MathDB
Problems
Contests
National and Regional Contests
New Zealand Contests
NZMOC Camp Selection Problems
2016 NZMOC Camp Selection Problems
2016 NZMOC Camp Selection Problems
Part of
NZMOC Camp Selection Problems
Subcontests
(8)
4
1
Hide problems
a karaka quadruple (p, a, b, c) exists with p + 2 = (a + b + c)/3
A quadruple
(
p
,
a
,
b
,
c
)
(p, a, b, c)
(
p
,
a
,
b
,
c
)
of positive integers is a karaka quadruple if
∙
\bullet
∙
p
p
p
is an odd prime number
∙
\bullet
∙
a
,
b
a, b
a
,
b
and
c
c
c
are distinct, and
∙
\bullet
∙
a
b
+
1
ab + 1
ab
+
1
,
b
c
+
1
bc + 1
b
c
+
1
and
c
a
+
1
ca + 1
c
a
+
1
are divisible by
p
p
p
. (a) Prove that for every karaka quadruple
(
p
,
a
,
b
,
c
)
(p, a, b, c)
(
p
,
a
,
b
,
c
)
we have
p
+
2
≤
a
+
b
+
c
3
p + 2 \le\frac{a + b + c}{3}
p
+
2
≤
3
a
+
b
+
c
. (b) Determine all numbers
p
p
p
for which a karaka quadruple
(
p
,
a
,
b
,
c
)
(p, a, b, c)
(
p
,
a
,
b
,
c
)
exists with
p
+
2
=
a
+
b
+
c
3
p + 2 =\frac{a + b + c}{3}
p
+
2
=
3
a
+
b
+
c
9
1
Hide problems
occasionally periodic n-tuple
An
n
n
n
-tuple
(
a
1
,
a
2
.
.
.
,
a
n
)
(a_1, a_2 . . . , a_n)
(
a
1
,
a
2
...
,
a
n
)
is occasionally periodic if there exist a non-negative integer
i
i
i
and a positive integer
p
p
p
satisfying
i
+
2
p
≤
n
i + 2p \le n
i
+
2
p
≤
n
and
a
i
+
j
=
a
i
+
j
+
p
a_{i+j} = a_{i+j+p}
a
i
+
j
=
a
i
+
j
+
p
for every
j
=
1
,
2
,
.
.
.
,
p
j = 1, 2, . . . , p
j
=
1
,
2
,
...
,
p
. Let
k
k
k
be a positive integer. Find the least positive integer
n
n
n
for which there exists an
n
n
n
-tuple
(
a
1
,
a
2
.
.
.
,
a
n
)
(a_1, a_2 . . . , a_n)
(
a
1
,
a
2
...
,
a
n
)
with elements from the set
{
1
,
2
,
.
.
.
,
k
}
\{1, 2, . . . , k\}
{
1
,
2
,
...
,
k
}
, which is not occasionally periodic but whose arbitrary extension
(
a
1
,
a
2
,
.
.
.
,
a
n
,
a
n
+
1
)
(a_1, a_2, . . . , a_n, a_{n+1})
(
a
1
,
a
2
,
...
,
a
n
,
a
n
+
1
)
is occasionally periodic for any
a
n
+
1
∈
{
1
,
2
,
.
.
.
,
k
}
a_{n+1} \in \{1, 2, . . . , k\}
a
n
+
1
∈
{
1
,
2
,
...
,
k
}
.
8
1
Hide problems
t/a_t = k if r / a_r = k + 1
Two positive integers
r
r
r
and
k
k
k
are given as is an infinite sequence of positive integers
a
1
≤
a
2
≤
a
3
≤
.
.
a_1 \le a_2 \le a_3 \le ..
a
1
≤
a
2
≤
a
3
≤
..
such that
r
a
r
=
k
+
1
\frac{r}{a_r}= k + 1
a
r
r
=
k
+
1
. Prove that there is a positive integer
t
t
t
such that
t
a
t
=
k
\frac{t}{a_t}= k
a
t
t
=
k
.
7
1
Hide problems
diophantine (x^2 + y^2)^n = (xy)^{2016} with no solutions
Find all positive integers
n
n
n
for which the equation
(
x
2
+
y
2
)
n
=
(
x
y
)
2016
(x^2 + y^2)^n = (xy)^{2016}
(
x
2
+
y
2
)
n
=
(
x
y
)
2016
has positive integer solutions.
5
1
Hide problems
Q(x) = (x + 1)P(x-1) -(x-1)P(x) is constant
Find all polynomials
P
(
x
)
P(x)
P
(
x
)
with real coefficients such that the polynomial
Q
(
x
)
=
(
x
+
1
)
P
(
x
−
1
)
−
(
x
−
1
)
P
(
x
)
Q(x) = (x + 1)P(x-1) -(x-1)P(x)
Q
(
x
)
=
(
x
+
1
)
P
(
x
−
1
)
−
(
x
−
1
)
P
(
x
)
is constant.
2
1
Hide problems
real numbers in a 5x5 table
We consider
5
×
5
5 \times 5
5
×
5
tables containing a real number in each of the
25
25
25
cells. The same number may occur in different cells, but no row or column contains five equal numbers. Such a table is balanced if the number in the middle cell of every row and column is the average of the numbers in that row or column. A cell is called small if the number in that cell is strictly smaller than the number in the cell in the very middle of the table. What is the least number of small cells that a balanced table can have?
1
1
Hide problems
equilateral triangle with vertices by same colors
Suppose that every point in the plane is coloured either black or white. Must there be an equilateral triangle such that all of its vertices are the same colour?
6
1
Hide problems
collinear wanted, altitudes , orthocenter, 2 circles with diameters
Altitudes
A
D
AD
A
D
and
B
E
BE
BE
of an acute triangle
A
B
C
ABC
A
BC
intersect at
H
H
H
. Let
P
≠
E
P \ne E
P
=
E
be the point of tangency of the circle with radius
H
E
HE
H
E
centred at
H
H
H
with its tangent line going through point
C
C
C
, and let
Q
≠
E
Q \ne E
Q
=
E
be the point of tangency of the circle with radius
B
E
BE
BE
centred at
B
B
B
with its tangent line going through
C
C
C
. Prove that the points
D
,
P
D, P
D
,
P
and
Q
Q
Q
are collinear.