MathDB
Problems
Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
1974 Canada National Olympiad
1974 Canada National Olympiad
Part of
Canada National Olympiad
Subcontests
(7)
7
1
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Bus route
A bus route consists of a circular road of circumference 10 miles and a straight road of length 1 mile which runs from a terminus to the point
Q
Q
Q
on the circular road (see diagram). 6763 It is served by two buses, each of which requires 20 minutes for the round trip. Bus No. 1, upon leaving the terminus, travels along the straight road, once around the circle clockwise and returns along the straight road to the terminus. Bus No. 2, reaching the terminus 10 minutes after Bus No. 1, has a similar route except that it proceeds counterclockwise around the circle. Both buses run continuously and do not wait at any point on the route except for a negligible amount of time to pick up and discharge passengers. A man plans to wait at a point
P
P
P
which is
x
x
x
miles (
0
≤
x
<
12
0\le x < 12
0
≤
x
<
12
) from the terminus along the route of Bus No. 1 and travel to the terminus on one of the buses. Assuming that he chooses to board that bus which will bring him to his destination at the earliest moment, there is a maximum time
w
(
x
)
w(x)
w
(
x
)
that his journey (waiting plus travel time) could take. Find
w
(
2
)
w(2)
w
(
2
)
; find
w
(
4
)
w(4)
w
(
4
)
. For what value of
x
x
x
will the time
w
(
x
)
w(x)
w
(
x
)
be the longest? Sketch a graph of
y
=
w
(
x
)
y = w(x)
y
=
w
(
x
)
for
0
≤
x
<
12
0\le x < 12
0
≤
x
<
12
.
6
1
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Stamps
An unlimited supply of 8-cent and 15-cent stamps is available. Some amounts of postage cannot be made up exactly, e.g., 7 cents, 29 cents. What is the largest unattainable amount, i.e., the amount, say
n
n
n
, of postage which is unattainable while all amounts larger than
n
n
n
are attainable? (Justify your answer.)
5
1
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Circle tangents
Given a circle with diameter
A
B
AB
A
B
and a point
X
X
X
on the circle different from
A
A
A
and
B
B
B
, let
t
a
t_{a}
t
a
,
t
b
t_{b}
t
b
and
t
x
t_{x}
t
x
be the tangents to the circle at
A
A
A
,
B
B
B
and
X
X
X
respectively. Let
Z
Z
Z
be the point where line
A
X
AX
A
X
meets
t
b
t_{b}
t
b
and
Y
Y
Y
the point where line
B
X
BX
BX
meets
t
a
t_{a}
t
a
. Show that the three lines
Y
Z
YZ
Y
Z
,
t
x
t_{x}
t
x
and
A
B
AB
A
B
are either concurrent (i.e., all pass through the same point) or parallel. 6762
4
1
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Sum of reals
Let
n
n
n
be a fixed positive integer. To any choice of real numbers satisfying 0\le x_{i}\le 1, i=1,2,\ldots, n, there corresponds the sum
∑
1
≤
i
<
j
≤
n
∣
x
i
−
x
j
∣
.
\sum_{1\le i<j\le n}|x_{i}-x_{j}|.
1
≤
i
<
j
≤
n
∑
∣
x
i
−
x
j
∣.
Let
S
(
n
)
S(n)
S
(
n
)
denote the largest possible value of this sum. Find
S
(
n
)
S(n)
S
(
n
)
.
3
1
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Coefficients of a polynomial
Let
f
(
x
)
=
a
0
+
a
1
x
+
a
2
x
2
+
⋯
+
a
n
x
n
f(x) = a_{0}+a_{1}x+a_{2}x^{2}+\cdots+a_{n}x^{n}
f
(
x
)
=
a
0
+
a
1
x
+
a
2
x
2
+
⋯
+
a
n
x
n
be a polynomial with coefficients satisfying the conditions: 0\le a_{i}\le a_{0}, i=1,2,\ldots,n. Let
b
0
,
b
1
,
…
,
b
2
n
b_{0},b_{1},\ldots,b_{2n}
b
0
,
b
1
,
…
,
b
2
n
be the coefficients of the polynomial \begin{align*}\left(f(x)\right)^{2}&= \left(a_{0}+a_{1}x+a_{2}x^{2}+\cdots a_{n}x^{n}\right)\\ &= b_{0}+b_{1}x+b_{2}x^{2}+\cdots+b_{2n}x^{2n}. \end{align*} Prove that
b
n
+
1
≤
1
2
(
f
(
1
)
)
2
b_{n+1}\le \frac{1}{2}\left(f(1)\right)^{2}
b
n
+
1
≤
2
1
(
f
(
1
)
)
2
.
2
1
Hide problems
Rectangle
Let
A
B
C
D
ABCD
A
BC
D
be a rectangle with
B
C
=
3
A
B
BC=3AB
BC
=
3
A
B
. Show that if
P
,
Q
P,Q
P
,
Q
are the points on side
B
C
BC
BC
with
B
P
=
P
Q
=
Q
C
BP = PQ = QC
BP
=
PQ
=
QC
, then
∠
D
B
C
+
∠
D
P
C
=
∠
D
Q
C
.
\angle DBC+\angle DPC = \angle DQC.
∠
D
BC
+
∠
D
PC
=
∠
D
QC
.
1
1
Hide problems
Two identities
i) If
x
=
(
1
+
1
n
)
n
x = \left(1+\frac{1}{n}\right)^{n}
x
=
(
1
+
n
1
)
n
and
y
=
(
1
+
1
n
)
n
+
1
y=\left(1+\frac{1}{n}\right)^{n+1}
y
=
(
1
+
n
1
)
n
+
1
, show that
y
x
=
x
y
y^{x}= x^{y}
y
x
=
x
y
. ii) Show that, for all positive integers
n
n
n
,
1
2
−
2
2
+
3
2
−
4
2
+
⋯
+
(
−
1
)
n
(
n
−
1
)
2
+
(
−
1
)
n
+
1
n
2
=
(
−
1
)
n
+
1
(
1
+
2
+
⋯
+
n
)
.
1^{2}-2^{2}+3^{2}-4^{2}+\cdots+(-1)^{n}(n-1)^{2}+(-1)^{n+1}n^{2}= (-1)^{n+1}(1+2+\cdots+n).
1
2
−
2
2
+
3
2
−
4
2
+
⋯
+
(
−
1
)
n
(
n
−
1
)
2
+
(
−
1
)
n
+
1
n
2
=
(
−
1
)
n
+
1
(
1
+
2
+
⋯
+
n
)
.