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National and Regional Contests
Norway Contests
Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Final Round
2003 Abels Math Contest (Norwegian MO)
2003 Abels Math Contest (Norwegian MO)
Part of
Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Final Round
Subcontests
(7)
4b
1
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more than m participants at a camp, friends related
Let
m
>
3
m> 3
m
>
3
be an integer. At a camp there are more than
m
m
m
participants. The camp manager discovers that every time he picks out the camp participants, they say they have exactly one mutual friend among the participants. Which is the largest possible number of participants at the camp? (If
A
A
A
is a friend of
B
,
B
B, B
B
,
B
is also a friend of
A
A
A
. A person is not considered a friend of himself.)
4a
1
Hide problems
25 boys and 25 girls sit around a table
25
25
25
boys and
25
25
25
girls sit around a table. Show that there is a person who has a girl sitting on either side of them.
3
1
Hide problems
< ACB +2 < ACD = 180^o , BC +CE = AE
Let
A
B
C
ABC
A
BC
be a triangle with
A
C
>
B
C
AC> BC
A
C
>
BC
, and let
S
S
S
be the circumscribed circle of the triangle.
A
B
AB
A
B
divides
S
S
S
into two arcs. Let
D
D
D
be the midpoint of the arc containing
C
C
C
. (a) Show that
∠
A
C
B
+
2
⋅
∠
A
C
D
=
18
0
o
\angle ACB +2 \cdot \angle ACD = 180^o
∠
A
CB
+
2
⋅
∠
A
C
D
=
18
0
o
. (b) Let
E
E
E
be the foot of the altitude from
D
D
D
on
A
C
AC
A
C
. Show that
B
C
+
C
E
=
A
E
BC +CE = AE
BC
+
CE
=
A
E
.
2b
1
Hide problems
\sum a_i^3 \ge t(\sum a_i )^2 when a_i \in N-{0}
Let
a
1
,
a
2
,
.
.
.
,
a
n
a_1,a_2,...,a_n
a
1
,
a
2
,
...
,
a
n
be
n
n
n
different positive integers where
n
≥
1
n\ge 1
n
≥
1
. Show that
∑
i
=
1
n
a
i
3
≥
(
∑
i
=
1
n
a
i
)
2
\sum_{i=1}^n a_i^3 \ge \left(\sum_{i=1}^n a_i\right)^2
i
=
1
∑
n
a
i
3
≥
(
i
=
1
∑
n
a
i
)
2
2a
1
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y^3+5=x(y^2+2) diophantine
Find all pairs
(
x
,
y
)
(x, y)
(
x
,
y
)
of integers numbers such that
y
3
+
5
=
x
(
y
2
+
2
)
y^3+5=x(y^2+2)
y
3
+
5
=
x
(
y
2
+
2
)
1b
1
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\sum x_i ^2 \le -nmM if \sum x_i = 0 and x_i \in [m,M]
Let
x
1
,
x
2
,
.
.
.
,
x
n
x_1,x_2,...,x_n
x
1
,
x
2
,
...
,
x
n
be real numbers in an interval
[
m
,
M
]
[m,M]
[
m
,
M
]
such that
∑
i
=
1
n
x
i
=
0
\sum_{i=1}^n x_i = 0
∑
i
=
1
n
x
i
=
0
. Show that
∑
i
=
1
n
x
i
2
≤
−
n
m
M
\sum_{i=1}^n x_i ^2 \le -nmM
∑
i
=
1
n
x
i
2
≤
−
nm
M
1a
1
Hide problems
x + y = 2 and x^3 + y^3 = 3, x^2+y^2 =?
Let
x
x
x
and
y
y
y
are real numbers such that
{
x
+
y
=
2
x
3
+
y
3
=
3
\begin{cases} x + y = 2 \\ x^3 + y^3 = 3\end{cases}
{
x
+
y
=
2
x
3
+
y
3
=
3
What is
x
2
+
y
2
x^2+y^2
x
2
+
y
2
?