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National and Regional Contests
Norway Contests
Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Final Round
2024 Abelkonkurransen Finale
2024 Abelkonkurransen Finale
Part of
Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Final Round
Subcontests
(8)
4b
1
Hide problems
Two cyclic pentagons with incenter condition implies concurrency
The pentagons
P
1
P
2
P
3
P
4
P
5
P_1P_2P_3P_4P_5
P
1
P
2
P
3
P
4
P
5
and
I
1
I
2
I
3
I
4
I
5
I_1I_2I_3I_4I_5
I
1
I
2
I
3
I
4
I
5
are cyclic, where
I
i
I_i
I
i
is the incentre of the triangle
P
i
−
1
P
i
P
i
+
1
P_{i-1}P_iP_{i+1}
P
i
−
1
P
i
P
i
+
1
(reckoned cyclically, that is
P
0
=
P
5
P_0=P_5
P
0
=
P
5
and
P
6
=
P
1
P_6=P_1
P
6
=
P
1
). Show that the lines
P
1
I
1
,
P
2
I
2
,
P
3
I
3
,
P
4
I
4
P_1I_1, P_2I_2, P_3I_3, P_4I_4
P
1
I
1
,
P
2
I
2
,
P
3
I
3
,
P
4
I
4
and
P
5
I
5
P_5I_5
P
5
I
5
meet in a single point.
4a
1
Hide problems
Determining alpha given angle condition with circumcenter
The triangle
A
B
C
ABC
A
BC
with
A
B
<
A
C
AB < AC
A
B
<
A
C
has an altitude
A
D
AD
A
D
. The points
E
E
E
and
A
A
A
lie on opposite sides of
B
C
BC
BC
, with
E
E
E
on the circumcircle of
A
B
C
ABC
A
BC
. Furthermore,
A
D
=
D
E
AD = DE
A
D
=
D
E
and
∠
A
D
O
=
∠
C
D
E
\angle ADO=\angle CDE
∠
A
D
O
=
∠
C
D
E
, where
O
O
O
is the circumcentre of
A
B
C
ABC
A
BC
. Determine
∠
B
A
C
\angle BAC
∠
B
A
C
.
3b
1
Hide problems
Maximizing red sum in even colouring of 2024-table
A
2024
2024
2024
-table is a table with two rows and
2024
2024
2024
columns containg all the numbers
1
,
2
,
…
,
4048
1,2,\dots,4048
1
,
2
,
…
,
4048
. Such a table is evenly coloured if exactly half of the numbers in each row, and one number in each column, is coloured red. The red sum in an evenly coloured
2024
2024
2024
-table is the sum of all the red numbers in the table. Let
N
N
N
be the largest number such that every
2024
2024
2024
-table has an even colouring with red sum
≥
N
\ge N
≥
N
. Determine
N
N
N
, and find the number of
2024
2024
2024
-tables such that every even colouring of the table has red sum
≤
N
\le N
≤
N
.
3a
1
Hide problems
Secret agents in Geostan and Combostan
Determine the smallest constant
N
N
N
so that the following may hold true: Geostan has deployed secret agents in Combostan. All pairs of agents can communicate, either directly or through other agents. The distance between two agents is the smallest number of agents in a communication chain between the two agents. Andreas and Edvard are among these agents, and Combostan has given Noah the task of determining the distance between Andreas and Edvard. Noah has a list of numbers, one for each agent. The number of an agent describes the longest of the two distances from the agent to Andreas and Edvard. However, Noah does not know which number corresponds to which agent, or which agents have direct contact. Given this information, he can write down
N
N
N
numbers and prove that the distance between Andreas and Edvard is one of these
N
N
N
numbers. The number
N
N
N
is independent of the agents’ communication network.
2b
1
Hide problems
Standard functional equation from R to R
Find all functions
f
:
R
→
R
f:\mathbb{R} \to \mathbb{R}
f
:
R
→
R
satisfying
x
f
(
f
(
x
)
+
y
)
=
f
(
x
y
)
+
x
2
xf(f(x)+y)=f(xy)+x^2
x
f
(
f
(
x
)
+
y
)
=
f
(
x
y
)
+
x
2
for all
x
,
y
∈
R
x,y \in \mathbb{R}
x
,
y
∈
R
.
2a
1
Hide problems
Finding smallest sequence with no prime difference
Positive integers
a
0
<
a
1
<
⋯
<
a
n
a_0<a_1<\dots<a_n
a
0
<
a
1
<
⋯
<
a
n
, are to be chosen so that
a
j
−
a
i
a_j-a_i
a
j
−
a
i
is not a prime for any
i
,
j
i,j
i
,
j
with
0
≤
i
<
j
≤
n
0 \le i <j \le n
0
≤
i
<
j
≤
n
. For each
n
≥
1
n \ge 1
n
≥
1
, determine the smallest possible value of
a
n
a_n
a
n
.
1b
1
Hide problems
Iterates of function give distinct residues
Find all functions
f
:
Z
→
Z
f:\mathbb{Z} \to \mathbb{Z}
f
:
Z
→
Z
such that the numbers
n
,
f
(
n
)
,
f
(
f
(
n
)
)
,
…
,
f
m
−
1
(
n
)
n, f(n),f(f(n)),\dots,f^{m-1}(n)
n
,
f
(
n
)
,
f
(
f
(
n
))
,
…
,
f
m
−
1
(
n
)
are distinct modulo
m
m
m
for all integers
n
,
m
n,m
n
,
m
with
m
>
1
m>1
m
>
1
. (Here
f
k
f^k
f
k
is defined by
f
0
(
n
)
=
n
f^0(n)=n
f
0
(
n
)
=
n
and
f
k
+
1
(
n
)
=
f
(
f
k
(
n
)
)
f^{k+1}(n)=f(f^{k}(n))
f
k
+
1
(
n
)
=
f
(
f
k
(
n
))
for
k
≥
0
k \ge 0
k
≥
0
.)
1a
1
Hide problems
Sum of coprime residues modulo n
Determine all integers
n
≥
2
n \ge 2
n
≥
2
such that
n
∣
s
n
−
t
n
n \mid s_n-t_n
n
∣
s
n
−
t
n
where
s
n
s_n
s
n
is the sum of all the integers in the interval
[
1
,
n
]
[1,n]
[
1
,
n
]
that are mutually prime to
n
n
n
, and
t
n
t_n
t
n
is the sum of the remaining integers in the same interval.