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Problems
Contests
National and Regional Contests
Finland Contests
Finnish National High School Mathematics Competition
2006 Finnish National High School Mathematics Competition
2006 Finnish National High School Mathematics Competition
Part of
Finnish National High School Mathematics Competition
Subcontests
(5)
5
1
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Sudoku type game
The game of Nelipe is played on a 16\times16-grid as follows: The two players write in turn numbers
1
,
2
,
.
.
.
,
16
1, 2,..., 16
1
,
2
,
...
,
16
in diff erent squares. The numbers on each row, column, and in every one of the 16 smaller squares have to be di fferent. The loser is the one who is not able to write a number. Which one of the players wins, if both play with an optimal strategy?
4
1
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Perpendicular medians
Two medians of a triangle are perpendicular. Prove that the medians of the triangle are the sides of a right-angled triangle.
3
1
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Three primes
The numbers
p
,
4
p
2
+
1
,
p, 4p^2 + 1,
p
,
4
p
2
+
1
,
and
6
p
2
+
1
6p^2 + 1
6
p
2
+
1
are primes. Determine
p
.
p.
p
.
2
1
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Inequality
Show that the inequality 3(1 + a^2 + a^4)\geq (1 + a + a^2)^2 holds for all real numbers
a
.
a.
a
.
1
1
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Simple equation
Determine all pairs
(
x
,
y
)
(x, y)
(
x
,
y
)
of positive integers for which the equation
x
+
y
+
x
y
=
2006
x + y + xy = 2006
x
+
y
+
x
y
=
2006
holds.