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National and Regional Contests
Norway Contests
Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Final Round
1993 Abels Math Contest (Norwegian MO)
1993 Abels Math Contest (Norwegian MO)
Part of
Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Final Round
Subcontests
(5)
1a
1
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equal sums of triangle areas, a+c = b+d, midpoints related
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral and
A
′
,
B
′
C
′
,
D
′
A',B'C',D'
A
′
,
B
′
C
′
,
D
′
be the midpoints of
A
B
,
B
C
,
C
D
,
D
A
AB,BC,CD,DA
A
B
,
BC
,
C
D
,
D
A
, respectively. Let
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
denote the areas of quadrilaterals into which lines
A
′
C
′
A'C'
A
′
C
′
and
B
′
D
′
B'D'
B
′
D
′
divide the quadrilateral
A
B
C
D
ABCD
A
BC
D
(where a corresponds to vertex
A
A
A
etc.). Prove that
a
+
c
=
b
+
d
a+c = b+d
a
+
c
=
b
+
d
.
1b
1
Hide problems
\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}< 2 when a,b,c sidelengths
Given a triangle with sides of lengths
a
,
b
,
c
a,b,c
a
,
b
,
c
, prove that
a
b
+
c
+
b
c
+
a
+
c
a
+
b
<
2
\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}< 2
b
+
c
a
+
c
+
a
b
+
a
+
b
c
<
2
.
2
1
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(a+b+c+d)^2 > 8(ac+bd) if b < c < d
If
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
are real numbers with
b
<
c
<
d
b < c < d
b
<
c
<
d
, prove that
(
a
+
b
+
c
+
d
)
2
>
8
(
a
c
+
b
d
)
(a+b+c+d)^2 > 8(ac+bd)
(
a
+
b
+
c
+
d
)
2
>
8
(
a
c
+
b
d
)
.
3
1
Hide problems
F_n = F_{n-1}F_{n-2}....F_1F_0 +2, coprime fermat numbers
The Fermat-numbers are defined by
F
n
=
2
2
n
+
1
F_n = 2^{2^n}+1
F
n
=
2
2
n
+
1
for
n
∈
N
n\in N
n
∈
N
. (a) Prove that
F
n
=
F
n
−
1
F
n
−
2
.
.
.
.
F
1
F
0
+
2
F_n = F_{n-1}F_{n-2}....F_1F_0 +2
F
n
=
F
n
−
1
F
n
−
2
....
F
1
F
0
+
2
for
n
>
0
n > 0
n
>
0
. (b) Prove that any two different Fermat numbers are coprime
4
1
Hide problems
+-1, at each of the 8 vertices of a given cube, product, sum
Each of the
8
8
8
vertices of a given cube is given a value
1
1
1
or
−
1
-1
−
1
. Each of the
6
6
6
faces is given the value of product of its four vertices. Let
A
A
A
be the sum of all the
14
14
14
values. Which are the possible values of
A
A
A
?