f
:
R
≥
0
→
R
≥
0
f: \mathbb{R}_{\ge 0}\to \mathbb{R}_{\ge 0}
f
:
R
≥
0
→
R
≥
0
, who satisfy the equation
f
(
f
(
x
)
+
f
(
y
)
)
=
y
f
(
1
+
y
f
(
x
)
)
f(f(x) + f(y)) = yf(1 + yf(x))
f
(
f
(
x
)
+
f
(
y
))
=
y
f
(
1
+
y
f
(
x
))
for all non-negative real numbers
x
x
x
and
y
y
y
.
3
2
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Slovenia 2019 TST1 P3
Let
n
n
n
be any positive integer and
M
M
M
a set that contains
n
n
n
positive integers. A sequence with
2
n
2^n
2
n
elements is christmassy if every element of the sequence is an element of
M
M
M
. Prove that, in any christmassy sequence there exist some successive elements, the product of whom is a perfect square.
Slovenia 2019 TST2 P3
Let
A
B
C
ABC
A
BC
be a non-right triangle and let
M
M
M
be the midpoint of
B
C
BC
BC
. Let
D
D
D
be a point on
A
M
AM
A
M
(D≠A, D≠M). Let ω1 be a circle through
D
D
D
that intersects
B
C
BC
BC
at
B
B
B
and let ω2 be a circle through
D
D
D
that intersects
B
C
BC
BC
at
C
C
C
. Let
A
B
AB
A
B
intersect ω1 at
B
B
B
and
E
E
E
, and let
A
C
AC
A
C
intersect ω2 at
C
C
C
and
F
F
F
. Prove, that the tangent on ω1 at
E
E
E
and the tangent on ω2 at
F
F
F
intersect on
A
M
AM
A
M
.