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National and Regional Contests
Norway Contests
Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Final Round
2011 Abels Math Contest (Norwegian MO)
2011 Abels Math Contest (Norwegian MO)
Part of
Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Final Round
Subcontests
(7)
2b
1
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equal products of areas in convex hecagon with concurrent diagonals
The diagonals
A
D
,
B
E
AD, BE
A
D
,
BE
, and
C
F
CF
CF
of a convex hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
intersect in a common point. Show that
a
(
A
B
E
)
a
(
C
D
A
)
a
(
E
F
C
)
=
a
(
B
C
E
)
a
(
D
E
A
)
a
(
F
A
C
)
a(ABE) a(CDA) a(EFC) = a(BCE) a(DEA) a(FAC)
a
(
A
BE
)
a
(
C
D
A
)
a
(
EFC
)
=
a
(
BCE
)
a
(
D
E
A
)
a
(
F
A
C
)
, where
a
(
K
L
M
)
a(KLM)
a
(
K
L
M
)
is the area of the triangle
K
L
M
KLM
K
L
M
. https://cdn.artofproblemsolving.com/attachments/0/a/bcbbddedde159150fe3c26b1f0a2bfc322aa1a.png
4b
1
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in 199 persons, each person is a friend of exactly 100 others
In a group of
199
199
199
persons, each person is a friend of exactly
100
100
100
other persons in the group. All friendships are mutual, and we do not count a person as a friend of himself/herself. For which integers
k
>
1
k > 1
k
>
1
is the existence of
k
k
k
persons, all being friends of each other, guaranteed?
4a
1
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n avenues south to north, n streets west to east, pay in every junction
In a town there are
n
n
n
avenues running from south to north. They are numbered
1
1
1
through
n
n
n
(from west to east). There are
n
n
n
streets running from west to east – they are also numbered
1
1
1
through
n
n
n
(from south to north). If you drive through the junction of the
k
k
k
th avenue and the
ℓ
\ell
ℓ
th street, you have to pay
k
ℓ
k\ell
k
ℓ
kroner. How much do you at least have to pay for driving from the junction of the
1
1
1
st avenue and the
1
1
1
st street to the junction of the nth avenue and the
n
n
n
th street? (You also pay for the starting and finishing junctions.)
3b
1
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functional inequality f(xy) <= 1/2 ( (f(x) + f(y) )
Find all functions
f
f
f
from the real numbers to the real numbers such that
f
(
x
y
)
≤
1
2
(
f
(
x
)
+
f
(
y
)
)
f(xy) \le \frac12 \left(f(x) + f(y) \right)
f
(
x
y
)
≤
2
1
(
f
(
x
)
+
f
(
y
)
)
for all real numbers
x
x
x
and
y
y
y
.
3a
1
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(m+n)a_{m+n }<= a_m +a_n , prove 1 / a_{200} > 4 x 10^7
The positive numbers
a
1
,
a
2
,
.
.
.
a_1, a_2,...
a
1
,
a
2
,
...
satisfy
a
1
=
1
a_1 = 1
a
1
=
1
and
(
m
+
n
)
a
m
+
n
≤
a
m
+
a
n
(m+n)a_{m+n }\le a_m +a_n
(
m
+
n
)
a
m
+
n
≤
a
m
+
a
n
for all positive integers
m
m
m
and
n
n
n
. Show that
1
a
200
>
4
⋅
1
0
7
\frac{1}{a_{200}} > 4 \cdot 10^7
a
200
1
>
4
⋅
1
0
7
. .
2a
1
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perpendicular diagonals wanted in a ABCD given sidelengths
In the quadrilateral
A
B
C
D
ABCD
A
BC
D
the side
A
B
AB
A
B
has length
7
,
B
C
7, BC
7
,
BC
length
14
,
C
D
14, CD
14
,
C
D
length
26
26
26
, and
D
A
DA
D
A
length
23
23
23
. Show that the diagonals are perpendicular.You may assume that the quadrilateral is convex (all internal angles are less than
18
0
o
180^o
18
0
o
).
1
1
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concatenating no, a_1 = n^2011, a_i is sum of the digits of a_{i-1}, a_4 ?
Let
n
n
n
be the number that is produced by concatenating the numbers
1
,
2
,
.
.
.
,
4022
1, 2,... , 4022
1
,
2
,
...
,
4022
, that is,
n
=
1234567891011...40214022
n = 1234567891011...40214022
n
=
1234567891011...40214022
. a. Show that
n
n
n
is divisible by
3
3
3
. b. Let
a
1
=
n
2011
a_1 = n^{2011}
a
1
=
n
2011
, and let
a
i
a_i
a
i
be the sum of the digits of
a
i
−
1
a_{i-1}
a
i
−
1
for
i
>
1
i > 1
i
>
1
. Find
a
4
a_4
a
4