MathDB
Problems
Contests
National and Regional Contests
New Zealand Contests
Auckland Mathematical Olympiad
2023 Auckland Mathematical Olympiad
2023 Auckland Mathematical Olympiad
Part of
Auckland Mathematical Olympiad
Subcontests
(10)
10
1
Hide problems
max ||...||x_1 - x_2|- x_3| -... | - x_{2023}| (2023 Auckland MO p10)
Find the maximum of the expression
∣
∣
.
.
.
∣
∣
x
1
−
x
2
∣
−
x
3
∣
−
.
.
.
∣
−
x
2023
∣
,
||...||x_1 - x_2|- x_3| -... | - x_{2023}|,
∣∣...∣∣
x
1
−
x
2
∣
−
x
3
∣
−
...∣
−
x
2023
∣
,
where
x
1
,
x
2
,
.
.
.
,
x
2023
x_1,x_2,..., x_{2023}
x
1
,
x
2
,
...
,
x
2023
are distinct natural numbers between
1
1
1
and
2023
2023
2023
.
9
1
Hide problems
d(O,AC)= 1/2 BC, cyclic ABCD, AC_|_BD (2023 Auckland MO p9)
Quadrillateral
A
B
C
D
ABCD
A
BC
D
is inscribed in a circle with centre
O
O
O
. Diagonals
A
C
AC
A
C
and
B
D
BD
B
D
are perpendicular. Prove that the distance from the centre
O
O
O
to
A
D
AD
A
D
is half the length of
B
C
BC
BC
.
8
1
Hide problems
1, 2,3,..., 2023 no numberis product of 2 others (2023 Auckland MO p8)
How few numbers is it possible to cross out from the sequence
1
,
2
,
3
,
.
.
.
,
2023
1, 2,3,..., 2023
1
,
2
,
3
,
...
,
2023
so that among those left no number is the product of any two (distinct) other numbers?
7
1
Hide problems
2024 polygons whose total area > 2023 (2023 Auckland MO p7)
In a square of area
1
1
1
there are situated
2024
2024
2024
polygons whose total area is greater than
2023
2023
2023
. Prove that they have a point in common.
6
1
Hide problems
infinite sequence of numbered lights (2023 Auckland MO p6)
Suppose there is an infi nite sequence of lights numbered
1
,
2
,
3
,
.
.
.
,
1, 2, 3,...,
1
,
2
,
3
,
...
,
and you know the following two rules about how the lights work:
∙
\bullet
∙
If the light numbered
k
k
k
is on, the lights numbered
2
k
2k
2
k
and
2
k
+
1
2k + 1
2
k
+
1
are also guaranteed to be on.
∙
\bullet
∙
If the light numbered
k
k
k
is off, then the lights numbered
4
k
+
1
4k + 1
4
k
+
1
and
4
k
+
3
4k + 3
4
k
+
3
are also guaranteed to be off. Suppose you notice that light number
2023
2023
2023
is on. Identify all the lights that are guaranteed to be on?
5
1
Hide problems
11 quadratics - 2player game (2023 Auckland MO p5)
There are
11
11
11
quadratic equations on the board, where each coefficient is replaced by a star. Initially, each of them looks like this
⋆
x
2
+
⋆
x
+
⋆
=
0.
\star x^2 + \star x + \star= 0.
⋆
x
2
+
⋆
x
+
⋆
=
0.
Two players are playing a game making alternating moves. In one move each ofthem replaces one star with a real nonzero number. The first player tries to make as many equations as possible without roots and the second player tries to make the number of equations without roots as small as possible. What is the maximal number of equations without roots that the fi rst player can achieve if the second player plays to her best? Describe the strategies of both players.
4
1
Hide problems
66...66x55...55 divisible by 7 (2023 Auckland MO p4)
Which digit must be substituted instead of the star so that the following large number
66...66
⏟
2023
⋆
55...55
⏟
2023
\underbrace{66...66}_{2023} \star \underbrace{55...55}_{2023}
2023
66...66
⋆
2023
55...55
is divisible by
7
7
7
?
3
1
Hide problems
0 or 1 checkers in 8x8 board (2023 Auckland MO p3)
Each square on an
8
×
8
8\times 8
8
×
8
checkers board contains either one or zero checkers. The number of checkers in each row is a multiple of
3
3
3
, the number of checkers in each column is a multiple of
5
5
5
. Assuming the top left corner of the board is shown below, how many checkers are used in total? https://cdn.artofproblemsolving.com/attachments/0/8/e46929e7ec3fff9be4892ef954ae299e0cb8c7.png
2
1
Hide problems
triangle by equal segments on triangles' extensions (2023 Auckland MO p2)
Triangle
A
B
C
ABC
A
BC
of area
1
1
1
is given. Point
A
′
A'
A
′
lies on the extension of side
B
C
BC
BC
beyond point
C
C
C
with
B
C
=
C
A
′
BC = CA'
BC
=
C
A
′
. Point
B
′
B'
B
′
lies on extension of side
C
A
CA
C
A
beyond
A
A
A
and
C
A
=
A
B
′
CA = AB'
C
A
=
A
B
′
.
C
′
C'
C
′
lies on extension of
A
B
AB
A
B
beyond
B
B
B
with
A
B
=
B
C
′
AB = BC'
A
B
=
B
C
′
. Find the area of triangle
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
.
1
1
Hide problems
7 adults or 11 children in a stadium's single section (2023 Auckland MO p1)
A single section at a stadium can hold either
7
7
7
adults or
11
11
11
children. When
N
N
N
sections are completely lled, an equal number of adults and children will be seated in them. What is the least possible value of
N
N
N
?