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Contests
National and Regional Contests
Canada Contests
Canadian Open Math Challenge
2018 Canadian Open Math Challenge
2018 Canadian Open Math Challenge
Part of
Canadian Open Math Challenge
Subcontests
(12)
C4
1
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2018 COMC C4
Source: 2018 Canadian Open Math Challenge Part C Problem 4 —--Given a positive integer
N
N
N
, Matt writes
N
N
N
in decimal on a blackboard, without writing any of the leading 0s. Every minute he takes two consicutive digits, erases them, and replaces them with the last digit of their product. Any leading zeroes created this way are also erased. He repeats this process for as long as he likes. We call the positive integer
M
M
M
obtainable from
N
N
N
if starting from
N
N
N
, there is a finite sequence of moves that Matt can make to produce the number
M
M
M
. For example, 10 is obtainible from 251023 via
2510
23
‾
→
25
‾
106
→
1
06
‾
→
10
2510\underline{23}\rightarrow\underline{25} 106\rightarrow 1\underline{06}\rightarrow 10
2510
23
→
25
106
→
1
06
→
10
(a)
\text{(a)}
(a)
Show that 2018 is obtainablefrom 2567777899.
(b)
\text{(b)}
(b)
Find two positive integers
A
A
A
and
B
B
B
for which there is no positive integer
C
C
C
(B.) such that both
A
A
A
and
B
B
B
are obtainablefrom
C
C
C
(c)
\text{(c)}
(c)
Let
S
S
S
be any finite set of positive integers, none of which contains the digit 5 (C.) in its decimal representation. Prove that there exists a positive integer
N
N
N
(C.) for which all elements of
S
S
S
are obtainable from
N
N
N
.
C3
1
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2018 COMC C3
Source: 2018 Canadian Open Math Challenge Part C Problem 3 —--Consider a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
. Let rays
B
A
BA
B
A
and
C
D
CD
C
D
intersect at
E
E
E
, rays
D
A
DA
D
A
and
C
B
CB
CB
intersect at
F
F
F
, and the diagonals
A
C
AC
A
C
and
B
D
BD
B
D
intersect at
G
G
G
. It is given that the triangles
D
B
F
DBF
D
BF
and
D
B
E
DBE
D
BE
have the same area.
(a)
\text{(a)}
(a)
Prove that
E
F
EF
EF
and
B
D
BD
B
D
are parallel.
(b)
\text{(b)}
(b)
Prove that
G
G
G
is the midpoint of
B
D
BD
B
D
.
(c)
\text{(c)}
(c)
Given that the area of triangle
A
B
D
ABD
A
B
D
is 4 and the area of triangle
C
B
D
CBD
CB
D
is 6, (C.)compute the area of triangle
E
F
G
EFG
EFG
.
C2
1
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2018 COMC C2
Source: 2018 Canadian Open Math Challenge Part C Problem 2 —--Alice has two boxes
A
A
A
and
B
B
B
. Initially box
a
a
a
contains
n
n
n
coins and box
B
B
B
is empty. On each turn, she may either move a coin from box
a
a
a
to box
B
B
B
, or remove
k
k
k
coins from box
A
A
A
, where
k
k
k
is the current number of coins in box
B
B
B
. She wins when box
A
A
A
is empty.
(a)
\text{(a)}
(a)
If initially box
A
A
A
contains 6 coins, show that Alice can win in 4 turns.
(b)
\text{(b)}
(b)
If initially box
A
A
A
contains 31 coins, show that Alice cannot win in 10 turns.
(c)
\text{(c)}
(c)
What is the minimum number of turns needed for Alice to win if box
A
A
A
initially contains 2018 coins?
C1
1
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2018 COMC C1
Source: 2018 Canadian Open Math Challenge Part C Problem 1 —--At Math-
e
e
e^e
e
e
-Mart, cans of cat food are arranged in an pentagonal pyramid of 15 layers high, with 1 can in the top layer, 5 cans in the second layer, 12 cans in the third layer, 22 cans in the fourth layer etc, so that the
k
th
k^{\text{th}}
k
th
layer is a pentagon with
k
k
k
cans on each side. https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNC9lLzA0NTc0MmM2OGUzMWIyYmE1OGJmZWQzMGNjMGY1NTVmNDExZjU2LnBuZw==&rn=YzFhLlBORw==https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYS9hLzA1YWJlYmE1ODBjMzYwZDFkYWQyOWQ1YTFhOTkzN2IyNzJlN2NmLnBuZw==&rn=YzFiLlBORw==
(a)
\text{(a)}
(a)
How many cans are on the bottom,
1
5
th
15^{\text{th}}
1
5
th
, (A.)layer of this pyramid?
(b)
\text{(b)}
(b)
The pentagonal pyramid is rearranged into a prism consisting of 15 identical layers. (B.)How many cans are on the bottom layer of the prism?
(c)
\text{(c)}
(c)
A triangular prism consist of indentical layers, each of which has a shape of a triangle. (C.)(the number of cans in a triangular layer is one of the triangular numbers: 1,3,6,10,...) (C.)For example, a prism could be composed of the following layers: https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvMi85L2NlZmE2M2Y3ODhiN2UzMTRkYzIxY2MzNjFmMDJkYmE0ZTJhMTcwLnBuZw==&rn=YzFjLlBORw== Prove that a pentagonal pyramid of cans with any number of layers
l
≥
2
l\ge 2
l
≥
2
can be rearranged (without a deficit or leftover) into a triangluar prism of cans with the same number of layers
l
l
l
.
