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National and Regional Contests
Norway Contests
Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Final Round
2013 Abels Math Contest (Norwegian MO) Final
2013 Abels Math Contest (Norwegian MO) Final
Part of
Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Final Round
Subcontests
(6)
1b
1
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a_{n+1} =(a_1 + a_2 + ...+ a_n)/n+1, for every b exists a_k : a_k < bk
The sequence
a
1
,
a
2
,
a
3
,
.
.
.
a_1, a_2, a_3,...
a
1
,
a
2
,
a
3
,
...
is defined so that
a
1
=
1
a_1 = 1
a
1
=
1
and
a
n
+
1
=
a
1
+
a
2
+
.
.
.
+
a
n
n
+
1
a_{n+1} =\frac{a_1 + a_2 + ...+ a_n}{n}+1
a
n
+
1
=
n
a
1
+
a
2
+
...
+
a
n
+
1
for
n
≥
1
n \ge 1
n
≥
1
. Show that for every positive real number
b
b
b
we can find
a
k
a_k
a
k
so that
a
k
<
b
k
a_k < bk
a
k
<
bk
.
1a
1
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if it is true that 3x^2 + y^2 >= -ax(x + y), find a
Find all real numbers
a
a
a
such that the inequality
3
x
2
+
y
2
≥
−
a
x
(
x
+
y
)
3x^2 + y^2 \ge -ax(x + y)
3
x
2
+
y
2
≥
−
a
x
(
x
+
y
)
holds for all real numbers
x
x
x
and
y
y
y
.
2
1
Hide problems
distance from point p to longest side h is less than or equal to s/\sqrt3
In a triangle
T
T
T
, all the angles are less than
9
0
o
90^o
9
0
o
, and the longest side has length
s
s
s
. Show that for every point
p
p
p
in
T
T
T
we can pick a corner
h
h
h
in
T
T
T
such that the distance from
p
p
p
to
h
h
h
is less than or equal to
s
/
3
s/\sqrt3
s
/
3
.
3
1
Hide problems
1/3+ 2/4+... +(p -3)/(p - 1)=a/b , p prime >=b, prove p /a
A prime number
p
≥
5
p \ge 5
p
≥
5
is given. Write
1
3
+
2
4
+
.
.
.
+
p
−
3
p
−
1
=
a
b
\frac13+\frac24+... +\frac{p -3}{p - 1}=\frac{a}{b}
3
1
+
4
2
+
...
+
p
−
1
p
−
3
=
b
a
for natural numbers
a
a
a
and
b
b
b
. Show that
p
p
p
divides
a
a
a
.
4b
1
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abc cubical boxes in a x b x c rectangular stack, bee flying inside the stack
A total of
a
⋅
b
⋅
c
a \cdot b \cdot c
a
⋅
b
⋅
c
cubical boxes are joined together in a
a
×
b
×
c
a \times b \times c
a
×
b
×
c
rectangular stack, where
a
,
b
,
c
≥
2
a, b, c \ge 2
a
,
b
,
c
≥
2
. A bee is found inside one of the boxes. It can fly from one box to another through a hole in the wall, but not through edges or corners. Also, it cannot fly outside the stack. For which triples
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
is it possible for the bee to fly through all of the boxes exactly once, and end up in the same box where it started?
4a
1
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crossing ordered quadruple of corners in a regular 2013-gon
An ordered quadruple
(
P
1
,
P
2
,
P
3
,
P
4
)
(P_1, P_2, P_3, P_4)
(
P
1
,
P
2
,
P
3
,
P
4
)
of corners in a regular
2013
2013
2013
-gon is called crossing if the four corners are all different, and the line segment from
P
1
P_1
P
1
to
P
2
P_2
P
2
intersects the line segment from
P
3
P_3
P
3
to
P
4
P_4
P
4
. How many crossing quadruples are there in the
2013
2013
2013
-gon?