MathDB
Problems
Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
1977 Canada National Olympiad
1977 Canada National Olympiad
Part of
Canada National Olympiad
Subcontests
(7)
7
1
Hide problems
Show that f(m, n) is less than or equal to 2^(mn)
A rectangular city is exactly
m
m
m
blocks long and
n
n
n
blocks wide (see diagram). A woman lives in the southwest corner of the city and works in the northeast corner. She walks to work each day but, on any given trip, she makes sure that her path does not include any intersection twice. Show that the number
f
(
m
,
n
)
f(m,n)
f
(
m
,
n
)
of different paths she can take to work satisfies
f
(
m
,
n
)
≤
2
m
n
f(m,n) \le 2^{mn}
f
(
m
,
n
)
≤
2
mn
.[asy] unitsize(0.4 cm);for(int i = 0; i <= 11; ++i) { draw((i,0)--(i,7)); }for(int j = 0; j <= 7; ++j) { draw((0,j)--(11,j)); }label("
⏟
\underbrace{\hspace{4.4 cm}}
", (11/2,-0.5)); label("\left. \begin{array}{c} \vspace{2.4 cm} \end{array} \right\}", (11,7/2)); label("
m
m
m
blocks", (11/2,-1.5)); label("
n
n
n
blocks", (14,7/2)); [/asy]
6
1
Hide problems
Show that every term of the sequence exceeds 1
Let
0
<
u
<
1
0 < u < 1
0
<
u
<
1
and define u_1 = 1 + u, u_2 = \frac{1}{u_1} + u, \dots, u_{n + 1} = \frac{1}{u_n} + u, n \ge 1. Show that
u
n
>
1
u_n > 1
u
n
>
1
for all values of
n
=
1
n = 1
n
=
1
, 2, 3,
…
\dots
…
.
5
1
Hide problems
Minimum distance from the point K
A right circular cone has base radius 1 cm and slant height 3 cm is given.
P
P
P
is a point on the circumference of the base and the shortest path from
P
P
P
around the cone and back to
P
P
P
is drawn (see diagram). What is the minimum distance from the vertex
V
V
V
to this path?[asy] import graph;unitsize(1 cm);filldraw(shift(-0.15,0.37)*rotate(17)*yscale(0.3)*xscale(1.41)*(Circle((0,0),1)),gray(0.9),nullpen); draw(yscale(0.3)*(arc((0,0),1.5,0,180)),dashed); draw(yscale(0.3)*(arc((0,0),1.5,180,360))); draw((1.5,0)--(0,4)--(-1.5,0)); draw((0,0)--(1.5,0),Arrows); draw(((1.5,0) + (0.3,0.1))--((0,4) + (0.3,0.1)),Arrows); draw(shift(-0.15,0.37)*rotate(17)*yscale(0.3)*xscale(1.41)*(arc((0,0),1,0,180)),dashed); draw(shift(-0.15,0.37)*rotate(17)*yscale(0.3)*xscale(1.41)*(arc((0,0),1,180,360)));label("
V
V
V
", (0,4), N); label("1 cm", (0.75,-0.5), N); label("
P
P
P
", (-1.5,0), SW); label("3 cm", (1.7,2)); [/asy]
4
1
Hide problems
Coefficients of two polynomials (mod 2)
Let
p
(
x
)
=
a
n
x
n
+
a
n
−
1
x
n
−
1
+
⋯
+
a
1
x
+
a
0
p(x) = a_n x^n + a_{n - 1} x^{n - 1} + \dots + a_1 x + a_0
p
(
x
)
=
a
n
x
n
+
a
n
−
1
x
n
−
1
+
⋯
+
a
1
x
+
a
0
and
q
(
x
)
=
b
m
x
m
+
a
m
−
1
x
m
−
1
+
⋯
+
b
1
x
+
b
0
q(x) = b_m x^m + a_{m - 1} x^{m - 1} + \dots + b_1 x + b_0
q
(
x
)
=
b
m
x
m
+
a
m
−
1
x
m
−
1
+
⋯
+
b
1
x
+
b
0
be two polynomials with integer coefficients. Suppose that all the coefficients of the product
p
(
x
)
⋅
q
(
x
)
p(x) \cdot q(x)
p
(
x
)
⋅
q
(
x
)
are even but not all of them are divisible by 4. Show that one of
p
(
x
)
p(x)
p
(
x
)
and
q
(
x
)
q(x)
q
(
x
)
has all even coefficients and the other has at least one odd coefficient.
3
1
Hide problems
Smallest natural b for which 7 + 7b + 7b^2 is a fourth power
N
N
N
is an integer whose representation in base
b
b
b
is
777
777
777
. Find the smallest integer
b
b
b
for which
N
N
N
is the fourth power of an integer.
2
1
Hide problems
Points P which maximize <OPX - [Canada MO 1977]
Let
O
O
O
be the centre of a circle and
A
A
A
a fixed interior point of the circle different from
O
O
O
. Determine all points
P
P
P
on the circumference of the circle such that the angle
O
P
A
OPA
OP
A
is a maximum.[asy] import graph;unitsize(2 cm);pair A, O, P;A = (0.5,0.2); O = (0,0); P = dir(80);draw(Circle(O,1)); draw(O--A--P--cycle);label("
A
A
A
", A, E); label("
O
O
O
", O, S); label("
P
P
P
", P, N); [/asy]
1
1
Hide problems
No naturals with 4m(m+1) = n(n+1) - [Canada MO 1977]
If
f
(
x
)
=
x
2
+
x
f(x) = x^2 + x
f
(
x
)
=
x
2
+
x
, prove that the equation
4
f
(
a
)
=
f
(
b
)
4f(a) = f(b)
4
f
(
a
)
=
f
(
b
)
has no solutions in positive integers
a
a
a
and
b
b
b
.