MathDB
Problems
Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
1995 Canada National Olympiad
1995 Canada National Olympiad
Part of
Canada National Olympiad
Subcontests
(5)
5
1
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Sequence - Proof of 0-value term
u
u
u
is a real parameter such that
0
<
u
<
1
0<u<1
0
<
u
<
1
. For
0
≤
x
≤
u
0\le x \le u
0
≤
x
≤
u
,
f
(
x
)
=
0
f(x)=0
f
(
x
)
=
0
. For
u
≤
x
≤
n
u\le x \le n
u
≤
x
≤
n
,
f
(
x
)
=
1
−
(
u
x
+
(
1
−
u
)
(
1
−
x
)
)
2
f(x)=1-\left(\sqrt{ux}+\sqrt{(1-u)(1-x)}\right)^2
f
(
x
)
=
1
−
(
ux
+
(
1
−
u
)
(
1
−
x
)
)
2
. The sequence
{
u
n
}
\{u_n\}
{
u
n
}
is define recursively as follows:
u
1
=
f
(
1
)
u_1=f(1)
u
1
=
f
(
1
)
and
u
n
=
f
(
u
n
−
1
)
u_n=f(u_{n-1})
u
n
=
f
(
u
n
−
1
)
∀
n
∈
N
,
n
≠
1
\forall n\in \mathbb{N}, n\neq 1
∀
n
∈
N
,
n
=
1
. Show that there exists a positive integer
k
k
k
for which
u
k
=
0
u_k=0
u
k
=
0
.
4
1
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Diophantine - Infinitely many solutions
Let
n
n
n
be a constant positive integer. Show that for only non-negative integers
k
k
k
, the Diophantine equation
∑
i
=
1
n
x
i
3
=
y
3
k
+
2
\sum_{i=1 }^{n}{ x_i ^3}=y^{3k+2}
∑
i
=
1
n
x
i
3
=
y
3
k
+
2
has infinitely many solutions in the positive integers
x
i
,
y
x_i, y
x
i
,
y
.
3
1
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Boomerang
Define a boomerang as a quadrilateral whose opposite sides do not intersect and one of whose internal angles is greater than
18
0
∘
180^{\circ}
18
0
∘
. Let
C
C
C
be a convex polygon with
s
s
s
sides. The interior region of
C
C
C
is the union of
q
q
q
quadrilaterals, none of whose interiors overlap each other.
b
b
b
of these quadrilaterals are boomerangs. Show that
q
≥
b
+
s
−
2
2
q\ge b+\frac{s-2}{2}
q
≥
b
+
2
s
−
2
.
2
1
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Exponential Inequality
Let
{
a
,
b
,
c
}
∈
R
+
\{a,b,c\}\in \mathbb{R}^{+}
{
a
,
b
,
c
}
∈
R
+
. Prove that
a
a
b
b
c
c
≥
(
a
b
c
)
a
+
b
+
c
3
a^a b^b c^c \ge (abc)^{\frac{a+b+c}{3}}
a
a
b
b
c
c
≥
(
ab
c
)
3
a
+
b
+
c
.
1
1
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Evaluate Sum
Let
f
(
x
)
=
9
x
9
x
+
3
f(x)=\frac{9^x}{9^x + 3}
f
(
x
)
=
9
x
+
3
9
x
. Evaluate
∑
i
=
1
1995
f
(
i
1996
)
\sum_{i=1}^{1995}{f\left(\frac{i}{1996}\right)}
∑
i
=
1
1995
f
(
1996
i
)
.