MathDB
Problems
Contests
National and Regional Contests
Finland Contests
Finnish National High School Mathematics Competition
1999 Finnish National High School Mathematics Competition
1999 Finnish National High School Mathematics Competition
Part of
Finnish National High School Mathematics Competition
Subcontests
(5)
5
1
Hide problems
Generalized domino tiles
An ordinary domino tile can be identifi ed as a pair
(
k
,
m
)
(k,m)
(
k
,
m
)
where numbers
k
k
k
and
m
m
m
can get values
0
,
1
,
2
,
3
,
4
,
5
0, 1, 2, 3, 4, 5
0
,
1
,
2
,
3
,
4
,
5
and
6.
6.
6.
Pairs
(
k
,
m
)
(k,m)
(
k
,
m
)
and
(
m
,
k
)
(m, k)
(
m
,
k
)
determine the same tile. In particular, the pair
(
k
,
k
)
(k, k)
(
k
,
k
)
determines one tile. We say that two domino tiles match, if they have a common component. Generalized n-domino tiles
m
m
m
and
k
k
k
can get values
0
,
1
,
.
.
.
,
n
.
0, 1,... , n.
0
,
1
,
...
,
n
.
What is the probability that two randomly chosen
n
n
n
-domino tiles match?
4
1
Hide problems
Intersecting circles
Three unit circles have a common point
O
.
O.
O
.
The other points of (pairwise) intersection are
A
,
B
A, B
A
,
B
and
C
C
C
. Show that the points
A
,
B
A, B
A
,
B
and
C
C
C
are located on some unit circle.
2
1
Hide problems
Arithmetic progression
Suppose that the positive numbers
a
1
,
a
2
,
.
.
,
a
n
a_1, a_2,.. , a_n
a
1
,
a
2
,
..
,
a
n
form an arithmetic progression; hence
a
k
+
1
−
a
k
=
d
,
a_{k+1}- a_k = d,
a
k
+
1
−
a
k
=
d
,
for
k
=
1
,
2
,
.
.
.
,
n
−
1.
k = 1, 2,... , n - 1.
k
=
1
,
2
,
...
,
n
−
1.
Prove that
1
a
1
a
2
+
1
a
2
a
3
+
.
.
.
+
1
a
n
−
1
a
n
=
n
−
1
a
1
a
n
.
\frac{1}{a_1a_2}+\frac{1}{a_2a_3}+...+\frac{1}{a_{n-1}a_n}=\frac{n-1}{a_1a_n}.
a
1
a
2
1
+
a
2
a
3
1
+
...
+
a
n
−
1
a
n
1
=
a
1
a
n
n
−
1
.
3
1
Hide problems
Primes in a seqeunce
Determine how many primes are there in the sequence
101
,
10101
,
1010101....
101, 10101, 1010101 ....
101
,
10101
,
1010101....
1
1
Hide problems
Equation has always a solution
Show that the equation
x
3
+
2
y
2
+
4
z
=
n
x^3 + 2y^2 + 4z = n
x
3
+
2
y
2
+
4
z
=
n
has an integral solution
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
for all integers
n
.
n.
n
.