MathDB
Problems
Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
1980 Canada National Olympiad
1980 Canada National Olympiad
Part of
Canada National Olympiad
Subcontests
(5)
5
1
Hide problems
Polyhedrons with a property of a parallelopiped
A parallelepiped has the property that all cross sections, which are parallel to any fixed face
F
F
F
, have the same perimeter as
F
F
F
. Determine whether or not any other polyhedron has this property.Typesetter's Note: I believe that proof of existence or non-existence suffices.
4
1
Hide problems
Gambling student tosses a coin, scores points
A gambling student tosses a fair coin. She gains
1
1
1
point for each head that turns up, and gains
2
2
2
points for each tail that turns up. Prove that the probability of the student scoring exactly
n
n
n
points is
1
3
⋅
(
2
+
(
−
1
2
)
n
)
\frac{1}{3}\cdot\left(2+\left(-\frac{1}{2}\right)^{n}\right)
3
1
⋅
(
2
+
(
−
2
1
)
n
)
.
3
1
Hide problems
Least perimeter of a triangle given an angle and inradius
Among all triangles having (i) a fixed angle
A
A
A
and (ii) an inscribed circle of fixed radius
r
r
r
, determine which triangle has the least minimum perimeter.
2
1
Hide problems
50 cards shuffled
The numbers from
1
1
1
to
50
50
50
are printed on cards. The cards are shuffled and then laid out face up in
5
5
5
rows of
10
10
10
cards each. The cards in each row are rearranged to make them increase from left to right. The cards in each column are then rearranged to make them increase from top to bottom. In the final arrangement, do the cards in the rows still increase from left to right?
1
1
Hide problems
Determine first and last digits
If
a
679
b
a679b
a
679
b
is the decimal expansion of a number in base
10
10
10
, such that it is divisible by
72
72
72
, determine
a
,
b
a,b
a
,
b
.