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Problems
Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
1990 Canada National Olympiad
1990 Canada National Olympiad
Part of
Canada National Olympiad
Subcontests
(5)
1
1
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Determine all pairs (n,k) - [Canada 1990 - P1]
A competition involving
n
≥
2
n\ge 2
n
≥
2
players was held over
k
k
k
days. In each day, the players received scores of
1
,
2
,
3
,
…
,
n
1,2,3,\ldots , n
1
,
2
,
3
,
…
,
n
points with no players receiving the same score. At the end of the
k
k
k
days, it was found that each player had exactly
26
26
26
points in total. Determine all pairs
(
n
,
k
)
(n,k)
(
n
,
k
)
for which this is possible.
5
1
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Show that 0 ≤ f(n+1) - f(n) ≤ 1 and find n s.t. f(n) = 1025
The function
f
:
N
→
R
f : \mathbb N \to \mathbb R
f
:
N
→
R
satisfies
f
(
1
)
=
1
,
f
(
2
)
=
2
f(1) = 1, f(2) = 2
f
(
1
)
=
1
,
f
(
2
)
=
2
and
f
(
n
+
2
)
=
f
(
n
+
2
−
f
(
n
+
1
)
)
+
f
(
n
+
1
−
f
(
n
)
)
.
f (n+2) = f(n+2 - f(n+1) ) + f(n+1 - f(n) ).
f
(
n
+
2
)
=
f
(
n
+
2
−
f
(
n
+
1
))
+
f
(
n
+
1
−
f
(
n
))
.
Show that
0
≤
f
(
n
+
1
)
−
f
(
n
)
≤
1
0 \leq f(n+1) - f(n) \leq 1
0
≤
f
(
n
+
1
)
−
f
(
n
)
≤
1
. Find all
n
n
n
for which
f
(
n
)
=
1025
f(n) = 1025
f
(
n
)
=
1025
.
3
1
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Sum of lengths of each pair of opposite sides of q is equal
The feet of the perpendiculars from the intersection point of the diagonals of a convex cyclic quadrilateral to the sides form a quadrilateral
q
q
q
. Show that the sum of the lengths of each pair of opposite sides of
q
q
q
is equal.
2
1
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Find the probablity - [Canada 1990 - P1]
n
(
n
+
1
)
2
\frac{n(n + 1)}{2}
2
n
(
n
+
1
)
distinct numbers are arranged at random into
n
n
n
rows. The first row has
1
1
1
number, the second has
2
2
2
numbers, the third has
3
3
3
numbers and so on. Find the probability that the largest number in each row is smaller than the largest number in each row with more numbers.
4
1
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particle
A particle can travel at speeds up to
2
m
s
\frac{2m}{s}
s
2
m
along the
x
x
x
-axis, and up to
1
m
s
\frac{1m}{s}
s
1
m
elsewhere in the plane. Provide a labelled sketch of the region which can be reached within one second by the particle starting at the origin.