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Contests
National and Regional Contests
Finland Contests
Finnish National High School Mathematics Competition
2005 Finnish National High School Mathematics Competition
2005 Finnish National High School Mathematics Competition
Part of
Finnish National High School Mathematics Competition
Subcontests
(5)
5
1
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Disorderly sequences
A finite sequence is said to be disorderly, if no two terms of the sequence have their average in between them. For example,
(
0
,
2
,
1
)
(0, 2, 1)
(
0
,
2
,
1
)
is disorderly, for
1
=
0
+
2
2
1 = \frac{0+2}{2}
1
=
2
0
+
2
is not in between
0
0
0
and
2
2
2
, and the other averages
0
+
1
2
=
1
2
\frac{0+1}{2} = \frac{1}{2}
2
0
+
1
=
2
1
and
2
+
1
2
=
1
1
2
\frac{2+1}{2} = 1\frac{1}{2}
2
2
+
1
=
1
2
1
do not even occur in the sequence. Prove that for every
n
∈
N
n \in \Bbb{N}
n
∈
N
there is a disorderly sequence enumerating the numbers
0
,
1
,
…
,
n
0, 1,\ldots , n
0
,
1
,
…
,
n
without repetitions.
4
1
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Permuting the digits we obtain an integer divisible by 7
The numbers
1
,
3
,
7
1, 3, 7
1
,
3
,
7
and
9
9
9
occur in the decimal representation of an integer. Show that permuting the order of digits one can obtain an integer divisible by
7.
7.
7.
3
1
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System of equations
Solve the group of equations:
{
(
x
+
y
)
3
=
z
(
y
+
z
)
3
=
x
(
z
+
x
)
3
=
y
\begin{cases} (x + y)^3 = z \\ (y + z)^3 = x \\ (z + x)^3 = y \end{cases}
⎩
⎨
⎧
(
x
+
y
)
3
=
z
(
y
+
z
)
3
=
x
(
z
+
x
)
3
=
y
2
1
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Arrangements in a round table
There are
12
12
12
seats at a round table in a restaurant. A group of five women and seven men arrives at the table. How many ways are there for choosing the sitting order, provided that every woman ought to be surrounded by two men and two orders are regarded as different, if at least one person has a different neighbour on one's right side.
1
1
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Perpendicular line segments
In the figure below, the centres of four squares have been connected by two line segments. Prove that these line segments are perpendicular.