MathDB
Problems
Contests
National and Regional Contests
Slovenia Contests
Slovenia Team Selection Tests
1998 Slovenia Team Selection Test
1998 Slovenia Team Selection Test
Part of
Slovenia Team Selection Tests
Subcontests
(6)
3
1
Hide problems
2 jars, each with 6 marbles which are labeled with unknown numbers
(a) Alenka has two jars, each with
6
6
6
marbles labeled with numbers
1
1
1
through
6
6
6
. She draws one marble from each jar at random. Denote by
p
n
p_n
p
n
the probability that the sum of the labels of the two drawn marbles is
n
n
n
. Compute pn for each
n
∈
N
n \in N
n
∈
N
. (b) Barbara has two jars, each with
6
6
6
marbles which are labeled with unknown numbers. The sets of labels in the two jars may differ and two marbles in the same jar can have the same label. If she draws one marble from each jar at random, the probability that the sum of the labels of the drawn marbles is
n
n
n
equals the probability
p
n
p_n
p
n
in Alenka’s case. Determine the labels of the marbles. Find all solution
5
1
Hide problems
fixed point and midpoint wanted
On a line
p
p
p
which does not meet a circle
K
K
K
with center
O
O
O
, point
P
P
P
is taken such that
O
P
⊥
p
OP \perp p
OP
⊥
p
. Let
X
≠
P
X \ne P
X
=
P
be an arbitrary point on
p
p
p
. The tangents from
X
X
X
to
K
K
K
touch it at
A
A
A
and
B
B
B
. Denote by
C
C
C
and
D
D
D
the orthogonal projections of
P
P
P
on
A
X
AX
A
X
and
B
X
BX
BX
respectively. (a) Prove that the intersection point
Y
Y
Y
of
A
B
AB
A
B
and
O
P
OP
OP
is independent of the location of
X
X
X
. (b) Lines
C
D
CD
C
D
and
O
P
OP
OP
meet at
Z
Z
Z
. Prove that
Z
Z
Z
is the midpoint of
P
P
P
.
6
1
Hide problems
a_{n+1} =\frac{a_n^2}{a_n +1} , prove [a_n] = 1994- n
Let
a
0
=
1998
a_0 = 1998
a
0
=
1998
and
a
n
+
1
=
a
n
2
a
n
+
1
a_{n+1} =\frac{a_n^2}{a_n +1}
a
n
+
1
=
a
n
+
1
a
n
2
for each nonnegative integer
n
n
n
. Prove that
[
a
n
]
=
1994
−
n
[a_n] = 1994- n
[
a
n
]
=
1994
−
n
for
0
≤
n
≤
1000
0 \le n \le 1000
0
≤
n
≤
1000
4
1
Hide problems
x+y^2+z^3 = xyz , where d=gcd(x,y)
Find all positive integers
x
x
x
and
y
y
y
such that
x
+
y
2
+
z
3
=
x
y
z
x+y^2+z^3 = xyz
x
+
y
2
+
z
3
=
x
yz
, where
z
z
z
is the greatest common divisor of
x
x
x
and
y
y
y
2
1
Hide problems
perpendicular wanted starting with a semicircle
A semicircle with center
O
O
O
and diameter
A
B
AB
A
B
is given. Point
M
M
M
on the extension of
A
B
AB
A
B
is taken so that
A
M
>
B
M
AM > BM
A
M
>
BM
. A line through
M
M
M
intersects the semicircle at
C
C
C
and
D
D
D
so that
C
M
<
D
M
CM < DM
CM
<
D
M
. The circumcircles of triangles
A
O
D
AOD
A
O
D
and
O
B
C
OBC
OBC
meet again at point
K
K
K
. Prove that
O
K
OK
O
K
and
K
M
KM
K
M
are perpendicular
1
1
Hide problems
f((x-y)^2)= f(x)^2 -2x f(y)+y^2
Find all functions
f
:
R
→
R
f : R \to R
f
:
R
→
R
that satisfy
f
(
(
x
−
y
)
2
)
=
f
(
x
)
2
−
2
x
f
(
y
)
+
y
2
f((x-y)^2)= f(x)^2 -2x f(y)+y^2
f
((
x
−
y
)
2
)
=
f
(
x
)
2
−
2
x
f
(
y
)
+
y
2
for all
x
,
y
∈
R
x,y \in R
x
,
y
∈
R