MathDB
Problems
Contests
National and Regional Contests
New Zealand Contests
NZMOC Camp Selection Problems
2017 NZMOC Camp Selection Problems
2017 NZMOC Camp Selection Problems
Part of
NZMOC Camp Selection Problems
Subcontests
(9)
8
1
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a + b + c = 51, abc = 4000, 0 < a <= 10 and c > 25
Find all possible real values for
a
,
b
a, b
a
,
b
and
c
c
c
such that (a)
a
+
b
+
c
=
51
a + b + c = 51
a
+
b
+
c
=
51
, (b)
a
b
c
=
4000
abc = 4000
ab
c
=
4000
, (c)
0
<
a
≤
10
0 < a \le 10
0
<
a
≤
10
and
c
≥
25
c \ge 25
c
≥
25
.
9
1
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a student who is friends with everyone else in a class of n students
Let
k
k
k
and
n
n
n
be positive integers, with
k
≤
n
k \le n
k
≤
n
. A certain class has n students, and among any
k
k
k
of them there is always one that is friends with the other
k
−
1
k- 1
k
−
1
. Find all values of
k
k
k
and
n
n
n
for which there must necessarily be a student who is friends with everyone else in the class.
7
1
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ac + bd is composite if a^4 + b^4 = c^4 + d^4 = e^5
Let
a
,
b
,
c
,
d
,
e
a, b, c, d, e
a
,
b
,
c
,
d
,
e
be distinct positive integers such that
a
4
+
b
4
=
c
4
+
d
4
=
e
5
.
a^4 + b^4 = c^4 + d^4 = e^5.
a
4
+
b
4
=
c
4
+
d
4
=
e
5
.
Show that
a
c
+
b
d
ac + bd
a
c
+
b
d
is composite.
5
1
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exactly 225 rectangles with odd sides in m x n grid
Find all pairs
(
m
,
n
)
(m, n)
(
m
,
n
)
of positive integers such that the
m
×
n
m \times n
m
×
n
grid contains exactly
225
225
225
rectangles whose side lengths are odd and whose edges lie on the lines of the grid.
4
1
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Ross wants to play solitaire with his deck of n playing cards
Ross wants to play solitaire with his deck of
n
n
n
playing cards, but he’s discovered that the deck is “boxed”: some cards are face up, and others are face down. He wants to turn them all face down again, by repeatedly choosing a block of consecutive cards, removing the block from the deck, turning it over, and replacing it back in the deck at the same point. What is the smallest number of such steps Ross needs in order to guarantee that he can turn all the cards face down again, regardless of how they start out?
3
1
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16p + 1 is a perfect cube
Find all prime numbers
p
p
p
such that
16
p
+
1
16p + 1
16
p
+
1
is a perfect cube.
1
1
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a < b < c < d < e with sums of pairs 32, 36 ,37 , 48 ,$51
Alice has five real numbers
a
<
b
<
c
<
d
<
e
a < b < c < d < e
a
<
b
<
c
<
d
<
e
. She takes the sum of each pair of numbers and writes down the ten sums. The three smallest sums are
32
32
32
,
36
36
36
and
37
37
37
, while the two largest sums are
48
48
48
and
51
51
51
. Determine
e
e
e
.
6
1
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concyclic wanted, 2 circumcircles related
Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral. The circumcircle of the triangle
A
B
C
ABC
A
BC
intersects the sides
C
D
CD
C
D
and
D
A
DA
D
A
in the points
P
P
P
and
Q
Q
Q
respectively, while the circumcircle of
C
D
A
CDA
C
D
A
intersects the sides
A
B
AB
A
B
and
B
C
BC
BC
in the points
R
R
R
and
S
S
S
. The lines
B
P
BP
BP
and
B
Q
BQ
BQ
intersect the line
R
S
RS
RS
in the points
M
M
M
and
N
N
N
respectively. Prove that the points
M
,
N
,
P
M, N, P
M
,
N
,
P
and
Q
Q
Q
lie on the same circle.
2
1
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isosceles wanted, AB = AH, CG = CB, ABCD # (2017 NZOMC Camp Sel. p2)
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram with an acute angle at
A
A
A
. Let
G
G
G
be the point on the line
A
B
AB
A
B
, distinct from
B
B
B
, such that
C
G
=
C
B
CG = CB
CG
=
CB
. Let
H
H
H
be the point on the line
B
C
BC
BC
, distinct from
B
B
B
, such that
A
B
=
A
H
AB = AH
A
B
=
A
H
. Prove that triangle
D
G
H
DGH
D
G
H
is isosceles.