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Contests
National and Regional Contests
Canada Contests
Canadian Open Math Challenge
2017 Canadian Open Math Challenge
2017 Canadian Open Math Challenge
Part of
Canadian Open Math Challenge
Subcontests
(12)
C4
1
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2017 COMC C4
Source: 2017 Canadian Open Math Challenge, Problem C4 —-- Let n be a positive integer and
S
n
=
{
1
,
2
,
.
.
.
,
2
n
−
1
,
2
n
}
S_n = \{1, 2, . . . , 2n - 1, 2n\}
S
n
=
{
1
,
2
,
...
,
2
n
−
1
,
2
n
}
. A perfect pairing of
S
n
S_n
S
n
is defined to be a partitioning of the
2
n
2n
2
n
numbers into
n
n
n
pairs, such that the sum of the two numbers in each pair is a perfect square. For example, if
n
=
4
n = 4
n
=
4
, then a perfect pairing of
S
4
S_4
S
4
is
(
1
,
8
)
,
(
2
,
7
)
,
(
3
,
6
)
,
(
4
,
5
)
(1, 8),(2, 7),(3, 6),(4, 5)
(
1
,
8
)
,
(
2
,
7
)
,
(
3
,
6
)
,
(
4
,
5
)
. It is not necessary for each pair to sum to the same perfect square. (a) Show that
S
8
S_8
S
8
has at least one perfect pairing. (b) Show that
S
5
S_5
S
5
does not have any perfect pairings. (c) Prove or disprove: there exists a positive integer
n
n
n
for which
S
n
S_n
S
n
has at least
2017
2017
2017
different perfect pairings. (Two pairings that are comprised of the same pairs written in a different order are considered the same pairing.)
C3
1
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2017 COMC C3
Source: 2017 Canadian Open Math Challenge, Problem C3 —-- Let
X
Y
Z
XYZ
X
Y
Z
be an acute-angled triangle. Let
s
s
s
be the side-length of the square which has two adjacent vertices on side
Y
Z
YZ
Y
Z
, one vertex on side
X
Y
XY
X
Y
and one vertex on side
X
Z
XZ
XZ
. Let
h
h
h
be the distance from
X
X
X
to the side
Y
Z
YZ
Y
Z
and let
b
b
b
be the distance from
Y
Y
Y
to
Z
Z
Z
.[asy] pair S, D; D = 1.27; S = 2.55; draw((2, 4)--(0, 0)--(7, 0)--cycle); draw((1.27,0)--(1.27+2.55,0)--(1.27+2.55,2.55)--(1.27,2.55)--cycle); label("
X
X
X
",(2,4),N); label("
Y
Y
Y
",(0,0),W); label("
Z
Z
Z
",(7,0),E); [/asy](a) If the vertices have coordinates
X
=
(
2
,
4
)
X = (2, 4)
X
=
(
2
,
4
)
,
Y
=
(
0
,
0
)
Y = (0, 0)
Y
=
(
0
,
0
)
and
Z
=
(
4
,
0
)
Z = (4, 0)
Z
=
(
4
,
0
)
, find
b
b
b
,
h
h
h
and
s
s
s
. (b) Given the height
h
=
3
h = 3
h
=
3
and
s
=
2
s = 2
s
=
2
, find the base
b
b
b
. (c) If the area of the square is
2017
2017
2017
, determine the minimum area of triangle
X
Y
Z
XYZ
X
Y
Z
.
C2
1
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2017 COMC C2
Source: 2017 Canadian Open Math Challenge, Problem C2 —-- A function
f
(
x
)
f(x)
f
(
x
)
is periodic with period
T
>
0
T > 0
T
>
0
if
f
(
x
+
T
)
=
f
(
x
)
f(x + T) = f(x)
f
(
x
+
T
)
=
f
(
x
)
for all
x
x
x
. The smallest such number
T
T
T
is called the least period. For example, the functions
sin
(
x
)
\sin(x)
sin
(
x
)
and
cos
(
x
)
\cos(x)
cos
(
x
)
are periodic with least period
2
π
2\pi
2
π
.
