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Contests
National and Regional Contests
Finland Contests
Finnish National High School Mathematics Competition
1997 Finnish National High School Mathematics Competition
1997 Finnish National High School Mathematics Competition
Part of
Finnish National High School Mathematics Competition
Subcontests
(5)
1
1
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Equation with parameter
Determine the real numbers
a
a
a
such that the equation
a
3
x
+
3
−
x
=
3
a 3^x + 3^{-x} = 3
a
3
x
+
3
−
x
=
3
has exactly one solution
x
.
x.
x
.
3
1
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Knights at a round table
12
12
12
knights are sitting at a round table. Every knight is an enemy with two of the adjacent knights but with none of the others.
5
5
5
knights are to be chosen to save the princess, with no enemies in the group. How many ways are there for the choice?
4
1
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Sum of the four-digit integers with only odd digits
Count the sum of the four-digit positive integers containing only odd digits in their decimal representation.
5
1
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Points on the plane
For an integer
n
≥
3
n\geq 3
n
≥
3
, place
n
n
n
points on the plane in such a way that all the distances between the points are at most one and exactly
n
n
n
of the pairs of points have the distance one.
2
1
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Tangent circles
Circles with radii
R
R
R
and
r
r
r
(
R
>
r
R > r
R
>
r
) are externally tangent. Another common tangent of the circles in drawn. This tangent and the circles bound a region inside which a circle as large as possible is drawn. What is the radius of this circle?