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National and Regional Contests
Norway Contests
Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Final Round
1999 Abels Math Contest (Norwegian MO)
1999 Abels Math Contest (Norwegian MO)
Part of
Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Final Round
Subcontests
(6)
4
1
Hide problems
sum of alternating sums of a subset
For every nonempty subset
R
R
R
of
S
=
{
1
,
2
,
.
.
.
,
10
}
S = \{1,2,...,10\}
S
=
{
1
,
2
,
...
,
10
}
, we define the alternating sum
A
(
R
)
A(R)
A
(
R
)
as follows: If
r
1
,
r
2
,
.
.
.
,
r
k
r_1,r_2,...,r_k
r
1
,
r
2
,
...
,
r
k
are the elements of
R
R
R
in the increasing order, then
A
(
R
)
=
r
k
−
r
k
−
1
+
r
k
−
2
−
.
.
.
+
(
−
1
)
k
−
1
r
1
A(R) = r_k -r_{k-1} +r_{k-2}- ... +(-1)^{k-1}r_1
A
(
R
)
=
r
k
−
r
k
−
1
+
r
k
−
2
−
...
+
(
−
1
)
k
−
1
r
1
. (a) Is it possible to partition
S
S
S
into two sets having the same alternating sum? (b) Determine the sum
∑
R
A
(
R
)
\sum_{R} A(R)
∑
R
A
(
R
)
, where
R
R
R
runs over all nonempty subsets of
S
S
S
.
3
1
Hide problems
equilateral wanted, triangle 30-75-75 given
An isosceles triangle
A
B
C
ABC
A
BC
with
A
B
=
A
C
AB = AC
A
B
=
A
C
and
∠
A
=
3
0
o
\angle A = 30^o
∠
A
=
3
0
o
is inscribed in a circle with center
O
O
O
. Point
D
D
D
lies on the shorter arc
A
C
AC
A
C
so that
∠
D
O
C
=
3
0
o
\angle DOC = 30^o
∠
D
OC
=
3
0
o
, and point
G
G
G
lies on the shorter arc
A
B
AB
A
B
so that
D
G
=
A
C
DG = AC
D
G
=
A
C
and
A
G
<
B
G
AG < BG
A
G
<
BG
. The line
B
G
BG
BG
intersects
A
C
AC
A
C
and
A
B
AB
A
B
at
E
E
E
and
F
F
F
, respectively. (a) Prove that triangle
A
F
G
AFG
A
FG
is equilateral. (b) Find the ratio between the areas of triangles
A
F
E
AFE
A
FE
and
A
B
C
ABC
A
BC
.
2b
1
Hide problems
b | a^3, c | b^3 , a | c^3 => abc | (a+b+c)^{13}
If
a
,
b
,
c
a,b,c
a
,
b
,
c
are positive integers such that
b
∣
a
3
,
c
∣
b
3
b | a^3, c | b^3
b
∣
a
3
,
c
∣
b
3
and
a
∣
c
3
a | c^3
a
∣
c
3
, prove that
a
b
c
∣
(
a
+
b
+
c
)
13
abc | (a+b+c)^{13}
ab
c
∣
(
a
+
b
+
c
)
13
2a
1
Hide problems
2m^2 +n^2 = 2mn+3n diophantine
Find all integers
m
m
m
and
n
n
n
such that
2
m
2
+
n
2
=
2
m
n
+
3
n
2m^2 +n^2 = 2mn+3n
2
m
2
+
n
2
=
2
mn
+
3
n
1b
1
Hide problems
a^2 +b^2 +c^2 +d^2 \ge a(b+c+d+e)
If
a
,
b
,
c
,
d
,
e
a,b,c,d,e
a
,
b
,
c
,
d
,
e
are real numbers, prove the inequality
a
2
+
b
2
+
c
2
+
d
2
+
e
2
≥
a
(
b
+
c
+
d
+
e
)
a^2 +b^2 +c^2 +d^2+e^2 \ge a(b+c+d+e)
a
2
+
b
2
+
c
2
+
d
2
+
e
2
≥
a
(
b
+
c
+
d
+
e
)
.
1a
1
Hide problems
f(t^2 +t +1) = t for all real t >= 0
Find a function
f
f
f
such that
f
(
t
2
+
t
+
1
)
=
t
f(t^2 +t +1) = t
f
(
t
2
+
t
+
1
)
=
t
for all real
t
≥
0
t \ge 0
t
≥
0