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Contests
National and Regional Contests
Canada Contests
Canadian Junior Mathematical Olympiad
2020 Canadian Junior Mathematical Olympiad
2020 Canadian Junior Mathematical Olympiad
Part of
Canadian Junior Mathematical Olympiad
Subcontests
(2)
2
1
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Ziquan makes a drawing in plane for art class, finite number of line segments
Ziquan makes a drawing in the plane for art class. He starts by placing his pen at the origin, and draws a series of line segments, such that the
n
t
h
n^{th}
n
t
h
line segment has length
n
n
n
. He is not allowed to lift his pen, so that the end of the
n
t
h
n^{th}
n
t
h
segment is the start of the
(
n
+
1
)
t
h
(n + 1)^{th}
(
n
+
1
)
t
h
segment. Line segments drawn are allowed to intersect and even overlap previously drawn segments. After drawing a finite number of line segments, Ziquan stops and hands in his drawing to his art teacher. He passes the course if the drawing he hands in is an
N
N
N
by
N
N
N
square, for some positive integer
N
N
N
, and he fails the course otherwise. Is it possible for Ziquan to pass the course?
1
1
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a_n >= 1/n if a_{n+1}^2 + a_{n+1} = a_n, a_1=1 , a_i>=0
Let
a
1
,
a
2
,
a
3
,
.
.
.
a_1, a_2, a_3, . . .
a
1
,
a
2
,
a
3
,
...
be a sequence of positive real numbers that satisfies
a
1
=
1
a_1 = 1
a
1
=
1
and
a
n
+
1
2
+
a
n
+
1
=
a
n
a^2_{n+1} + a_{n+1} = a_n
a
n
+
1
2
+
a
n
+
1
=
a
n
for every natural number
n
n
n
. Prove that
a
n
≥
1
n
a_n \ge \frac{1}{n}
a
n
≥
n
1
for every natural number
n
n
n
.