MathDB
Problems
Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
1978 Canada National Olympiad
1978 Canada National Olympiad
Part of
Canada National Olympiad
Subcontests
(6)
6
1
Hide problems
Sketch the graph
Sketch the graph of
x
3
+
x
y
+
y
3
=
3
x^3 + xy + y^3 = 3
x
3
+
x
y
+
y
3
=
3
.
5
1
Hide problems
Eve and Odette play a game - checkerboard
Eve and Odette play a game on a
3
×
3
3\times 3
3
×
3
checkerboard, with black checkers and white checkers. The rules are as follows:
I.
\text{I.}
I.
They play alternately.
II.
\text{II.}
II.
A turn consists of placing one checker on an unoccupied square of the board.
III.
\text{III.}
III.
In her turn, a player may select either a white checker or a black checker and need not always use the same colour.
IV.
\text{IV.}
IV.
When the board is full, Eve obtains one point for every row, column or diagonal that has an even number of black checkers, and Odette obtains one point for very row, column or diagonal that has an odd number of black checkers.
V.
\text{V.}
V.
The player obtaining at least five of the eight points WINS.
(a)
\text{(a)}
(a)
Is a
4
−
4
4-4
4
−
4
tie possible? Explain.
(b)
\text{(b)}
(b)
Describe a winning strategy for the girl who is first to play.
4
1
Hide problems
Ratio of areas - triangle:quadrilateral
The sides
A
D
AD
A
D
and
B
C
BC
BC
of a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
are extended to meet at
E
E
E
. Let
H
H
H
and
G
G
G
be the midpoints of
B
D
BD
B
D
and
A
C
AC
A
C
, respectively. Find the ratio of the area of the triangle
E
H
G
EHG
E
H
G
to that of the quadrilateral
A
B
C
D
ABCD
A
BC
D
.
3
1
Hide problems
Find largest z from system
Determine the largest real number
z
z
z
such that \begin{align*} x + y + z = 5 \\ xy + yz + xz = 3 \end{align*} and
x
x
x
,
y
y
y
are also real.
1
1
Hide problems
Find units digit of n^2 from tens digit
Let
n
n
n
be an integer. If the tens digit of
n
2
n^2
n
2
is 7, what is the units digit of
n
2
n^2
n
2
?
2
1
Hide problems
Diophantine - squared and cubed terms
Find all pairs of
a
a
a
,
b
b
b
of positive integers satisfying the equation
2
a
2
=
3
b
3
2a^2 = 3b^3
2
a
2
=
3
b
3
.