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National and Regional Contests
Norway Contests
Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Final Round
2008 Abels Math Contest (Norwegian MO) Final
2008 Abels Math Contest (Norwegian MO) Final
Part of
Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Final Round
Subcontests
(6)
2a
1
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n boards of width b, on a floor with width B, B/B is an integer
We wish to lay down boards on a floor with width
B
B
B
in the direction across the boards. We have
n
n
n
boards of width
b
b
b
, and
B
/
b
B/b
B
/
b
is an integer, and
n
b
≤
B
nb \le B
nb
≤
B
. There are enough boards to cover the floor, but the boards may have different lengths. Show that we can cut the boards in such a way that every board length on the floor has at most one join where two boards meet end to end. https://cdn.artofproblemsolving.com/attachments/f/f/24ce8ae05d85fd522da0e18c0bb8017ca3c8e8.png
2b
1
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a game on a square board consisting of n x n white tiles, coloring black
A and B play a game on a square board consisting of
n
×
n
n \times n
n
×
n
white tiles, where
n
≥
2
n \ge 2
n
≥
2
. A moves first, and the players alternate taking turns. A move consists of picking a square consisting of
2
×
2
2\times 2
2
×
2
or
3
×
3
3\times 3
3
×
3
white tiles and colouring all these tiles black. The first player who cannot find any such squares has lost. Show that A can always win the game if A plays the game right.
3
1
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1/x+ 1/y +1/z + 9/4 <= 1/x^2 + 1/y^2 + 1/z^2 when x,y,z >0 with x + y +z= 2
a) Let
x
x
x
and
y
y
y
be positive numbers such that
x
+
y
=
2
x + y = 2
x
+
y
=
2
. Show that
1
x
+
1
y
≤
1
x
2
+
1
y
2
\frac{1}{x}+\frac{1}{y} \le \frac{1}{x^2}+\frac{1}{y^2}
x
1
+
y
1
≤
x
2
1
+
y
2
1
b) Let
x
,
y
x,y
x
,
y
and
z
z
z
be positive numbers such that
x
+
y
+
z
=
2
x + y +z= 2
x
+
y
+
z
=
2
. Show that
1
x
+
1
y
+
1
z
+
9
4
≤
1
x
2
+
1
y
2
+
1
z
2
\frac{1}{x}+\frac{1}{y} +\frac{1}{z} +\frac{9}{4} \le \frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}
x
1
+
y
1
+
z
1
+
4
9
≤
x
2
1
+
y
2
1
+
z
2
1
.
1
1
Hide problems
s(n) = \frac16 n^3 - \frac12 n^2 + \frac13 n is integer when n is integer
Let
s
(
n
)
=
1
6
n
3
−
1
2
n
2
+
1
3
n
s(n) = \frac16 n^3 - \frac12 n^2 + \frac13 n
s
(
n
)
=
6
1
n
3
−
2
1
n
2
+
3
1
n
. (a) Show that
s
(
n
)
s(n)
s
(
n
)
is an integer whenever
n
n
n
is an integer. (b) How many integers
n
n
n
with
0
<
n
≤
2008
0 < n \le 2008
0
<
n
≤
2008
are such that
s
(
n
)
s(n)
s
(
n
)
is divisible by
4
4
4
?
4b
1
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equal segments given and wanted, circumcenters related
A point
D
D
D
lies on the side
B
C
BC
BC
, and a point
E
E
E
on the side
A
C
AC
A
C
, of the triangle
A
B
C
ABC
A
BC
, and
B
D
BD
B
D
and
A
E
AE
A
E
have the same length. The line through the centres of the circumscribed circles of the triangles
A
D
C
ADC
A
D
C
and
B
E
C
BEC
BEC
crosses
A
C
AC
A
C
in
K
K
K
and
B
C
BC
BC
in
L
L
L
. Show that
K
C
KC
K
C
and
L
C
LC
L
C
have the same length.
4a
1
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circle, triangles with equal areas, angles of a triangle wanted
Three distinct points
A
,
B
A, B
A
,
B
, and
C
C
C
lie on a circle with centre at
O
O
O
. The triangles
A
O
B
,
B
O
C
AOB, BOC
A
OB
,
BOC
, and
C
O
A
COA
CO
A
have equal area. What are the possible measures of the angles of the triangle
A
B
C
ABC
A
BC
?