MathDB
Problems
Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
2008 Canada National Olympiad
2008 Canada National Olympiad
Part of
Canada National Olympiad
Subcontests
(5)
5
1
Hide problems
Problem on a chessboard
A self-avoiding rook walk on a chessboard (a rectangular grid of unit squares) is a path traced by a sequence of moves parallel to an edge of the board from one unit square to another, such that each begins where the previous move ended and such that no move ever crosses a square that has previously been crossed, i.e., the rook's path is non-self-intersecting. Let
R
(
m
,
n
)
R(m, n)
R
(
m
,
n
)
be the number of self-avoiding rook walks on an
m
×
n
m \times n
m
×
n
(
m
m
m
rows,
n
n
n
columns) chessboard which begin at the lower-left corner and end at the upper-left corner. For example, R(m, 1) \equal{} 1 for all natural numbers
m
m
m
; R(2, 2) \equal{} 2; R(3, 2) \equal{} 4; R(3, 3) \equal{} 11. Find a formula for
R
(
3
,
n
)
R(3, n)
R
(
3
,
n
)
for each natural number
n
n
n
.
4
1
Hide problems
Function on the natural numbers
Determine all functions
f
f
f
defined on the natural numbers that take values among the natural numbers for which (f(n))^p \equiv n {\rm mod}\; f(p) for all
n
∈
N
n \in {\bf N}
n
∈
N
and all prime numbers
p
p
p
.
3
1
Hide problems
An inequality with condition a+b+c=1.
Let
a
a
a
,
b
b
b
,
c
c
c
be positive real numbers for which a \plus{} b \plus{} c \equal{} 1. Prove that {{a\minus{}bc}\over{a\plus{}bc}} \plus{} {{b\minus{}ca}\over{b\plus{}ca}} \plus{} {{c\minus{}ab}\over{c\plus{}ab}} \leq {3 \over 2}.
2
1
Hide problems
Determine all functions
Determine all functions
f
f
f
defined on the set of rational numbers that take rational values for which f(2f(x) \plus{} f(y)) \equal{} 2x \plus{} y, for each
x
x
x
and
y
y
y
.
1
1
Hide problems
Segments bisect the area of quadrilateral.
A
B
C
D
ABCD
A
BC
D
is a convex quadrilateral for which
A
B
AB
A
B
is the longest side. Points
M
M
M
and
N
N
N
are located on sides
A
B
AB
A
B
and
B
C
BC
BC
respectively, so that each of the segments
A
N
AN
A
N
and
C
M
CM
CM
divides the quadrilateral into two parts of equal area. Prove that the segment
M
N
MN
MN
bisects the diagonal
B
D
BD
B
D
.