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Problems
Contests
National and Regional Contests
New Zealand Contests
NZMOC Camp Selection Problems
2013 NZMOC Camp Selection Problems
2013 NZMOC Camp Selection Problems
Part of
NZMOC Camp Selection Problems
Subcontests
(12)
10
1
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((x + y)^2- 6)((x-y)^2 + 8)/(x-y)^2 >= C
Find the largest possible real number
C
C
C
such that for all pairs
(
x
,
y
)
(x, y)
(
x
,
y
)
of real numbers with
x
≠
y
x \ne y
x
=
y
and
x
y
=
2
xy = 2
x
y
=
2
,
(
(
x
+
y
)
2
−
6
)
)
(
x
−
y
)
2
+
8
)
)
(
x
−
y
)
2
≥
C
.
\frac{((x + y)^2- 6))(x-y)^2 + 8))}{(x-y)^2} \ge C.
(
x
−
y
)
2
((
x
+
y
)
2
−
6
))
(
x
−
y
)
2
+
8
))
≥
C
.
Also determine for which pairs
(
x
,
y
)
(x, y)
(
x
,
y
)
equality holds.
12
1
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p(m-1) < p(m) < p(m + 1), p(n) is the largest prime divisor of n
For a positive integer
n
n
n
, let
p
(
n
)
p(n)
p
(
n
)
denote the largest prime divisor of
n
n
n
. Show that there exist infinitely many positive integers m such that
p
(
m
−
1
)
<
p
(
m
)
<
p
(
m
+
1
)
p(m-1) < p(m) < p(m + 1)
p
(
m
−
1
)
<
p
(
m
)
<
p
(
m
+
1
)
.
11
1
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171 binary sequences each of length 12
Show that we cannot find
171
171
171
binary sequences (sequences of
0
0
0
’s and
1
1
1
’s), each of length
12
12
12
such that any two of them differ in at least four positions.
8
1
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c = a + b/a -1/b is perfect square.
Suppose that
a
a
a
and
b
b
b
are positive integers such that
c
=
a
+
b
a
−
1
b
c = a +\frac{b}{a} -\frac{1}{b}
c
=
a
+
a
b
−
b
1
is an integer. Prove that
c
c
c
is a perfect square.
5
1
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f(xy) = f(x)f(y), f(30) = 1, for any n whose last digit is 7, f(n) = 1
Consider functions
f
f
f
from the whole numbers (non-negative integers) to the whole numbers that have the following properties:
∙
\bullet
∙
For all
x
x
x
and
y
y
y
,
f
(
x
y
)
=
f
(
x
)
f
(
y
)
f(xy) = f(x)f(y)
f
(
x
y
)
=
f
(
x
)
f
(
y
)
,
∙
\bullet
∙
f
(
30
)
=
1
f(30) = 1
f
(
30
)
=
1
, and
∙
\bullet
∙
for any
n
n
n
whose last digit is
7
7
7
,
f
(
n
)
=
1
f(n) = 1
f
(
n
)
=
1
. Obviously, the function whose value at
n
n
n
is
1
1
1
for all
n
n
n
is one such function. Are there any others? If not, why not, and if so, what are they?
7
1
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max no of inversions in a sequence of positive integers whose sum is 2014
In a sequence of positive integers an inversion is a pair of positions such that the element in the position to the left is greater than the element in the position to the right. For instance the sequence
2
,
5
,
3
,
1
,
3
2,5,3,1,3
2
,
5
,
3
,
1
,
3
has five inversions - between the first and fourth positions, the second and all later positions, and between the third and fourth positions. What is the largest possible number of inversions in a sequence of positive integers whose sum is
2014
2014
2014
?
3
1
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n distinct positive integers, the sum of whose reciprocals is equal to 1
Prove that for any positive integer
n
>
2
n > 2
n
>
2
we can find
n
n
n
distinct positive integers, the sum of whose reciprocals is equal to
1
1
1
.
2
1
Hide problems
primes that can be written both as a sum and as a difference of two primes
Find all primes that can be written both as a sum and as a difference of two primes (note that
1
1
1
is not a prime).
1
1
Hide problems
5 weights, together with a balance that compares the weight of 2 things
You have a set of five weights, together with a balance that allows you to compare the weight of two things. The weights are known to be
10
10
10
,
20
20
20
,
30
30
30
,
40
40
40
and
50
50
50
grams, but are otherwise identical except for their labels. The
10
10
10
and
50
50
50
gram weights are clearly labelled, but the labels have been erased on the remaining weights. Using the balance exactly once, is it possible to determine what one of the three unlabelled weights is? If so, explain how, and if not, explain why not.
9
1
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# wanted, 3 right isosceles on sides of ABC (2013 NZOMC Camp Sel. p9)
Let
A
B
C
ABC
A
BC
be a triangle with
∠
C
A
B
>
4
5
o
\angle CAB > 45^o
∠
C
A
B
>
4
5
o
and
∠
C
B
A
>
4
5
o
\angle CBA > 45^o
∠
CB
A
>
4
5
o
. Construct an isosceles right angled triangle
R
A
B
RAB
R
A
B
with
A
B
AB
A
B
as its hypotenuse and
R
R
R
inside
A
B
C
ABC
A
BC
. Also construct isosceles right angled triangles
A
C
Q
ACQ
A
CQ
and
B
C
P
BCP
BCP
having
A
C
AC
A
C
and
B
C
BC
BC
respectively as their hypotenuses and lying entirely outside
A
B
C
ABC
A
BC
. Show that
C
Q
R
P
CQRP
CQRP
is a parallelogram.
6
1
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criterion for bicentric ABCD to be a square (2013 NZOMC Camp Sel. p6)
A
B
C
D
ABCD
A
BC
D
is a quadrilateral having both an inscribed circle (one tangent to all four sides) with center
I
,
I,
I
,
and a circumscribed circle with center
O
O
O
. Let
S
S
S
be the point of intersection of the diagonals of
A
B
C
D
ABCD
A
BC
D
. Show that if any two of
S
,
I
S, I
S
,
I
and
O
O
O
coincide, then
A
B
C
D
ABCD
A
BC
D
is a square (and hence all three coincide).
4
1
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ratio of sides of cubes, octahedron (2013 NZOMC Camp Sel. p4)
Let
C
C
C
be a cube. By connecting the centres of the faces of
C
C
C
with lines we form an octahedron
O
O
O
. By connecting the centers of each face of
O
O
O
with lines we get a smaller cube
C
′
C'
C
′
. What is the ratio between the side length of
C
C
C
and the side length of
C
′
C'
C
′
?