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National and Regional Contests
Norway Contests
Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Final Round
2012 Abels Math Contest (Norwegian MO) Final
2012 Abels Math Contest (Norwegian MO) Final
Part of
Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Final Round
Subcontests
(7)
4b
1
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a_m = s a_{m-1} + b_m, Sum a^2_i <= 100/(1-s^2)
Positive numbers
b
1
,
b
2
,
.
.
.
,
b
n
b_1, b_2,..., b_n
b
1
,
b
2
,
...
,
b
n
are given so that
b
1
+
b
2
+
.
.
.
+
b
n
≤
10
b_1 + b_2 + ...+ b_n \le 10
b
1
+
b
2
+
...
+
b
n
≤
10
. Further,
a
1
=
b
1
a_1 = b_1
a
1
=
b
1
and
a
m
=
s
a
m
−
1
+
b
m
a_m = sa_{m-1} + b_m
a
m
=
s
a
m
−
1
+
b
m
for
m
>
1
m > 1
m
>
1
, where
0
≤
s
<
1
0 \le s < 1
0
≤
s
<
1
. Show that
a
1
2
+
a
2
2
+
.
.
.
+
a
n
2
≤
100
1
−
s
2
a^2_1 + a^2_2 + ... + a^2_n \le \frac{100}{1 - s^2}
a
1
2
+
a
2
2
+
...
+
a
n
2
≤
1
−
s
2
100
4a
1
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(1 +x/y)^3 +(1 +y/x)^3 >= 16 for x,y>0
Two positive numbers
x
x
x
and
y
y
y
are given. Show that
(
1
+
x
y
)
3
+
(
1
+
y
x
)
3
≥
16
\left(1 +\frac{x}{y} \right)^3 + \left(1 +\frac{y}{x}\right)^3 \ge 16
(
1
+
y
x
)
3
+
(
1
+
x
y
)
3
≥
16
.
3b
1
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k^m - 1 is divisible by 2^m
Which positive integers
m
m
m
are such that
k
m
−
1
k^m - 1
k
m
−
1
is divisible by
2
m
2^m
2
m
for all odd numbers
k
≥
3
k \ge 3
k
≥
3
?
3a
1
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last three digits in the product 1 · 3 · 5 · 7 · . . . · 2009 · 2011
Find the last three digits in the product
1
⋅
3
⋅
5
⋅
7
⋅
.
.
.
⋅
2009
⋅
2011
1 \cdot 3\cdot 5\cdot 7 \cdot . . . \cdot 2009 \cdot 2011
1
⋅
3
⋅
5
⋅
7
⋅
...
⋅
2009
⋅
2011
.
2
1
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1/R_12 +1/R_23+1/R_34+1/R_14 = 2 (1/R_13+1/R_24 )
(a)Two circles
S
1
S_1
S
1
and
S
2
S_2
S
2
are placed so that they do not overlap each other, neither completely nor partially. They have centres in
O
1
O_1
O
1
and
O
2
O_2
O
2
, respectively. Further,
L
1
L_1
L
1
and
M
1
M_1
M
1
are different points on
S
1
S_1
S
1
so that
O
2
L
1
O_2L_1
O
2
L
1
and
O
2
M
1
O_2M_1
O
2
M
1
are tangent to
S
1
S_1
S
1
, and similarly
L
2
L_2
L
2
and
M
2
M_2
M
2
are different points on
S
2
S_2
S
2
so that
O
1
L
2
O_1L_2
O
1
L
2
and
O
1
M
2
O_1M_2
O
1
M
2
are tangent to
S
2
S_2
S
2
. Show that there exists a unique circle which is tangent to the four line segments
O
2
L
1
,
O
2
M
1
,
O
1
L
2
O_2L_1, O_2M_1, O_1L_2
O
2
L
1
,
O
2
M
1
,
O
1
L
2
, and
O
1
M
2
O_1M_2
O
1
M
2
.(b) Four circles
S
1
,
S
2
,
S
3
S_1, S_2, S_3
S
1
,
S
2
,
S
3
and
S
4
S_4
S
4
are placed so that none of them overlap each other, neither completely nor partially. They have centres in
O
1
,
O
2
,
O
3
O_1, O_2, O_3
O
1
,
O
2
,
O
3
, and
O
4
O_4
O
4
, respectively. For each pair
(
S
i
,
S
j
)
(S_i, S_j )
(
S
i
,
S
j
)
of circles, with
1
≤
i
<
j
≤
4
1 \le i < j \le 4
1
≤
i
<
j
≤
4
, we find a circle
S
i
j
S_{ij}
S
ij
as in part a. The circle
S
i
j
S_{ij}
S
ij
has radius
R
i
j
R_{ij}
R
ij
. Show that
1
R
12
+
1
R
23
+
1
R
34
+
1
R
14
=
2
(
1
R
13
+
1
R
24
)
\frac{1}{R_{12}} + \frac{1}{R_{23}}+\frac{1}{R_{34}}+\frac{1}{R_{14}}= 2 \left(\frac{1}{R_{13}} +\frac{1}{R_{24}}\right)
R
12
1
+
R
23
1
+
R
34
1
+
R
14
1
=
2
(
R
13
1
+
R
24
1
)
1b
1
Hide problems
bw painting integers, m white => m + 20 white, k black => k + 35 black
Every integer is painted white or black, so that if
m
m
m
is white then
m
+
20
m + 20
m
+
20
is also white, and if
k
k
k
is black then
k
+
35
k + 35
k
+
35
is also black. For which
n
n
n
can exactly
n
n
n
of the numbers
1
,
2
,
.
.
.
,
50
1, 2, ..., 50
1
,
2
,
...
,
50
be white?
1a
1
Hide problems
Berit has 11 twenty-kroner , 14 ten-kroner and 12 five kroner coins
Berit has
11
11
11
twenty kroner coins,
14
14
14
ten kroner coins, and
12
12
12
five kroner coins. An exchange machine can exchange three ten kroner coins into one twenty kroner coin and two five kroner coins, and the reverse. It can also exchange two twenty kroner coins into three ten kroner coins and two five kroner coins, and the reverse. (i) Can Berit get the same number of twenty kroner and ten kroner coins, but no five kroner coins? (ii) Can she get the same number each of twenty kroner, ten kroner, and five kroner coins?