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Problems
Contests
National and Regional Contests
New Zealand Contests
New Zealand MO
2019 New Zealand MO
2019 New Zealand MO
Part of
New Zealand MO
Subcontests
(8)
8
1
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\prod_{i=1}^{X_n} \frac{x_i}{s + 1 - x_i} + \prod_{i=1}^{Y_n} (1 - x_i) \le 1
Suppose that
x
1
,
x
2
,
x
3
,
.
.
.
x
n
x_1, x_2, x_3, . . . x_n
x
1
,
x
2
,
x
3
,
...
x
n
are real numbers between
0
0
0
and
1
1
1
with sum
s
s
s
. Prove that
∏
i
=
1
n
x
i
s
+
1
−
x
i
+
∏
i
=
1
n
(
1
−
x
i
)
≤
1.
\prod_{i=1}^{n} \frac{x_i}{s + 1 - x_i} + \prod_{i=1}^{n} (1 - x_i) \le 1.
i
=
1
∏
n
s
+
1
−
x
i
x
i
+
i
=
1
∏
n
(
1
−
x
i
)
≤
1.
6
1
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max of m(U) /|U|, U subset of V , set of vertices of regular 21-gon
Let
V
V
V
be the set of vertices of a regular
21
21
21
-gon. Given a non-empty subset
U
U
U
of
V
V
V
, let
m
(
U
)
m(U)
m
(
U
)
be the number of distinct lengths that occur between two distinct vertices in
U
U
U
. What is the maximum value of
m
(
U
)
∣
U
∣
\frac{m(U)}{|U|}
∣
U
∣
m
(
U
)
as
U
U
U
varies over all non-empty subsets of
V
V
V
?
5
2
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n^4 - n^3 + 3n^2 + 5 is a perfect square
Find all positive integers
n
n
n
such that
n
4
−
n
3
+
3
n
2
+
5
n^4 - n^3 + 3n^2 + 5
n
4
−
n
3
+
3
n
2
+
5
is a perfect square.
equilateral triangle is partitioned into smaller equilateral triangular pieces
An equilateral triangle is partitioned into smaller equilateral triangular pieces. Prove that two of the pieces are the same size.
4
2
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122^n - 102^n - 21^n is one less than a multiple of 2020
Show that the number
12
2
n
−
10
2
n
−
2
1
n
122^n - 102^n - 21^n
12
2
n
−
10
2
n
−
2
1
n
is always one less than a multiple of
2020
2020
2020
, for any positive integer
n
n
n
.
n2^k -7 is a perfect square
Show that for all positive integers
k
k
k
, there exists a positive integer n such that
n
2
k
−
7
n2^k -7
n
2
k
−
7
is a perfect square.
2
2
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(x^2 + 3x + 1)^{x^2-x-6} = 1
Find all real solutions to the equation
(
x
2
+
3
x
+
1
)
x
2
−
x
−
6
=
1
(x^2 + 3x + 1)^{x^2-x-6} = 1
(
x
2
+
3
x
+
1
)
x
2
−
x
−
6
=
1
.
angle bisector wanted, 2 right angles given, intersections of diagonals of ABCD
Let
X
X
X
be the intersection of the diagonals
A
C
AC
A
C
and
B
D
BD
B
D
of convex quadrilateral
A
B
C
D
ABCD
A
BC
D
. Let
P
P
P
be the intersection of lines
A
B
AB
A
B
and
C
D
CD
C
D
, and let
Q
Q
Q
be the intersection of lines
P
X
PX
PX
and
A
D
AD
A
D
. Suppose that
∠
A
B
X
=
∠
X
C
D
=
9
0
o
\angle ABX = \angle XCD = 90^o
∠
A
BX
=
∠
XC
D
=
9
0
o
. Prove that
Q
P
QP
QP
is the angle bisector of
∠
B
Q
C
\angle BQC
∠
BQC
.
7
1
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collinear wanted, convex ABCDEF , 2 congruent rectangles
Let
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
be a convex hexagon containing a point
P
P
P
in its interior such that
P
A
B
C
PABC
P
A
BC
and
P
D
E
F
PDEF
P
D
EF
are congruent rectangles with
P
A
=
B
C
=
P
D
=
E
F
PA = BC = P D = EF
P
A
=
BC
=
P
D
=
EF
(and
A
B
=
P
C
=
D
E
=
P
F
AB = PC = DE = PF
A
B
=
PC
=
D
E
=
PF
). Let
ℓ
\ell
ℓ
be the line through the midpoint of
A
F
AF
A
F
and the circumcentre of
P
C
D
PCD
PC
D
. Prove that
ℓ
\ell
ℓ
passes through
P
P
P
.
3
2
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computational from New Zealand, segments (2019 NZMO 1.3 )
In triangle
A
B
C
ABC
A
BC
, points
D
D
D
and
E
E
E
lie on the interior of segments
A
B
AB
A
B
and
A
C
AC
A
C
, respectively,such that
A
D
=
1
AD = 1
A
D
=
1
,
D
B
=
2
DB = 2
D
B
=
2
,
B
C
=
4
BC = 4
BC
=
4
,
C
E
=
2
CE = 2
CE
=
2
and
E
A
=
3
EA = 3
E
A
=
3
. Let
D
E
DE
D
E
intersect
B
C
BC
BC
at
F
F
F
. Determine the length of
C
F
CF
CF
.
a^a + b^b + c^c >= 3 if a,b,c>0 with a + b + c = 3
Let
a
,
b
a, b
a
,
b
and
c
c
c
be positive real numbers such that
a
+
b
+
c
=
3
a + b + c = 3
a
+
b
+
c
=
3
. Prove that
a
a
+
b
b
+
c
c
≥
3
a^a + b^b + c^c \ge 3
a
a
+
b
b
+
c
c
≥
3
1
2
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numbers < 2019 divisible by either 18 or 21, but not both
How many positive integers less than
2019
2019
2019
are divisible by either
18
18
18
or
21
21
21
, but not both?
Combinatorics problem
A positive integer is called sparkly if it has exactly 9 digits, and for any n between 1 and 9 (inclusive), the nth digit is a positive multiple of n. How many positive integers are sparkly?