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Problems
Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
1982 Canada National Olympiad
1982 Canada National Olympiad
Part of
Canada National Olympiad
Subcontests
(5)
5
1
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The centroid of A"B"C"D" is the same as the centroid of ABCD
The altitudes of a tetrahedron
A
B
C
D
ABCD
A
BC
D
are extended externally to points
A
′
A'
A
′
,
B
′
B'
B
′
,
C
′
C'
C
′
, and
D
′
D'
D
′
, where
A
A
′
=
k
/
h
a
AA' = k/h_a
A
A
′
=
k
/
h
a
,
B
B
′
=
k
/
h
b
BB' = k/h_b
B
B
′
=
k
/
h
b
,
C
C
′
=
k
/
h
c
CC' = k/h_c
C
C
′
=
k
/
h
c
, and
D
D
′
=
k
/
h
d
DD' = k/h_d
D
D
′
=
k
/
h
d
. Here,
k
k
k
is a constant and
h
a
h_a
h
a
denotes the length of the altitude of
A
B
C
D
ABCD
A
BC
D
from vertex
A
A
A
, etc. Prove that the centroid of tetrahedron
A
′
B
′
C
′
D
′
A'B'C'D'
A
′
B
′
C
′
D
′
coincides with the centroid of
A
B
C
D
ABCD
A
BC
D
.
4
1
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Number of permutations [Canada MO 1982 - P4]
Let
p
p
p
be a permutation of the set
S
n
=
{
1
,
2
,
…
,
n
}
S_n = \{1, 2, \dots, n\}
S
n
=
{
1
,
2
,
…
,
n
}
. An element
j
∈
S
n
j \in S_n
j
∈
S
n
is called a fixed point of
p
p
p
if
p
(
j
)
=
j
p(j) = j
p
(
j
)
=
j
. Let
f
n
f_n
f
n
be the number of permutations having no fixed points, and
g
n
g_n
g
n
be the number with exactly one fixed point. Show that
∣
f
n
−
g
n
∣
=
1
|f_n - g_n| = 1
∣
f
n
−
g
n
∣
=
1
.
3
1
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Points with irrational distance from at least one of points
Let
R
n
\mathbb{R}^n
R
n
be the
n
n
n
-dimensional Euclidean space. Determine the smallest number
g
(
n
)
g(n)
g
(
n
)
of a points of a set in
R
n
\mathbb{R}^n
R
n
such that every point in
R
n
\mathbb{R}^n
R
n
is an irrational distance from at least one point in that set.
1
1
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Show that the area of A'B'C'D' is twice that of ABCD
In the diagram,
O
B
i
OB_i
O
B
i
is parallel and equal in length to
A
i
A
i
+
1
A_i A_{i + 1}
A
i
A
i
+
1
for
i
=
1
i = 1
i
=
1
, 2, 3, and 4 (
A
5
=
A
1
A_5 = A_1
A
5
=
A
1
). Show that the area of
B
1
B
2
B
3
B
4
B_1 B_2 B_3 B_4
B
1
B
2
B
3
B
4
is twice that of
A
1
A
2
A
3
A
4
A_1 A_2 A_3 A_4
A
1
A
2
A
3
A
4
.[asy] unitsize(1 cm);pair O; pair[] A, B;O = (0,0); A[1] = (0.5,-3); A[2] = (2,0); A[3] = (-0.2,0.5); A[4] = (-1,0); B[1] = A[2] - A[1]; B[2] = A[3] - A[2]; B[3] = A[4] - A[3]; B[4] = A[1] - A[4];draw(A[1]--A[2]--A[3]--A[4]--cycle); draw(B[1]--B[2]--B[3]--B[4]--cycle); draw(O--B[1]); draw(O--B[2]); draw(O--B[3]); draw(O--B[4]);label("
A
1
A_1
A
1
", A[1], S); label("
A
2
A_2
A
2
", A[2], E); label("
A
3
A_3
A
3
", A[3], N); label("
A
4
A_4
A
4
", A[4], W); label("
B
1
B_1
B
1
", B[1], NE); label("
B
2
B_2
B
2
", B[2], W); label("
B
3
B_3
B
3
", B[3], SW); label("
B
4
B_4
B
4
", B[4], S); label("
O
O
O
", O, E); [/asy]
2
1
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On the roots of x^3 = x^2 + x + 1 - [Canada MO 1982 - P2]
If
a
a
a
,
b
b
b
and
c
c
c
are the roots of the equation
x
3
−
x
2
−
x
−
1
=
0
x^3 - x^2 - x - 1 = 0
x
3
−
x
2
−
x
−
1
=
0
, (i) show that
a
a
a
,
b
b
b
and
c
c
c
are distinct: (ii) show that
a
1982
−
b
1982
a
−
b
+
b
1982
−
c
1982
b
−
c
+
c
1982
−
a
1982
c
−
a
\frac{a^{1982} - b^{1982}}{a - b} + \frac{b^{1982} - c^{1982}}{b - c} + \frac{c^{1982} - a^{1982}}{c - a}
a
−
b
a
1982
−
b
1982
+
b
−
c
b
1982
−
c
1982
+
c
−
a
c
1982
−
a
1982
is an integer.