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Problems
Contests
National and Regional Contests
New Zealand Contests
NZMOC Camp Selection Problems
2010 NZMOC Camp Selection Problems
2010 NZMOC Camp Selection Problems
Part of
NZMOC Camp Selection Problems
Subcontests
(6)
6
2
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each person knew exactly 22 others at a strange party
At a strange party, each person knew exactly
22
22
22
others. For any pair of people
X
X
X
and
Y
Y
Y
who knew one another, there was no other person at the party that they both knew. For any pair of people
X
X
X
and
Y
Y
Y
who did not know each other, there were exactly six other people that they both knew. How many people were at the party?
3 of 8 integers from {1, 2, . . . , 16, 17} such that a_i - a_j = k
Suppose
a
1
,
a
2
,
.
.
.
,
a
8
a_1, a_2, . . . , a_8
a
1
,
a
2
,
...
,
a
8
are eight distinct integers from
{
1
,
2
,
.
.
.
,
16
,
17
}
\{1, 2, . . . , 16, 17\}
{
1
,
2
,
...
,
16
,
17
}
. Show that there is an integer
k
>
0
k > 0
k
>
0
such that there are at least three different (not necessarily disjoint) pairs
(
i
,
j
)
(i, j)
(
i
,
j
)
such that
a
i
−
a
j
=
k
a_i - a_j = k
a
i
−
a
j
=
k
. Also find a set of seven distinct integers from
{
1
,
2
,
.
.
.
,
16
,
17
}
\{1, 2, . . . , 16, 17\}
{
1
,
2
,
...
,
16
,
17
}
such that there is no integer
k
>
0
k > 0
k
>
0
with that property.
3
2
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n^5 + n + 1 is prime
Find all positive integers n such that
n
5
+
n
+
1
n^5 + n + 1
n
5
+
n
+
1
is prime.
diophantine x^3 + y^3 - 3xy = p -1
Let
p
p
p
be a prime number. Find all pairs
(
x
,
y
)
(x, y)
(
x
,
y
)
of positive integers such that
x
3
+
y
3
−
3
x
y
=
p
−
1
x^3 + y^3 - 3xy = p -1
x
3
+
y
3
−
3
x
y
=
p
−
1
.
1
2
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numbers 1 to 8 in a 8x8 board
We number both the rows and the columns of an
8
×
8
8 \times 8
8
×
8
chessboard with the numbers
1
1
1
to
8
8
8
. Some grains of rice are placed on each square, in such a way that the number of grains on each square is equal to the product of the row and column numbers of the square. How many grains of rice are there on the entire chessboard?
x_{2010} =? if x_{n+1} =\frac{4 \max{x_n, 4}}{x_{n-1}}
For any two positive real numbers
x
0
>
0
x_0 > 0
x
0
>
0
,
x
1
>
0
x_1 > 0
x
1
>
0
, a sequence of real numbers is defined recursively by
x
n
+
1
=
4
max
{
x
n
,
4
}
x
n
−
1
x_{n+1} =\frac{4 \max\{x_n, 4\}}{x_{n-1}}
x
n
+
1
=
x
n
−
1
4
max
{
x
n
,
4
}
for
n
≥
1
n \ge 1
n
≥
1
. Find
x
2010
x_{2010}
x
2010
.
5
2
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ratio BC:AD wanted, AB=CE, BE= AD, <AED=<BAD (2010 NZOMC Camp Sel. J5)
The diagonals of quadrilateral
A
B
C
D
ABCD
A
BC
D
intersect in point
E
E
E
. Given that
∣
A
B
∣
=
∣
C
E
∣
|AB| =|CE|
∣
A
B
∣
=
∣
CE
∣
,
∣
B
E
∣
=
∣
A
D
∣
|BE| = |AD|
∣
BE
∣
=
∣
A
D
∣
, and
∠
A
E
D
=
∠
B
A
D
\angle AED = \angle BAD
∠
A
E
D
=
∠
B
A
D
, determine the ratio
∣
B
C
∣
:
∣
A
D
∣
|BC|:|AD|
∣
BC
∣
:
∣
A
D
∣
.
\sqrt{\frac{9n - 1}{n + 7}} is rational
Determine the values of the positive integer
n
n
n
for which
A
=
9
n
−
1
n
+
7
A =\sqrt{\frac{9n - 1}{n + 7}}
A
=
n
+
7
9
n
−
1
is rational.
2
2
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square inscribed in sector of a circle (2010 NZOMC Camp Sel. J2)
A
B
AB
A
B
is a chord of length
6
6
6
in a circle of radius
5
5
5
and centre
O
O
O
. A square is inscribed in the sector
O
A
B
OAB
O
A
B
with two vertices on the circumference and two sides parallel to
A
B
AB
A
B
. Find the area of the square.
pentagon has 4 triangles of same area (2010 NZOMC Camp Sel. S2)
In a convex pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
the areas of the triangles
A
B
C
,
A
B
D
,
A
C
D
ABC, ABD, ACD
A
BC
,
A
B
D
,
A
C
D
and
A
D
E
ADE
A
D
E
are all equal to the same value x. What is the area of the triangle
B
C
E
BCE
BCE
?
4
2
Hide problems
1/a+1/b+2007/\lcm (a, b)= 1/\gcd (a, b), 1/a+1/b+2010/\lcm (a, b)= 1/\gcd (a, b)
Find all positive integer solutions
(
a
,
b
)
(a, b)
(
a
,
b
)
to the equation
1
a
+
1
b
+
n
l
c
m
(
a
,
b
)
=
1
g
c
d
(
a
,
b
)
\frac{1}{a}+\frac{1}{b}+ \frac{n}{lcm(a,b)}=\frac{1}{gcd(a, b)}
a
1
+
b
1
+
l
c
m
(
a
,
b
)
n
=
g
c
d
(
a
,
b
)
1
for (i)
n
=
2007
n = 2007
n
=
2007
; (ii)
n
=
2010
n = 2010
n
=
2010
.
1/QD = 1/QB + 1/QC, inscribed equilateral (2012 UNSW S5 Australia)
A line drawn from the vertex
A
A
A
of the equilateral triangle
A
B
C
ABC
A
BC
meets the side
B
C
BC
BC
at
D
D
D
and the circumcircle of the triangle at point
Q
Q
Q
. Prove that
1
Q
D
=
1
Q
B
+
1
Q
C
\frac{1}{QD} = \frac{1}{QB} + \frac{1}{QC}
Q
D
1
=
QB
1
+
QC
1
.