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National and Regional Contests
New Zealand Contests
NZMOC Camp Selection Problems
2011 NZMOC Camp Selection Problems
2011 NZMOC Camp Selection Problems
Part of
NZMOC Camp Selection Problems
Subcontests
(6)
6
2
Hide problems
3 x 2^m + 1 = n^2
Find all pairs of non-negative integers
m
m
m
and
n
n
n
that satisfy
3
⋅
2
m
+
1
=
n
2
.
3 \cdot 2^m + 1 = n^2.
3
⋅
2
m
+
1
=
n
2
.
2011^2 lattice points (x, y), subset with >= 4x2011x\sqrt{2011} points
Consider the set
G
G
G
of
201
1
2
2011^2
201
1
2
points
(
x
,
y
)
(x, y)
(
x
,
y
)
in the plane where
x
x
x
and
y
y
y
are both integers between
1
1
1
and
2011
2011
2011
inclusive. Let
A
A
A
be any subset of
G
G
G
containing at least
4
×
2011
×
2011
4\times 2011\times \sqrt{2011}
4
×
2011
×
2011
points. Show that there are at least
201
1
2
2011^2
201
1
2
parallelograms whose vertices lie in
A
A
A
and all of whose diagonals meet at a single point.
3
2
Hide problems
16 competitors in a tournament
There are
16
16
16
competitors in a tournament, all of whom have different playing strengths and in any match between two players the stronger player always wins. Show that it is possible to find the strongest and second strongest players in
18
18
18
matches.
2-player game on a equilateral grid
Chris and Michael play a game on a board which is a rhombus of side length
n
n
n
(a positive integer) consisting of two equilateral triangles, each of which has been divided into equilateral triangles of side length
1
1
1
. Each has a single token, initially on the leftmost and rightmost squares of the board, called the “home” squares (the illustration shows the case
n
=
4
n = 4
n
=
4
). https://cdn.artofproblemsolving.com/attachments/e/b/8135203c22ce77c03c144850099ad1c575edb8.png A move consists of moving your token to an adjacent triangle (two triangles are adjacent only if they share a side). To win the game, you must either capture your opponent’s token (by moving to the triangle it occupies), or move on to your opponent’s home square. Supposing that Chris moves first, which, if any, player has a winning strategy?
1
2
Hide problems
numbers in a 3x3 square
A three by three square is filled with positive integers. Each row contains three different integers, the sums of each row are all the same, and the products of each row are all different. What is the smallest possible value for the sum of each row?
m! + n! = m^n
Find all pairs of positive integers
m
m
m
and
n
n
n
such that
m
!
+
n
!
=
m
n
.
m! + n! = m^n.
m
!
+
n
!
=
m
n
.
.
4
2
Hide problems
(m + 1)! + (n + 1)! = m^2n
Find all pairs of positive integers
m
m
m
and
n
n
n
such that
(
m
+
1
)
!
+
(
n
+
1
)
!
=
m
2
n
.
(m + 1)! + (n + 1)! = m^2n.
(
m
+
1
)!
+
(
n
+
1
)!
=
m
2
n
.
ABxAD=BPxDP+APxCP if <APB+<CP D=180^o, # (2011 NZOMC Camp Sel. S4)
Let a point
P
P
P
inside a parallelogram
A
B
C
D
ABCD
A
BC
D
be given such that
∠
A
P
B
+
∠
C
P
D
=
18
0
o
\angle APB +\angle CPD = 180^o
∠
A
PB
+
∠
CP
D
=
18
0
o
. Prove that
A
B
⋅
A
D
=
B
P
⋅
D
P
+
A
P
⋅
C
P
AB \cdot AD = BP \cdot DP + AP \cdot CP
A
B
⋅
A
D
=
BP
⋅
D
P
+
A
P
⋅
CP
.
5
2
Hide problems
computational with a square, collinear given (2011 NZOMC Camp Sel. J5)
Let a square
A
B
C
D
ABCD
A
BC
D
with sides of length
1
1
1
be given. A point
X
X
X
on
B
C
BC
BC
is at distance
d
d
d
from
C
C
C
, and a point
Y
Y
Y
on
C
D
CD
C
D
is at distance
d
d
d
from
C
C
C
. The extensions of:
A
B
AB
A
B
and
D
X
DX
D
X
meet at
P
P
P
,
A
D
AD
A
D
and
B
Y
BY
B
Y
meet at
Q
,
A
X
Q, AX
Q
,
A
X
and
D
C
DC
D
C
meet at
R
R
R
, and
A
Y
AY
A
Y
and
B
C
BC
BC
meet at
S
S
S
. If points
P
,
Q
,
R
P, Q, R
P
,
Q
,
R
and
S
S
S
are collinear, determine
d
d
d
.
[(a^2-b^2)^3 + (b^2- c^2)^3 + (c^2 - a2)^3][(a-b)^3 + (b-c)^3 + (c-a)^3]> 8sbc
Prove that for any three distinct positive real numbers
a
,
b
a, b
a
,
b
and
c
c
c
:
(
a
2
−
b
2
)
3
+
(
b
2
−
c
2
)
3
+
(
c
2
−
a
2
)
3
(
a
−
b
)
3
+
(
b
−
c
)
3
+
(
c
−
a
)
3
>
8
a
b
c
.
\frac{(a^2 - b^2)^3 + (b^2 - c^2)^3 + (c^2 - a^2)^3}{(a - b)^3 + (b - c)^3 + (c - a)^3}> 8abc.
(
a
−
b
)
3
+
(
b
−
c
)
3
+
(
c
−
a
)
3
(
a
2
−
b
2
)
3
+
(
b
2
−
c
2
)
3
+
(
c
2
−
a
2
)
3
>
8
ab
c
.
2
2
Hide problems
intersection of circles lies in 3rd side (2011 NZOMC Camp Sel. J2)
Let an acute angled triangle
A
B
C
ABC
A
BC
be given. Prove that the circles whose diameters are
A
B
AB
A
B
and
A
C
AC
A
C
have a point of intersection on
B
C
BC
BC
.
max angle, altitude tangent to circumcircle (2011 NZOMC Camp Sel. S2)
In triangle
A
B
C
ABC
A
BC
, the altitude from
B
B
B
is tangent to the circumcircle of
A
B
C
ABC
A
BC
. Prove that the largest angle of the triangle is between
9
0
o
90^o
9
0
o
and
13
5
o
135^o
13
5
o
. If the altitudes from both
B
B
B
and from
C
C
C
are tangent to the circumcircle, then what are the angles of the triangle?