MathDB
Problems
Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
1992 Canada National Olympiad
1992 Canada National Olympiad
Part of
Canada National Olympiad
Subcontests
(5)
5
1
Hide problems
Deck of Card
A deck of 2n\plus{}1 cards consists of a joker and, for each number between 1 and
n
n
n
inclusive, two cards marked with that number. The 2n\plus{}1 cards are placed in a row, with the joker in the middle. For each
k
k
k
with
1
≤
k
≤
n
,
1 \leq k \leq n,
1
≤
k
≤
n
,
the two cards numbered
k
k
k
have exactly k\minus{}1 cards between them. Determine all the values of
n
n
n
not exceeding 10 for which this arrangement is possible. For which values of
n
n
n
is it impossible?
4
1
Hide problems
Equation
Solve the equation x^2 \plus{} \frac{x^2}{(x\plus{}1)^2} \equal{} 3
3
1
Hide problems
Locations of U and V
In the diagram,
A
B
C
D
ABCD
A
BC
D
is a square, with
U
U
U
and
V
V
V
interior points of the sides
A
B
AB
A
B
and
C
D
CD
C
D
respectively. Determine all the possible ways of selecting
U
U
U
and
V
V
V
so as to maximize the area of the quadrilateral
P
U
Q
V
PUQV
P
U
Q
V
. http://i250.photobucket.com/albums/gg265/geometry101/CMO1992Number3.jpg
2
1
Hide problems
Inequality
For
x
,
y
,
z
≥
0
,
x,y,z \geq 0,
x
,
y
,
z
≥
0
,
establish the inequality x(x\minus{}z)^2 \plus{} y(y\minus{}z)^2 \geq (x\minus{}z)(y\minus{}z)(x\plus{}y\minus{}z) and determine when equality holds.
1
1
Hide problems
Natural Number and Prime
Prove that the product of the first
n
n
n
natural numbers is divisible by the sum of the first
n
n
n
natural numbers if and only if n\plus{}1 is not an odd prime.