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Contests
National and Regional Contests
Canada Contests
Canadian Senior Mathematics Contest
2018 Canadian Senior Mathematics Contest
2018 Canadian Senior Mathematics Contest
Part of
Canadian Senior Mathematics Contest
Subcontests
(9)
B3
1
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CSMC 2018 Part B Problem 3
A string of length
n
n
n
is a sequence of
n
n
n
characters from a specified set. For example,
B
C
A
A
B
BCAAB
BC
AA
B
is a string of length 5 with characters from the set
{
A
,
B
,
C
}
\{A,B,C\}
{
A
,
B
,
C
}
. A substring of a given string is a string of characters that occur consecutively and in order in the given string. For example, the string
C
A
CA
C
A
is a substring of
B
C
A
A
B
BCAAB
BC
AA
B
but
B
A
BA
B
A
is not a substring of
B
C
A
A
B
BCAAB
BC
AA
B
. [*]List all strings of length 4 with characters from the set
{
A
,
B
,
C
}
\{A,B,C\}
{
A
,
B
,
C
}
in which both the strings
A
B
AB
A
B
and
B
A
BA
B
A
occur as substrings. (For example, the string
A
B
A
C
ABAC
A
B
A
C
should appear in your list.) [*]Determine the number of strings of length 7 with characters from the set
{
A
,
B
,
C
}
\{A,B,C\}
{
A
,
B
,
C
}
in which
C
C
CC
CC
occures as a substring. [*]Let
f
(
n
)
f(n)
f
(
n
)
be the number of strings of length
n
n
n
with characters from the set
{
A
,
B
,
C
}
\{A,B,C\}
{
A
,
B
,
C
}
such that [*]
C
C
CC
CC
occurs as a substring, and[*]if either
A
B
AB
A
B
or
B
A
BA
B
A
occurs as a substring then there is an occurrence of the substring
C
C
CC
CC
to its left. (for example, when
n
=
6
n\;=\;6
n
=
6
, the strings
C
C
A
A
B
C
CCAABC
CC
AA
BC
and
A
C
C
B
B
B
ACCBBB
A
CCBBB
and
C
C
A
B
C
C
CCABCC
CC
A
BCC
satisfy the requirements, but the strings
B
A
C
C
A
B
BACCAB
B
A
CC
A
B
and
A
C
B
B
A
B
ACBBAB
A
CBB
A
B
and
A
C
B
C
A
C
ACBCAC
A
CBC
A
C
do not). Prove that
f
(
2097
)
f(2097)
f
(
2097
)
is a multiple of
97
97
97
.
B2
1
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CSMC 2018 Part B Problem 2
[*]Determine the positive integer
x
x
x
for which
1
4
−
1
x
=
1
6
.
\dfrac14-\dfrac{1}{x}=\dfrac16.
4
1
−
x
1
=
6
1
.
[*]Determine all pairs of positive integers
(
a
,
b
)
(a,b)
(
a
,
b
)
for which
a
b
−
b
+
a
−
1
=
4.
ab-b+a-1=4.
ab
−
b
+
a
−
1
=
4.
[*]Determine the number of pairs of positive integers
(
y
,
z
)
(y,z)
(
y
,
z
)
for which
1
y
−
1
z
=
1
12
.
\dfrac{1}{y}-\dfrac{1}{z}=\dfrac{1}{12}.
y
1
−
z
1
=
12
1
.
[*]Prove that, for every prime number
p
p
p
, there are at least two pairs
(
r
,
s
)
(r,s)
(
r
,
s
)
of positive integers for which
1
r
−
1
s
=
1
p
2
.
\dfrac{1}{r}-\dfrac{1}{s}=\dfrac{1}{p^2}.
r
1
−
s
1
=
p
2
1
.
B1
1
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CSMC 2018 Part B Problem 1
Alexandra draws a letter A which stands on the
x
x
x
-axis. [*]The left side of the letter A lies along the line with equation
y
=
3
x
+
6
y=3x+6
y
=
3
x
+
6
. What is the
x
x
x
-intercept of the line with equation
y
=
3
x
+
6
y=3x+6
y
=
3
x
+
6
? [*]The right side of the letter A lies along the line
L
2
L_2
L
2
and the leter is symmetric about the
y
y
y
-axis. What is the equation of line
L
2
L_2
L
2
? [*]Determine the are of the triangle formed by the
x
x
x
axis and the left and right sides of the letter A. [*]Alexandra completes the letter A by adding to Figure 1. She draws the horizontal part of the letter A along the line
y
=
c
y=c
y
=
c
, as in Figure 2. The area of the shaded region inside the letter A and above the line with equation
y
=
c
y=c
y
=
c
is
4
9
\frac49
9
4
of the total area of the region above the
x
x
x
axis and between the left and right sides. Determine the value of
c
c
c
. Figure 1 [asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(10cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -4.8408113739622465, xmax = 5.491811096383217, ymin = -3.0244242161812847, ymax = 8.241467380517944; /* image dimensions */ pen cqcqcq = rgb(0.7529411764705882,0.7529411764705882,0.7529411764705882); Label laxis; laxis.p = fontsize(10); xaxis(xmin, xmax, EndArrow(6), above = true); yaxis(ymin, ymax, EndArrow(8.25),above = true); /* draws axes; NoZero hides '0' label */ /* draw figures */ draw((0,6)--(-2,0), linewidth(2)); draw((-2,0)--(2,0), linewidth(2)); draw((2,0)--(0,6), linewidth(2)); label("
y
=
3
x
+
6
y=3x+6
y
=
3
x
+
6
",(-2.874280000573916,3.508459668295191),SE*labelscalefactor); label("
L
2
L_2
L
2
",(1.3754276283584919,3.5917872688624928),SE*labelscalefactor); label("
O
O
O
