MathDB
Problems
Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
1973 Canada National Olympiad
1973 Canada National Olympiad
Part of
Canada National Olympiad
Subcontests
(7)
7
1
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Sum of fractions
Observe that \frac{1}{1}= \frac{1}{2}+\frac{1}{2}; \frac{1}{2}=\frac{1}{3}+\frac{1}{6}; \frac{1}{3}=\frac{1}{4}+\frac{1}{12}; \frac{1}{4}= \frac{1}{5}+\frac{1}{20}. State a general law suggested by these examples, and prove it. Prove that for any integer
n
n
n
greater than 1 there exist positive integers
i
i
i
and
j
j
j
such that
1
n
=
1
i
(
i
+
1
)
+
1
(
i
+
1
)
(
i
+
2
)
+
1
(
i
+
2
)
(
i
+
3
)
+
⋯
+
1
j
(
j
+
1
)
.
\frac{1}{n}= \frac{1}{i(i+1)}+\frac{1}{(i+1)(i+2)}+\frac{1}{(i+2)(i+3)}+\cdots+\frac{1}{j(j+1)}.
n
1
=
i
(
i
+
1
)
1
+
(
i
+
1
)
(
i
+
2
)
1
+
(
i
+
2
)
(
i
+
3
)
1
+
⋯
+
j
(
j
+
1
)
1
.
[hide="Remark."] It seems that this is a two-part problem.
6
1
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Circle
If
A
A
A
and
B
B
B
are fixed points on a given circle not collinear with centre
O
O
O
of the circle, and if
X
Y
XY
X
Y
is a variable diameter, find the locus of
P
P
P
(the intersection of the line through
A
A
A
and
X
X
X
and the line through
B
B
B
and
Y
Y
Y
).
5
1
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Harmonic
For every positive integer
n
n
n
, let
h
(
n
)
=
1
+
1
2
+
1
3
+
⋯
+
1
n
.
h(n) = 1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}.
h
(
n
)
=
1
+
2
1
+
3
1
+
⋯
+
n
1
.
For example,
h
(
1
)
=
1
h(1) = 1
h
(
1
)
=
1
,
h
(
2
)
=
1
+
1
2
h(2) = 1+\frac{1}{2}
h
(
2
)
=
1
+
2
1
,
h
(
3
)
=
1
+
1
2
+
1
3
h(3) = 1+\frac{1}{2}+\frac{1}{3}
h
(
3
)
=
1
+
2
1
+
3
1
. Prove that for
n
=
2
,
3
,
4
,
…
n=2,3,4,\ldots
n
=
2
,
3
,
4
,
…
n
+
h
(
1
)
+
h
(
2
)
+
h
(
3
)
+
⋯
+
h
(
n
−
1
)
=
n
h
(
n
)
.
n+h(1)+h(2)+h(3)+\cdots+h(n-1) = nh(n).
n
+
h
(
1
)
+
h
(
2
)
+
h
(
3
)
+
⋯
+
h
(
n
−
1
)
=
nh
(
n
)
.
4
1
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Convex polygon with nine vertices
The figure shows a (convex) polygon with nine vertices. The six diagonals which have been drawn dissect the polygon into the seven triangles:
P
0
P
1
P
3
P_{0}P_{1}P_{3}
P
0
P
1
P
3
,
P
0
P
3
P
6
P_{0}P_{3}P_{6}
P
0
P
3
P
6
,
P
0
P
6
P
7
P_{0}P_{6}P_{7}
P
0
P
6
P
7
,
P
0
P
7
P
8
P_{0}P_{7}P_{8}
P
0
P
7
P
8
,
P
1
P
2
P
3
P_{1}P_{2}P_{3}
P
1
P
2
P
3
,
P
3
P
4
P
6
P_{3}P_{4}P_{6}
P
3
P
4
P
6
,
P
4
P
5
P
6
P_{4}P_{5}P_{6}
P
4
P
5
P
6
. In how many ways can these triangles be labeled with the names
△
1
\triangle_{1}
△
1
,
△
2
\triangle_{2}
△
2
,
△
3
\triangle_{3}
△
3
,
△
4
\triangle_{4}
△
4
,
△
5
\triangle_{5}
△
5
,
△
6
\triangle_{6}
△
6
,
△
7
\triangle_{7}
△
7
so that
P
i
P_{i}
P
i
is a vertex of triangle
△
i
\triangle_{i}
△
i
for
i
=
1
,
2
,
3
,
4
,
5
,
6
,
7
i = 1, 2, 3, 4, 5, 6, 7
i
=
1
,
2
,
3
,
4
,
5
,
6
,
7
? Justify your answer. 6740
3
1
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Primes
Prove that if
p
p
p
and
p
+
2
p+2
p
+
2
are prime integers greater than 3, then 6 is a factor of
p
+
1
p+1
p
+
1
.
2
1
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Absolute value equation
Find all real numbers that satisfy the equation
∣
x
+
3
∣
−
∣
x
−
1
∣
=
x
+
1
|x+3|-|x-1|=x+1
∣
x
+
3∣
−
∣
x
−
1∣
=
x
+
1
. (Note:
∣
a
∣
=
a
|a| = a
∣
a
∣
=
a
if
a
≥
0
a\ge 0
a
≥
0
;
∣
a
∣
=
−
a
|a|=-a
∣
a
∣
=
−
a
if
a
<
0
a<0
a
<
0
.)
1
1
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A few quick ones...
(i) Solve the simultaneous inequalities,
x
<
1
4
x
x<\frac{1}{4x}
x
<
4
x
1
and
x
<
0
x<0
x
<
0
; i.e. find a single inequality equivalent to the two simultaneous inequalities. (ii) What is the greatest integer that satisfies both inequalities
4
x
+
13
<
0
4x+13 < 0
4
x
+
13
<
0
and
x
2
+
3
x
>
16
x^{2}+3x > 16
x
2
+
3
x
>
16
. (iii) Give a rational number between
11
/
24
11/24
11/24
and
6
/
13
6/13
6/13
. (iv) Express 100000 as a product of two integers neither of which is an integral multiple of 10. (v) Without the use of logarithm tables evaluate
1
log
2
36
+
1
log
3
36
.
\frac{1}{\log_{2}36}+\frac{1}{\log_{3}36}.
lo
g
2
36
1
+
lo
g
3
36
1
.