\qquad
(a) Let a function
g
(
x
)
g(x)
g
(
x
)
be periodic with the least period
T
=
π
T = \pi
T
=
π
. Determine the least period of
g
(
x
/
3
)
g(x/3)
g
(
x
/3
)
.
\qquad
(b) Determine the least period of
H
(
x
)
=
s
i
n
(
8
x
)
+
c
o
s
(
4
x
)
H(x) = sin(8x) + cos(4x)
H
(
x
)
=
s
in
(
8
x
)
+
cos
(
4
x
)
\qquad
(c) Determine the least periods of each of
G
(
x
)
=
s
i
n
(
c
o
s
(
x
)
)
G(x) = sin(cos(x))
G
(
x
)
=
s
in
(
cos
(
x
))
and
F
(
x
)
=
c
o
s
(
s
i
n
(
x
)
)
F(x) = cos(sin(x))
F
(
x
)
=
cos
(
s
in
(
x
))
.
C1
1
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2017 COMC C1
Source: 2017 Canadian Open Math Challenge, Problem C1 —-- For a positive integer
n
n
n
, we define function
P
(
n
)
P(n)
P
(
n
)
to be the sum of the digits of
n
n
n
plus the number of digits of
n
n
n
. For example,
P
(
45
)
=
4
+
5
+
2
=
11
P(45) = 4 + 5 + 2 = 11
P
(
45
)
=
4
+
5
+
2
=
11
. (Note that the first digit of
n
n
n
reading from left to right, cannot be
0
0
0
).
\qquad
(a) Determine
P
(
2017
)
P(2017)
P
(
2017
)
.
\qquad
(b) Determine all numbers
n
n
n
such that
P
(
n
)
=
4
P(n) = 4
P
(
n
)
=
4
.
\qquad
(c) Determine with an explanation whether there exists a number
n
n
n
for which
P
(
n
)
−
P
(
n
+
1
)
>
50
P(n) - P(n + 1) > 50
P
(
n
)
−
P
(
n
+
1
)
>
50
.
B4
1
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2017 COMC B4
Source: 2017 Canadian Open Math Challenge, Problem B4 —-- Numbers
a
a
a
,
b
b
b
and
c
c
c
form an arithmetic sequence if
b
−
a
=
c
−
b
b - a = c - b
b
−
a
=
c
−
b
. Let
a
a
a
,
b
b
b
,
c
c
c
be positive integers forming an arithmetic sequence with
a
<
b
<
c
a < b < c
a
<
b
<
c
. Let
f
(
x
)
=
a
x
2
+
b
x
+
c
f(x) = ax2 + bx + c
f
(
x
)
=
a
x
2
+
b
x
+
c
. Two distinct real numbers
r
r
r
and
s
s
s
satisfy
f
(
r
)
=
s
f(r) = s
f
(
r
)
=
s
and
f
(
s
)
=
r
f(s) = r
f
(
s
)
=
r
. If
r
s
=
2017
rs = 2017
rs
=
2017
, determine the smallest possible value of
a
a
a
.
B3
1
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2017 COMC B3
Source: 2017 Canadian Open Math Challenge, Problem B3 —-- Regular decagon (10-sided polygon)
A
B
C
D
E
F
G
H
I
J
ABCDEFGHIJ
A
BC
D
EFG
H
I
J
has area
2017
2017
2017
square units. Determine the area (in square units) of the rectangle
C
D
H
I
CDHI
C
DH
I
.[asy] pair A,B,C,D,E,F,G,H,I,J; A = (0.809016994375, 0.587785252292); B = (0.309016994375, 0.951056516295); C = (-0.309016994375, 0.951056516295); D = (-0.809016994375, 0.587785252292); E = (-1, 0); F = (-0.809016994375, -0.587785252292); G = (-0.309016994375, -0.951056516295); H = (0.309016994375, -0.951056516295); I = (0.809016994375, -0.587785252292); J = (1, 0); label("
A
A
A
",A,NE); label("
B
B
B
",B,NE); label("
C
C
C
",C,NW); label("
D
D
D
",D,NW); label("
E
E
E
",E,E); label("
F
F
F
",F,E); label("
G
G
G
",G,SW); label("
H
H
H
",H,S); label("
I
I
I
",I,SE); label("
J
J
J
",J,2*dir(0)); fill(C--D--H--I--cycle,mediumgrey); draw(polygon(10)); [/asy]
B2
1
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2017 COMC B2
Source: 2017 Canadian Open Math Challenge, Problem B2 —-- There are twenty people in a room, with
a
a
a
men and
b
b
b
women. Each pair of men shakes hands, and each pair of women shakes hands, but there are no handshakes between a man and a woman. The total number of handshakes is
106
106
106
. Determine the value of
a
⋅
b
a \cdot b
a
⋅
b
.