",(0,0),SW*labelscalefactor); /* dots and labels */ clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy] Figure 2 [asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(10cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -4.707487213054563, xmax = 5.6251352572909, ymin = -3.4577277391312538, ymax = 7.808163857567977; /* image dimensions */ pen cqcqcq = rgb(0.7529411764705882,0.7529411764705882,0.7529411764705882); draw((-1.114884596113444,2.6553462116596678)--(1.1148845961134441,2.6553462116596678)--(0,6)--cycle, linewidth(2)); Label laxis; laxis.p = fontsize(10); xaxis(xmin, xmax, EndArrow(6), above = true); yaxis(ymin, ymax, EndArrow(6), above = true); /* draws axes; NoZero hides '0' label */ /* draw figures */ draw((0,6)--(-2,0), linewidth(2)); draw((-2,0)--(2,0), linewidth(2)); draw((2,0)--(0,6), linewidth(2)); label("
O
O
O
",(0,0),SW*labelscalefactor); draw((-1.114884596113444,2.6553462116596678)--(1.1148845961134441,2.6553462116596678), linewidth(2)); draw((-1.114884596113444,2.6553462116596678)--(1.1148845961134441,2.6553462116596678), linewidth(2)); draw((1.1148845961134441,2.6553462116596678)--(0,6), linewidth(2)); draw((0,6)--(-1.114884596113444,2.6553462116596678), linewidth(2)); fill((0,6)--(-1.114884596113444,2.6553462116596678)--(1.1148845961134441,2.6553462116596678)--cycle,black); label("
y
=
c
y=c
y
=
c
",(1.4920862691527148,3.1251527056856054),SE*labelscalefactor); /* dots and labels */ clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* yes i used geogebra fight me*/ [/asy]
A6
1
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CSMC 2018 Part A Problem 6
Suppose that
0
∘
<
A
<
9
0
∘
0^\circ < A < 90^\circ
0
∘
<
A
<
9
0
∘
and
0
∘
<
B
<
9
0
∘
0^\circ < B < 90^\circ
0
∘
<
B
<
9
0
∘
and
(
4
+
tan
2
A
)
(
5
+
tan
2
B
)
=
320
tan
A
tan
B
\left(4+\tan^2 A\right)\left(5+\tan^2 B\right) = \sqrt{320}\tan A\tan B
(
4
+
tan
2
A
)
(
5
+
tan
2
B
)
=
320
tan
A
tan
B
Determine all possible values of
cos
A
sin
B
\cos A\sin B
cos
A
sin
B
.
A5
1
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CSMC 2018 Part A Problem 5
In the diagram,
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
is a regular hexagon with side length 2. Points
E
E
E
and
F
F
F
are on the
x
x
x
axis and points
A
A
A
,
B
B
B
,
C
C
C
, and
D
D
D
lie on a parabola. What is the distance between the two
x
x
x
intercepts of the parabola?[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(6cm); real labelscalefactor = 0.5; pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); pen dotstyle = black; real xmin = -3.3215445204635294, xmax = 7.383669550094284, ymin = -4.983460515387094, ymax = 6.688676116382409; pen zzttqq = rgb(0.6,0.2,0); pen cqcqcq = rgb(0.7529411764705882,0.7529411764705882,0.7529411764705882); draw((2,0)--(4,0)--(5,1.7320508075688774)--(4,3.4641016151377553)--(2,3.4641016151377557)--(1,1.732050807568879)--cycle, linewidth(1)); Label laxis; laxis.p = fontsize(10); xaxis(xmin, xmax, EndArrow(6), above = true); yaxis(ymin, ymax, EndArrow(6), above = true);draw((2,0)--(4,0), linewidth(1)); draw((4,0)--(5,1.7320508075688774), linewidth(1)); draw((5,1.7320508075688774)--(4,3.4641016151377553), linewidth(1)); draw((4,3.4641016151377553)--(2,3.4641016151377557), linewidth(1)); draw((2,3.4641016151377557)--(1,1.732050807568879), linewidth(1)); draw((1,1.732050807568879)--(2,0), linewidth(1)); real f1 (real x) {return -0.58*x^(2)+3.46*x-1.15;} draw(graph(f1,-3.3115445204635297,7.373669550094284), linewidth(1)); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /*yes i used geogebra fight me*/ [/asy]
A4
1
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CSMC 2018 Part A Problem 4
Suppose that
n
n
n
is a positive integer and that
a
a
a
is the integer equal to
1
0
2
n
−
1
3
(
1
0
n
+
1
)
.
\frac{10^{2n}-1}{3\left(10^n+1\right)}.
3
(
1
0
n
+
1
)
1
0
2
n
−
1
.
If the sum of the digits of
a
a
a
is 567, what is the value of
n
n
n
?
A3
1
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CSMC 2018 Part A Problem 3
A jar contains 6 crayons, of which 3 are red, 2 are blue, and 1 is green. Jakob reaches into the jar and randomly removes 2 of the crayons. What is the probability that both of these crayons are red?
A2
1
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CSMC 2018 Part A Problem 2
A rabbit, a skunk and a turtle are running a race. The skunk finishes the race in 6 minutes. The rabbit runs 3 times as quickly as the skunk. The rabbit runs 5 times as quickly as the turtle. How long does the turtle take to finish the race?
A1
1
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CSMC 2018 Part A Problem 1
Paul has 6 boxes, each of which contains 12 trays. Paul also has 4 extra trays. If each tray can hold 8 apples, what is the largest possible number of apples that can be held by the 6 boxes and 4 extra trays?