MathDB
Problems
Contests
National and Regional Contests
Norway Contests
Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Final Round
2017 Abels Math Contest (Norwegian MO) Final
2017 Abels Math Contest (Norwegian MO) Final
Part of
Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Final Round
Subcontests
(6)
3b
1
Hide problems
x and o game, in an infinite grid of regular triangles
In an infinite grid of regular triangles, Niels and Henrik are playing a game they made up. Every other time, Niels picks a triangle and writes
×
\times
×
in it, and every other time, Henrik picks a triangle where he writes a
o
o
o
. If one of the players gets four in a row in some direction (see figure), he wins the game. Determine whether one of the players can force a victory. https://cdn.artofproblemsolving.com/attachments/6/e/5e80f60f110a81a74268fded7fd75a71e07d3a.png
3a
1
Hide problems
How many cards can Nils show you without revealing his 8-digit number?
Nils has a telephone number with eight different digits. He has made
28
28
28
cards with statements of the type “The digit
a
a
a
occurs earlier than the digit
b
b
b
in my telephone number” – one for each pair of digits appearing in his number. How many cards can Nils show you without revealing his number?
1b
1
Hide problems
functional for starters II f(x)f(y) = f(x + y) + xy
Find all functions
f
:
R
→
R
f : R \to R
f
:
R
→
R
which satisfy
f
(
x
)
f
(
y
)
=
f
(
x
+
y
)
+
x
y
f(x)f(y) = f(x + y) + xy
f
(
x
)
f
(
y
)
=
f
(
x
+
y
)
+
x
y
for all
x
,
y
∈
R
x, y \in R
x
,
y
∈
R
.
1a
1
Hide problems
functional for starters f(x)f(y) = f(xy) + xy
Find all functions
f
:
R
→
R
f : R \to R
f
:
R
→
R
which satisfy
f
(
x
)
f
(
y
)
=
f
(
x
y
)
+
x
y
f(x)f(y) = f(xy) + xy
f
(
x
)
f
(
y
)
=
f
(
x
y
)
+
x
y
for all
x
,
y
∈
R
x, y \in R
x
,
y
∈
R
.
2
1
Hide problems
a_{n+2 }= 15a_{n+1} + 16a_n, infinite integers k such that 269 \ a_k
Let the sequence an be defined by
a
0
=
2
,
a
1
=
15
a_0 = 2, a_1 = 15
a
0
=
2
,
a
1
=
15
, and
a
n
+
2
=
15
a
n
+
1
+
16
a
n
a_{n+2 }= 15a_{n+1} + 16a_n
a
n
+
2
=
15
a
n
+
1
+
16
a
n
for
n
≥
0
n \ge 0
n
≥
0
. Show that there are infinitely many integers
k
k
k
such that
269
∣
a
k
269 | a_k
269∣
a
k
.
4
1
Hide problems
length of AP depends only on BC=a and < BAC = \alpha
Let
a
>
0
a > 0
a
>
0
and
0
<
α
<
π
0 < \alpha <\pi
0
<
α
<
π
be given. Let
A
B
C
ABC
A
BC
be a triangle with
B
C
=
a
BC = a
BC
=
a
and
∠
B
A
C
=
α
\angle BAC = \alpha
∠
B
A
C
=
α
, and call the cicumcentre
O
O
O
, and the orthocentre
H
H
H
. The point
P
P
P
lies on the ray from
A
A
A
through
O
O
O
. Let
S
S
S
be the mirror image of
P
P
P
through
A
C
AC
A
C
, and
T
T
T
the mirror image of
P
P
P
through
A
B
AB
A
B
. Assume that
S
A
T
H
SATH
S
A
T
H
is cyclic. Show that the length
A
P
AP
A
P
depends only on
a
a
a
and
α
\alpha
α
.