MathDB
Problems
Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
1983 Canada National Olympiad
1983 Canada National Olympiad
Part of
Canada National Olympiad
Subcontests
(5)
2
1
Hide problems
Unchanged functions after transformation
For each
r
∈
R
r\in\mathbb{R}
r
∈
R
let
T
r
T_r
T
r
be the transformation of the plane that takes the point
(
x
,
y
)
(x, y)
(
x
,
y
)
into the point
(
2
r
x
;
r
2
r
x
+
2
r
y
)
(2^r x; r2^r x+2^r y)
(
2
r
x
;
r
2
r
x
+
2
r
y
)
. Let
F
F
F
be the family of all such transformations (i.e.
F
=
{
T
r
:
r
∈
R
}
F = \{T_r : r\in\mathbb{R}\}
F
=
{
T
r
:
r
∈
R
}
). Find all curves
y
=
f
(
x
)
y = f(x)
y
=
f
(
x
)
whose graphs remain unchanged by every transformation in
F
F
F
.
3
1
Hide problems
Is volume of tetrahedron dependent on area of faces?
The area of a triangle is determined by the lengths of its sides. Is the volume of a tetrahedron determined by the areas of its faces?
4
1
Hide problems
Prime p divides infinitely many 2^n - n
Prove that for every prime number
p
p
p
, there are infinitely many positive integers
n
n
n
such that
p
p
p
divides
2
n
−
n
2^n - n
2
n
−
n
.
5
1
Hide problems
GM of set = GM of the GMs of all non-zero subsets
The geometric mean (G.M.) of
k
k
k
positive integers
a
1
a_1
a
1
,
a
2
a_2
a
2
,
…
\dots
…
,
a
k
a_k
a
k
is defined to be the (positive)
k
k
k
-th root of their product. For example, the G.M. of 3, 4, 18 is 6. Show that the G.M. of a set
S
S
S
of
n
n
n
positive numbers is equal to the G.M. of the G.M.'s of all non-empty subsets of
S
S
S
.
1
1
Hide problems
Diophantine Equation with Factorials
Find all positive integers
w
w
w
,
x
x
x
,
y
y
y
and
z
z
z
which satisfy
w
!
=
x
!
+
y
!
+
z
!
w! = x! + y! + z!
w
!
=
x
!
+
y
!
+
z
!
.