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Contests
National and Regional Contests
Norway Contests
Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Final Round
2001 Abels Math Contest (Norwegian MO)
2001 Abels Math Contest (Norwegian MO)
Part of
Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Final Round
Subcontests
(6)
4
1
Hide problems
2-day team competition in chess, 3 schools with 15 pupils each attend
At a two-day team competition in chess, three schools with
15
15
15
pupils each attend. Each student plays one game against each player on the other two teams, ie a total of
30
30
30
chess games per student. a) Is it possible for each student to play exactly
15
15
15
games after the first day? b) Show that it is possible for each student to play exactly
16
16
16
games after the first day. c) Assume that each student has played exactly
16
16
16
games after the first day. Show that there are three students, one from each school, who have played their three parties
3b
1
Hide problems
\sqrt{ F_1 }+\sqrt{ F_2}\le \sqrt{ F}, areas inside quadrilateral
The diagonals
A
C
AC
A
C
and
B
D
BD
B
D
in the convex quadrilateral
A
B
C
D
ABCD
A
BC
D
intersect in
S
S
S
. Let
F
1
F_1
F
1
and
F
2
F_2
F
2
be the areas of
△
A
B
S
\vartriangle ABS
△
A
BS
and
△
C
S
D
\vartriangle CSD
△
CS
D
. and let
F
F
F
be the area of the quadrilateral
A
B
C
D
ABCD
A
BC
D
. Show that
F
1
+
F
2
≤
F
\sqrt{ F_1 }+\sqrt{ F_2}\le \sqrt{ F}
F
1
+
F
2
≤
F
3a
1
Hide problems
maxarea of a quadrilateral with sidelengths 1, 4, 7,8
What is the largest possible area of a quadrilateral with sidelengths
1
,
4
,
7
1, 4, 7
1
,
4
,
7
and
8
8
8
?
2
1
Hide problems
strong subsets of powerset, union and intersections of sets
Let
A
A
A
be a set, and let
P
(
A
)
P (A)
P
(
A
)
be the powerset of all non-empty subsets of
A
A
A
. (For example,
A
=
{
1
,
2
,
3
}
A = \{1,2,3\}
A
=
{
1
,
2
,
3
}
, then
P
(
A
)
=
{
{
1
}
,
{
2
}
,
{
3
}
,
{
1
,
2
}
,
{
1
,
3
}
,
{
2
,
3
}
,
{
1
,
2
,
3
}
}
P (A) = \{\{1\},\{2\} ,\{3\},\{1,2\}, \{1,3\},\{2,3\}, \{1,2,3\}\}
P
(
A
)
=
{{
1
}
,
{
2
}
,
{
3
}
,
{
1
,
2
}
,
{
1
,
3
}
,
{
2
,
3
}
,
{
1
,
2
,
3
}}
.) A subset
F
F
F
of P
(
A
)
(A)
(
A
)
is called strong if the following is true: If
B
1
B_1
B
1
and
B
2
B_2
B
2
are elements of
F
F
F
, then
B
1
∪
B
2
B_1 \cup B_2
B
1
∪
B
2
is also an element of
F
F
F
. Suppose that
F
F
F
and
G
G
G
are strong subsets of
P
(
A
)
P (A)
P
(
A
)
. a) Is the union
F
∪
G
F \cup G
F
∪
G
necessarily strong? b) Is the intersection
F
∩
G
F \cap G
F
∩
G
necessarily strong?
1b
1
Hide problems
x^3, y^3 , x + y are rational, then xy, x^2+y^2, x, y are also rational
Suppose that
x
x
x
and
y
y
y
are positive real numbers such that
x
3
,
y
3
x^3, y^3
x
3
,
y
3
and
x
+
y
x + y
x
+
y
are all rational numbers. Show that the numbers
x
y
,
x
2
+
y
2
,
x
xy, x^2+y^2, x
x
y
,
x
2
+
y
2
,
x
and
y
y
y
are also rational
1a
1
Hide problems
α + b + c> 0, ax^2 + bx + c = 0$ has no real solutions, prove c>0
Suppose that
a
,
b
,
c
a, b, c
a
,
b
,
c
are real numbers such that
a
+
b
+
c
>
0
a + b + c> 0
a
+
b
+
c
>
0
, and so the equation
a
x
2
+
b
x
+
c
=
0
ax^2 + bx + c = 0
a
x
2
+
b
x
+
c
=
0
has no real solutions. Show that
c
>
0
c> 0
c
>
0
.