MathDB
Problems
Contests
National and Regional Contests
New Zealand Contests
NZMOC Camp Selection Problems
2018 NZMOC Camp Selection Problems
2018 NZMOC Camp Selection Problems
Part of
NZMOC Camp Selection Problems
Subcontests
(10)
7
1
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N is a multiple of 3^{25}, but not a multiple of 3^{26}
Let
N
N
N
be the number of ways to colour each cell in a
2
×
50
2 \times 50
2
×
50
rectangle either red or blue such that each
2
×
2
2 \times 2
2
×
2
block contains at least one blue cell. Show that
N
N
N
is a multiple of
3
25
3^{25}
3
25
, but not a multiple of
3
26
3^{26}
3
26
5
1
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abc >= 4038^3 if 1/(a + 2019)+1/(b + 2019)+1/(c + 2019)=1/2019
Let
a
,
b
a, b
a
,
b
and
c
c
c
be positive real numbers satisfying
1
a
+
2019
+
1
b
+
2019
+
1
c
+
2019
=
1
2019
.
\frac{1}{a + 2019}+\frac{1}{b + 2019}+\frac{1}{c + 2019}=\frac{1}{2019}.
a
+
2019
1
+
b
+
2019
1
+
c
+
2019
1
=
2019
1
.
Prove that
a
b
c
≥
403
8
3
abc \ge 4038^3
ab
c
≥
403
8
3
.
10
1
Hide problems
f(x)f(y) = f(xy + 1) + f(x - y) - 2
Find all functions
f
:
R
→
R
f : R \to R
f
:
R
→
R
such that
f
(
x
)
f
(
y
)
=
f
(
x
y
+
1
)
+
f
(
x
−
y
)
−
2
f(x)f(y) = f(xy + 1) + f(x - y) - 2
f
(
x
)
f
(
y
)
=
f
(
x
y
+
1
)
+
f
(
x
−
y
)
−
2
for all
x
,
y
∈
R
x, y \in R
x
,
y
∈
R
.
9
1
Hide problems
x^n + y^n = p^k , if n > 1 is odd, and p is an odd prime, then n is a power of p
Let
x
,
y
,
p
,
n
,
k
x, y, p, n, k
x
,
y
,
p
,
n
,
k
be positive integers such that
x
n
+
y
n
=
p
k
.
x^n + y^n = p^k.
x
n
+
y
n
=
p
k
.
Prove that if
n
>
1
n > 1
n
>
1
is odd, and
p
p
p
is an odd prime, then
n
n
n
is a power of
p
p
p
.
3
1
Hide problems
2 in 8 points whose distnce <5 inside a 7x12 rectangle
Show that amongst any
8
8
8
points in the interior of a
7
×
12
7 \times 12
7
×
12
rectangle, there exists a pair whose distance is less than
5
5
5
.Note: The interior of a rectangle excludes points lying on the sides of the rectangle.
2
1
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diophantine a^2 + ab - b = 2018
Find all pairs of integers
(
a
,
b
)
(a, b)
(
a
,
b
)
such that
a
2
+
a
b
−
b
=
2018.
a^2 + ab - b = 2018.
a
2
+
ab
−
b
=
2018.
1
1
Hide problems
(a - b)(a - c)(a - d)(b - c)(b - d)(c - d) is multiple of 12
Suppose that
a
,
b
,
c
a, b, c
a
,
b
,
c
and
d
d
d
are four different integers. Explain why
(
a
−
b
)
(
a
−
c
)
(
a
−
d
)
(
b
−
c
)
(
b
−
d
)
(
c
−
d
)
(a - b)(a - c)(a - d)(b - c)(b -d)(c - d)
(
a
−
b
)
(
a
−
c
)
(
a
−
d
)
(
b
−
c
)
(
b
−
d
)
(
c
−
d
)
must be a multiple of
12
12
12
.
8
1
Hide problems
AC, BD and \lambd concurrent or //, 4 circles related
Let
λ
\lambda
λ
be a line and let
M
,
N
M, N
M
,
N
be two points on
λ
\lambda
λ
. Circles
α
\alpha
α
and
β
\beta
β
centred at
A
A
A
and
B
B
B
respectively are both tangent to
λ
\lambda
λ
at
M
M
M
, with
A
A
A
and
B
B
B
being on opposite sides of
λ
\lambda
λ
. Circles
γ
\gamma
γ
and
δ
\delta
δ
centred at
C
C
C
and
D
D
D
respectively are both tangent to
λ
\lambda
λ
at
N
N
N
, with
C
C
C
and
D
D
D
being on opposite sides of
λ
\lambda
λ
. Moreover
A
A
A
and
C
C
C
are on the same side of
λ
\lambda
λ
. Prove that if there exists a circle tangent to all circles
α
,
β
,
γ
,
δ
\alpha, \beta, \gamma, \delta
α
,
β
,
γ
,
δ
containing all of them in its interior, then the lines
A
C
,
B
D
AC, BD
A
C
,
B
D
and
λ
\lambda
λ
are either concurrent or parallel.
4
1
Hide problems
<PXY=90^o wanted, <CPA=90^o, <CBP=<CAP, midpoints (2018 NZOMC Camp Sel. p4)
Let
P
P
P
be a point inside triangle
A
B
C
ABC
A
BC
such that
∠
C
P
A
=
9
0
o
\angle CPA = 90^o
∠
CP
A
=
9
0
o
and
∠
C
B
P
=
∠
C
A
P
\angle CBP = \angle CAP
∠
CBP
=
∠
C
A
P
. Prove that
∠
P
X
Y
=
9
0
o
\angle P XY = 90^o
∠
PX
Y
=
9
0
o
, where
X
X
X
and
Y
Y
Y
are the midpoints of
A
B
AB
A
B
and
A
C
AC
A
C
respectively.
6
1
Hide problems
Bound on pentagon formed from cube
The intersection of a cube and a plane is a pentagon. Prove the length of at least oneside of the pentagon differs from 1 metre by at least 20 centimetres.