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Problems
Contests
National and Regional Contests
Finland Contests
Finnish National High School Mathematics Competition
1998 Finnish National High School Mathematics Competition
1998 Finnish National High School Mathematics Competition
Part of
Finnish National High School Mathematics Competition
Subcontests
(5)
5
1
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Covering a 15x36 checkerboard
15\times 36-checkerboard is covered with square tiles. There are two kinds of tiles, with side
7
7
7
or
5.
5.
5.
Tiles are supposed to cover whole squares of the board and be non-overlapping. What is the maximum number of squares to be covered?
4
1
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110 points in a unit square!
There are
110
110
110
points in a unit square. Show that some four of these points reside in a circle whose radius is
1
/
8.
1/8.
1/8.
3
1
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Subsequence of a geometric seqeunce
Consider the geometric sequence
1
/
2
,
1
/
4
,
1
/
8
,
.
.
.
1/2, \ 1 / 4, \ 1 / 8,...
1/2
,
1/4
,
1/8
,
...
Can one choose a subsequence, finite or infinite, for which the ratio of consecutive terms is not
1
1
1
and whose sum is
1
/
5
?
1/5?
1/5
?
2
1
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Keys and a safe
There are
11
11
11
members in the competetion committee. The problem set is kept in a safe having several locks. The committee members have been provided with keys in such a way that every six members can open the safe, but no fi ve members can do that. What is the smallest possible number of locks, and how many keys are needed in that case?
1
1
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Points and areas
Show that points
A
,
B
,
C
A, B, C
A
,
B
,
C
and
D
D
D
can be placed on the plane in such a way that the quadrilateral
A
B
C
D
ABCD
A
BC
D
has an area which is twice the area of the quadrilateral
A
D
B
C
.
ADBC.
A
D
BC
.