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Contests
National and Regional Contests
Canada Contests
Canadian Junior Mathematical Olympiad
2023 Canadian Junior Mathematical Olympiad
2023 Canadian Junior Mathematical Olympiad
Part of
Canadian Junior Mathematical Olympiad
Subcontests
(2)
1
1
Hide problems
s_{i+1} = |s_i - s_{i-1}|
Let
a
a
a
and
b
b
b
be non-negative integers. Consider a sequence
s
1
s_1
s
1
,
s
2
s_2
s
2
,
s
3
s_3
s
3
,
.
.
.
. . .
...
such that
s
1
=
a
s_1 = a
s
1
=
a
,
s
2
=
b
s_2 = b
s
2
=
b
, and
s
i
+
1
=
∣
s
i
−
s
i
−
1
∣
s_{i+1} = |s_i - s_{i-1}|
s
i
+
1
=
∣
s
i
−
s
i
−
1
∣
for
i
≥
2
i \ge 2
i
≥
2
. Prove that there is some
i
i
i
for which
s
i
=
0
s_i = 0
s
i
=
0
.
2
1
Hide problems
Proving 4 points concyclic in a right triangle
An acute triangle is a triangle that has all angles less that
9
0
∘
90^{\circ}
9
0
∘
(
9
0
∘
90^{\circ}
9
0
∘
is a Right Angle). Let
A
B
C
ABC
A
BC
be a right-angled triangle with
∠
A
C
B
=
9
0
∘
.
\angle ACB =90^{\circ}.
∠
A
CB
=
9
0
∘
.
Let
C
D
CD
C
D
be the altitude from
C
C
C
to
A
B
,
AB,
A
B
,
and let
E
E
E
be the intersection of the angle bisector of
∠
A
C
D
\angle ACD
∠
A
C
D
with
A
D
.
AD.
A
D
.
Let
E
F
EF
EF
be the altitude from
E
E
E
to
B
C
.
BC.
BC
.
Prove that the circumcircle of
B
E
F
BEF
BEF
passes through the midpoint of
C
E
.
CE.
CE
.