MathDB
Problems
Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
1986 Canada National Olympiad
1986 Canada National Olympiad
Part of
Canada National Olympiad
Subcontests
(5)
5
1
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Infinitely many terms divisible by 1986
Let
u
1
u_1
u
1
,
u
2
u_2
u
2
,
u
3
u_3
u
3
,
…
\dots
…
be a sequence of integers satisfying the recurrence relation
u
n
+
2
=
u
n
+
1
2
−
u
n
u_{n + 2} = u_{n + 1}^2 - u_n
u
n
+
2
=
u
n
+
1
2
−
u
n
. Suppose
u
1
=
39
u_1 = 39
u
1
=
39
and
u
2
=
45
u_2 = 45
u
2
=
45
. Prove that 1986 divides infinitely many terms of the sequence.
4
1
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Divisibility related to powers
For all positive integers
n
n
n
and
k
k
k
, define
F
(
n
,
k
)
=
∑
r
=
1
n
r
2
k
−
1
F(n,k) = \sum_{r = 1}^n r^{2k - 1}
F
(
n
,
k
)
=
∑
r
=
1
n
r
2
k
−
1
. Prove that
F
(
n
,
1
)
F(n,1)
F
(
n
,
1
)
divides
F
(
n
,
k
)
F(n,k)
F
(
n
,
k
)
.
3
1
Hide problems
Constant chord implies constant angle
A chord
S
T
ST
ST
of constant length slides around a semicircle with diameter
A
B
AB
A
B
.
M
M
M
is the midpoint of
S
T
ST
ST
and
P
P
P
is the foot of the perpendicular from
S
S
S
to
A
B
AB
A
B
. Prove that
∠
S
P
M
\angle SPM
∠
SPM
is constant for all positions of
S
T
ST
ST
.
2
1
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A Mathlon = competition with M events
A Mathlon is a competition in which there are
M
M
M
athletic events. Such a competition was held in which only
A
A
A
,
B
B
B
, and
C
C
C
participated. In each event
p
1
p_1
p
1
points were awarded for first place,
p
2
p_2
p
2
for second and
p
3
p_3
p
3
for third, where
p
1
>
p
2
>
p
3
>
0
p_1 > p_2 > p_3 > 0
p
1
>
p
2
>
p
3
>
0
and
p
1
p_1
p
1
,
p
2
p_2
p
2
,
p
3
p_3
p
3
are integers. The final scores for
A
A
A
was 22, for
B
B
B
was 9 and for
C
C
C
was also 9.
B
B
B
won the 100 metres. What is the value of
M
M
M
and who was second in the high jump?
1
1
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Simple cevian-related problem
In the diagram line segments
A
B
AB
A
B
and
C
D
CD
C
D
are of length 1 while angles
A
B
C
ABC
A
BC
and
C
B
D
CBD
CB
D
are
9
0
∘
90^\circ
9
0
∘
and
3
0
∘
30^\circ
3
0
∘
respectively. Find
A
C
AC
A
C
.[asy] import geometry; import graph;unitsize(1.5 cm);pair A, B, C, D;B = (0,0); D = (3,0); A = 2*dir(120); C = extension(B,dir(30),A,D);draw(A--B--D--cycle); draw(B--C); draw(arc(B,0.5,0,30));label("
A
A
A
", A, NW); label("
B
B
B
", B, SW); label("
C
C
C
", C, NE); label("
D
D
D
", D, SE); label("
3
0
∘
30^\circ
3
0
∘
", (0.8,0.2)); label("
9
0
∘
90^\circ
9
0
∘
", (0.1,0.5));perpendicular(B,NE,C-B); [/asy]