B4
1
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2018 COMC B4
Source: 2018 Canadian Open Math Challenge Part B Problem 4 —--Determine the number of
5
5
5
-tuples of integers
(
x
1
,
x
2
,
x
3
,
x
4
,
x
5
)
(x_1,x_2,x_3,x_4,x_5)
(
x
1
,
x
2
,
x
3
,
x
4
,
x
5
)
such that
(a)
\text{(a)}
(a)
x
i
≥
i
x_i\ge i
x
i
≥
i
for
1
≤
i
≤
5
1\le i \le 5
1
≤
i
≤
5
;
(b)
\text{(b)}
(b)
∑
i
=
1
5
x
i
=
25
\sum_{i=1}^5 x_i = 25
∑
i
=
1
5
x
i
=
25
.
B3
1
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2018 COMC B3
Source: 2018 Canadian Open Math Challenge Part B Problem 3 —--The doubling sum function is defined by
D
(
a
,
n
)
=
a
+
2
a
+
4
a
+
8
a
+
.
.
.
⏞
n terms
.
D(a,n)=\overbrace{a+2a+4a+8a+...}^{\text{n terms}}.
D
(
a
,
n
)
=
a
+
2
a
+
4
a
+
8
a
+
...
n terms
.
For example, we have
D
(
5
,
3
)
=
5
+
10
+
20
=
35
D(5,3)=5+10+20=35
D
(
5
,
3
)
=
5
+
10
+
20
=
35
and
D
(
11
,
5
)
=
11
+
22
+
44
+
88
+
176
=
341.
D(11,5)=11+22+44+88+176=341.
D
(
11
,
5
)
=
11
+
22
+
44
+
88
+
176
=
341.
Determine the smallest positive integer
n
n
n
such that for every integer
i
i
i
between
1
1
1
and
6
6
6
, inclusive, there exists a positive integer
a
i
a_i
a
i
such that
D
(
a
i
,
i
)
=
n
.
D(a_i,i)=n.
D
(
a
i
,
i
)
=
n
.
B2
1
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2018 COMC B2
Source: 2018 Canadian Open Math Challenge Part B Problem 2 —--Let ABCD be a square with side length 1. Points
X
X
X
and
Y
Y
Y
are on sides
B
C
BC
BC
and
C
D
CD
C
D
respectively such that the areas of triangels
A
B
X
ABX
A
BX
,
X
C
Y
XCY
XC
Y
, and
Y
D
A
YDA
Y
D
A
are equal. Find the ratio of the area of
△
A
X
Y
\triangle AXY
△
A
X
Y
to the area of
△
X
C
Y
\triangle XCY
△
XC
Y
. https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvZi9lLzAzZjhhYzU0N2U0MGY2NGZlODM4MWI4Njg2MmEyMjhlY2M3ZjgzLnBuZw==&rn=YjIuUE5H
B1
1
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2018 COMC B1
Source: 2018 Canadian Open Math Challenge Part B Problem 1 —--Let
(
1
+
2
)
5
=
a
+
b
2
(1+\sqrt2)^5 = a+b\sqrt2
(
1
+
2
)
5
=
a
+
b
2
, where
a
a
a
and
b
b
b
are positive integers. Determine the value of
a
+
b
.
a+b.
a
+
b
.
A4
1
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2018 COMC A4
Source: 2018 Canadian Open Math Challenge Part A Problem 4 —--In the sequence of positive integers, starting with
2018
,
121
,
16
,
.
.
.
2018, 121, 16, ...
2018
,
121
,
16
,
...
each term is the square of the sum of digits of the previous term. What is the
201
8
th
2018^{\text{th}}
201
8
th
term of the sequence?
A3
1
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2018 COMC A3
Source: 2018 Canadian Open Math Challenge Part A Problem 3 —--Points
(
0
,
0
)
(0,0)
(
0
,
0
)
and
(
3
7
,
7
3
)
(3\sqrt7,7\sqrt3)
(
3
7
,
7
3
)
are the endpoints of a diameter of circle
Γ
.
\Gamma.
Γ.
Determine the other
x
x
x
intercept of
Γ
.
\Gamma.
Γ.
A2
1
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2018 COMC A2
Source: 2018 Canadian Open Math Challenge Part A Problem 2 —--Let
v
v
v
,
w
w
w
,
x
x
x
,
y
y
y
, and
z
z
z
be five distinct integers such that
45
=
v
×
w
×
x
×
y
×
z
.
45 = v\times w\times x\times y\times z.
45
=
v
×
w
×
x
×
y
×
z
.
What is the sum of the integers?
A1
1
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2018 COMC A1
Source: 2018 Canadian Open Math Challenge Part A Problem 1 —--Suppose
x
x
x
is a real number such that
x
(
x
+
3
)
=
154.
x(x+3)=154.
x
(
x
+
3
)
=
154.
Determine the value of
(
x
+
1
)
(
x
+
2
)
(x+1)(x+2)
(
x
+
1
)
(
x
+
2
)
.