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Contests
National and Regional Contests
New Zealand Contests
Auckland Mathematical Olympiad
2019 Auckland Mathematical Olympiad
2019 Auckland Mathematical Olympiad
Part of
Auckland Mathematical Olympiad
Subcontests
(5)
5
2
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2player game with 2019 coins (2019 Auckland MO J5)
2019
2019
2019
coins are on the table. Two students play the following game making alternating moves. The first player can in one move take the odd number of coins from
1
1
1
to
99
99
99
, the second player in one move can take an even number of coins from
2
2
2
to
100
100
100
. The player who can not make a move is lost. Who has the winning strategy in this game?
coloring a map by 2019 circles (2019 Auckland MO S5)
2019
2019
2019
circles split a plane into a number of parts whose boundaries are arcs of those circles. How many colors are needed to color this geographic map if any two neighboring parts must be coloured with different colours?
4
2
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a_n = a_{n-1} - 1/n+ 1/(n + 1) - 1/(n + 2) (2019 Auckland MO J4)
Suppose
a
1
=
1
6
a_1 =\frac16
a
1
=
6
1
and
a
n
=
a
n
−
1
−
1
n
+
2
n
+
1
−
1
n
+
2
a_n = a_{n-1} - \frac{1}{n}+ \frac{2}{n + 1} - \frac{1}{n + 2}
a
n
=
a
n
−
1
−
n
1
+
n
+
1
2
−
n
+
2
1
for
n
>
1
n > 1
n
>
1
. Find
a
100
a_{100}
a
100
.
min pos. integer not in form (2^a - 2^b/(2^c - 2^d) (2019 Auckland MO S4)
Find the smallest positive integer that cannot be expressed in the form
2
a
−
2
b
2
c
−
2
d
\frac{2^a - 2^b}{2^c - 2^d}
2
c
−
2
d
2
a
−
2
b
, where
a
a
a
,
b
b
b
,
c
c
c
,
d
d
d
are non-negative integers.
2
2
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any 2 of 2109 segments intersect (2019 Auckland MO J2)
There are
2019
2019
2019
segments
[
a
1
,
b
1
]
[a_1, b_1]
[
a
1
,
b
1
]
,
.
.
.
...
...
,
[
a
2019
,
b
2019
]
[a_{2019}, b_{2019}]
[
a
2019
,
b
2019
]
on the line. It is known that any two of them intersect. Prove that they all have a point in common.
exist a,b among 43 pos. integers 100 divides a^2-b^2 (2019 Auckland MO S2)
Prove that among any
43
43
43
positive integers there exist two
a
a
a
and
b
b
b
such that
a
2
−
b
2
a^2 - b^2
a
2
−
b
2
is divisible by
100
100
100
.
3
2
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min pos. integer in form (2^a - 2^b)/(2^c - 2^d) is odd (2019 Auckland MO J3)
Let
x
x
x
be the smallest positive integer that cannot be expressed in the form
2
a
−
2
b
2
c
−
2
d
\frac{2^a - 2^b}{2^c - 2^d}
2
c
−
2
d
2
a
−
2
b
, where
a
a
a
,
b
b
b
,
c
c
c
,
d
d
d
are non-negative integers. Prove that
x
x
x
is odd.
each 2 of n polygons have a common point (2019 Auckland MO S3)
There is a finite number of polygons in a plane and each two of them have a point in common. Prove that there exists a line which crosses every polygon.
1
2
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<ACB=? <BAC=20,<CAD=60, <ADB=<50, <BDC =10 (2019 Auckland MO J1)
Given a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
in which
∠
B
A
C
=
2
0
o
\angle BAC = 20^o
∠
B
A
C
=
2
0
o
,
∠
C
A
D
=
6
0
o
\angle CAD = 60^o
∠
C
A
D
=
6
0
o
,
∠
A
D
B
=
5
0
o
\angle ADB = 50^o
∠
A
D
B
=
5
0
o
, and
∠
B
D
C
=
1
0
o
\angle BDC = 10^o
∠
B
D
C
=
1
0
o
. Find
∠
A
C
B
\angle ACB
∠
A
CB
.
f(sin x) = sin(17x) if f(cos x) = cos(17x) (2019 Auckland MO S1)
Function
f
f
f
satisfies the equation
f
(
cos
x
)
=
cos
(
17
x
)
f(\cos x) = \cos (17x)
f
(
cos
x
)
=
cos
(
17
x
)
. Prove that it also satisfies the equation
f
(
sin
x
)
=
sin
(
17
x
)
f(\sin x) = \sin (17x)
f
(
sin
x
)
=
sin
(
17
x
)
.