B1
1
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2017 COMC B1
Source: 2017 Canadian Open Math Challenge, Problem B1 —-- Andrew and Beatrice practice their free throws in basketball. One day, they attempted a total of
105
105
105
free throws between them, with each person taking at least one free throw. If Andrew made exactly
1
/
3
1/3
1/3
of his free throw attempts and Beatrice made exactly
3
/
5
3/5
3/5
of her free throw attempts, what is the highest number of successful free throws they could have made between them?
A4
1
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2017 COMC A4
Source: 2017 Canadian Open Math Challenge, Problem A4 —-- Three positive integers
a
a
a
,
b
b
b
,
c
c
c
satisfy
4
a
⋅
5
b
⋅
6
c
=
8
8
⋅
9
9
⋅
1
0
10
.
4^a \cdot 5^b \cdot 6^c = 8^8 \cdot 9^9 \cdot 10^{10}.
4
a
⋅
5
b
⋅
6
c
=
8
8
⋅
9
9
⋅
1
0
10
.
Determine the sum of
a
+
b
+
c
a + b + c
a
+
b
+
c
.
A3
1
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2017 COMC A3
Source: 2017 Canadian Open Math Challenge, Problem A3 —-- Two
1
1
1
×
1
1
1
squares are removed from a
5
5
5
×
5
5
5
grid as shown.[asy] size(3cm); for(int i = 0; i < 6; ++i) { for(int j = 0; j < 6; ++j) { if(j < 5) { draw((i, j)--(i, j + 1)); } } } draw((0,1)--(5,1)); draw((0,2)--(5,2)); draw((0,3)--(5,3)); draw((0,4)--(5,4)); draw((0,5)--(1,5)); draw((2,5)--(5,5)); draw((0,0)--(2,0)); draw((3,0)--(5,0)); [/asy]Determine the total number of squares of various sizes on the grid.
A2
1
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2017 COMC A2
Source: 2017 Canadian Open Math Challenge, Problem A2 —-- An equilateral triangle has sides of length
4
4
4
cm. At each vertex, a circle with radius
2
2
2
cm is drawn, as shown in the figure below. The total area of the shaded regions of the three circles is
a
⋅
π
cm
2
a\cdot \pi \text{cm}^2
a
⋅
π
cm
2
. Determine
a
a
a
.[asy] size(2.5cm); draw(circle((0,2sqrt(3)/3),1)); draw(circle((1,-sqrt(3)/3),1)); draw(circle((-1,-sqrt(3)/3),1)); draw((0,2sqrt(3)/3) -- arc((0,2sqrt(3)/3), 1, 240, 300) -- cycle); fill(((0,2sqrt(3)/3) -- arc((0,2sqrt(3)/3), 1, 240, 300) -- cycle),mediumgray); draw((1,-sqrt(3)/3) -- arc((1,-sqrt(3)/3), 1, 180, 120) -- cycle); fill(((1,-sqrt(3)/3) -- arc((1,-sqrt(3)/3), 1, 180, 120) -- cycle),mediumgray); draw((-1,-sqrt(3)/3) -- arc((-1,-sqrt(3)/3), 1, 0, 60) -- cycle); fill(((-1,-sqrt(3)/3) -- arc((-1,-sqrt(3)/3), 1, 0, 60) -- cycle),mediumgray); [/asy]
A1
1
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2017 COMC A1
Source: 2017 Canadian Open Math Challenge, Problem A1 —-- The average of the numbers
2
2
2
,
5
5
5
,
x
x
x
,
14
14
14
,
15
15
15
is
x
x
x
. Determine the value of
x
x
x
.