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National and Regional Contests
Norway Contests
Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Final Round
2016 Abels Math Contest (Norwegian MO) Final
2016 Abels Math Contest (Norwegian MO) Final
Part of
Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Final Round
Subcontests
(6)
3a
1
Hide problems
X, Y,C' lie on a line iff AC = BC, 3 circles, 3 common tangents, 2 right angles
Three circles
S
A
,
S
B
S_A, S_B
S
A
,
S
B
, and
S
C
S_C
S
C
in the plane with centers in
A
,
B
A, B
A
,
B
, and
C
C
C
, respectively, are mutually tangential on the outside. The touchpoint between
S
A
S_A
S
A
and
S
B
S_B
S
B
we call
C
′
C'
C
′
, the one
S
A
S_A
S
A
between
S
C
S_C
S
C
we call
B
′
B'
B
′
, and the one between
S
B
S_B
S
B
and
S
C
S_C
S
C
we call
A
′
A'
A
′
. The common tangent between
S
A
S_A
S
A
and
S
C
S_C
S
C
(passing through B') we call
ℓ
B
\ell_B
ℓ
B
, and the common tangent between
S
B
S_B
S
B
and
S
C
S_C
S
C
(passing through
A
′
A'
A
′
) we call
ℓ
A
\ell_A
ℓ
A
. The intersection point of
ℓ
A
\ell_A
ℓ
A
and
ℓ
B
\ell_B
ℓ
B
is called
X
X
X
. The point
Y
Y
Y
is located so that
∠
X
B
Y
\angle XBY
∠
XB
Y
and
∠
Y
A
X
\angle YAX
∠
Y
A
X
are both right angles. Show that the points
X
,
Y
X, Y
X
,
Y
, and
C
′
C'
C
′
lie on a line if and only if
A
C
=
B
C
AC = BC
A
C
=
BC
.
3b
1
Hide problems
norwegian parallel segments, symmetry points, angle bisectors related
Let
A
B
C
ABC
A
BC
be an acute triangle with
A
B
<
A
C
AB < AC
A
B
<
A
C
. The points
A
1
A_1
A
1
and
A
2
A_2
A
2
are located on the line
B
C
BC
BC
so that
A
A
1
AA_1
A
A
1
and
A
A
2
AA_2
A
A
2
are the inner and outer angle bisectors at
A
A
A
for the triangle
A
B
C
ABC
A
BC
. Let
A
3
A_3
A
3
be the mirror image
A
2
A_2
A
2
with respect to
C
C
C
, and let
Q
Q
Q
be a point on
A
A
1
AA_1
A
A
1
such that
∠
A
1
Q
A
3
=
9
0
o
\angle A_1QA_3 = 90^o
∠
A
1
Q
A
3
=
9
0
o
. Show that
Q
C
/
/
A
B
QC // AB
QC
//
A
B
.
2b
1
Hide problems
diophantine with factorial x^3 + 2y^3 + 4z^3 = 9!
Find all non-negative integers
x
,
y
x, y
x
,
y
and
z
z
z
such that
x
3
+
2
y
3
+
4
z
3
=
9
!
x^3 + 2y^3 + 4z^3 = 9!
x
3
+
2
y
3
+
4
z
3
=
9
!
2a
1
Hide problems
diophantine system, a + b = cd , c + d = ab when 0<a<=b, 0<c<=d
Find all positive integers
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
with
a
≤
b
a \le b
a
≤
b
and
c
≤
d
c \le d
c
≤
d
such that
{
a
+
b
=
c
d
c
+
d
=
a
b
\begin{cases} a + b = cd \\ c + d = ab \end{cases}
{
a
+
b
=
c
d
c
+
d
=
ab
.
4
1
Hide problems
Function with absol value
Find all functions
f
:
R
→
R
f : \mathbb{R} \to \mathbb{R}
f
:
R
→
R
such that
f
(
x
)
f
(
y
)
=
∣
x
−
y
∣
⋅
f
(
x
y
+
1
x
−
y
)
f(x) f(y) = |x - y| \cdot f \left( \frac{xy + 1}{x - y} \right)
f
(
x
)
f
(
y
)
=
∣
x
−
y
∣
⋅
f
(
x
−
y
x
y
+
1
)
Holds for all
x
≠
y
∈
R
x \not= y \in \mathbb{R}
x
=
y
∈
R
1
1
Hide problems
walking sequence
A "walking sequence" is a sequence of integers with
a
i
+
1
=
a
i
±
1
a_{i+1} = a_i \pm 1
a
i
+
1
=
a
i
±
1
for every
i
i
i
.Show that there exists a sequence
b
1
,
b
2
,
.
.
.
,
b
2016
b_1, b_2, . . . , b_{2016}
b
1
,
b
2
,
...
,
b
2016
such that for every walking sequence
a
1
,
a
2
,
.
.
.
,
a
2016
a_1, a_2, . . . , a_{2016}
a
1
,
a
2
,
...
,
a
2016
where 1 \leq a_i \leq1010, there is for some
j
j
j
for which
a
j
=
b
j
a_j = b_j
a
j
=
b
j
.