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Problems
Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
1979 Canada National Olympiad
1979 Canada National Olympiad
Part of
Canada National Olympiad
Subcontests
(5)
5
1
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Show that 2^n < f(n) ≤ 4 . 3^(n-1) - [Canada MO 1979 - P5]
A walk consists of a sequence of steps of length 1 taken in the directions north, south, east, or west. A walk is self-avoiding if it never passes through the same point twice. Let
f
(
n
)
f(n)
f
(
n
)
be the number of
n
n
n
-step self-avoiding walks which begin at the origin. Compute
f
(
1
)
f(1)
f
(
1
)
,
f
(
2
)
f(2)
f
(
2
)
,
f
(
3
)
f(3)
f
(
3
)
,
f
(
4
)
f(4)
f
(
4
)
, and show that
2
n
<
f
(
n
)
≤
4
⋅
3
n
−
1
.
2^n < f(n) \le 4 \cdot 3^{n - 1}.
2
n
<
f
(
n
)
≤
4
⋅
3
n
−
1
.
4
1
Hide problems
Dog and rabbit [Canada MO 1979 - P4]
A dog standing at the centre of a circular arena sees a rabbit at the wall. The rabbit runs round the wall and the dog pursues it along a unique path which is determined by running at the same speed and staying on the radial line joining the centre of the arena to the rabbit. Show that the dog overtakes the rabbit just as it reaches a point one-quarter of the way around the arena.
3
1
Hide problems
Show that 1/[a, b] + 1/[b, c] + 1/[c, d] + 1/[d, e] ≤ 15/16
Let
a
a
a
,
b
b
b
,
c
c
c
,
d
d
d
,
e
e
e
be integers such that
1
≤
a
<
b
<
c
<
d
<
e
1 \le a < b < c < d < e
1
≤
a
<
b
<
c
<
d
<
e
. Prove that
1
[
a
,
b
]
+
1
[
b
,
c
]
+
1
[
c
,
d
]
+
1
[
d
,
e
]
≤
15
16
,
\frac{1}{[a,b]} + \frac{1}{[b,c]} + \frac{1}{[c,d]} + \frac{1}{[d,e]} \le \frac{15}{16},
[
a
,
b
]
1
+
[
b
,
c
]
1
+
[
c
,
d
]
1
+
[
d
,
e
]
1
≤
16
15
,
where
[
m
,
n
]
[m,n]
[
m
,
n
]
denotes the least common multiple of
m
m
m
and
n
n
n
(e.g.
[
4
,
6
]
=
12
[4,6] = 12
[
4
,
6
]
=
12
).
2
1
Hide problems
Sum of angles in two tetrahedras are not always the same
It is known in Euclidean geometry that the sum of the angles of a triangle is constant. Prove, however, that the sum of the dihedral angles of a tetrahedron is not constant.
1
1
Hide problems
a,b,c,d in AP and a,h,k,d in GP --> bc ≥ hk - [Canada 1979]
Given: (i)
a
a
a
,
b
>
0
b > 0
b
>
0
; (ii)
a
a
a
,
A
1
A_1
A
1
,
A
2
A_2
A
2
,
b
b
b
is an arithmetic progression; (iii)
a
a
a
,
G
1
G_1
G
1
,
G
2
G_2
G
2
,
b
b
b
is a geometric progression. Show that
A
1
A
2
≥
G
1
G
2
.
A_1 A_2 \ge G_1 G_2.
A
1
A
2
≥
G
1
G
2
.