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National and Regional Contests
Canada Contests
Canadian Senior Mathematics Contest
2018 Canadian Senior Mathematics Contest
B2
B2
Part of
2018 Canadian Senior Mathematics Contest
Problems
(1)
CSMC 2018 Part B Problem 2
Source:
11/24/2018
[*]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
.
CSMC
CSMC 2018