MathDB
Problems
Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
1999 Canada National Olympiad
1999 Canada National Olympiad
Part of
Canada National Olympiad
Subcontests
(5)
4
1
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17 numbers
Suppose
a
1
,
a
2
,
⋯
,
a
8
a_1,a_2,\cdots,a_8
a
1
,
a
2
,
⋯
,
a
8
are eight distinct integers from
{
1
,
2
,
⋯
,
16
,
17
}
\{1,2,\cdots,16,17\}
{
1
,
2
,
⋯
,
16
,
17
}
. Show that there is an integer
k
>
0
k > 0
k
>
0
such that the equation
a
i
−
a
j
=
k
a_i - a_j = k
a
i
−
a
j
=
k
has at least three different solutions. Also, find a specific set of 7 distinct integers from
{
1
,
2
,
…
,
16
,
17
}
\{1,2,\ldots,16,17\}
{
1
,
2
,
…
,
16
,
17
}
such that the equation
a
i
−
a
j
=
k
a_i - a_j = k
a
i
−
a
j
=
k
does not have three distinct solutions for any
k
>
0
k > 0
k
>
0
.
2
1
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Constant length
Let
A
B
C
ABC
A
BC
be an equilateral triangle of altitude 1. A circle with radius 1 and center on the same side of
A
B
AB
A
B
as
C
C
C
rolls along the segment
A
B
AB
A
B
. Prove that the arc of the circle that is inside the triangle always has the same length.
3
1
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Find n if n = (d(n))^2
Determine all positive integers
n
n
n
with the property that
n
=
(
d
(
n
)
)
2
n = (d(n))^2
n
=
(
d
(
n
)
)
2
. Here
d
(
n
)
d(n)
d
(
n
)
denotes the number of positive divisors of
n
n
n
.
1
1
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Another [x] equation
Find all real solutions to the equation
4
x
2
−
40
⌊
x
⌋
+
51
=
0
4x^2 - 40 \lfloor x \rfloor + 51 = 0
4
x
2
−
40
⌊
x
⌋
+
51
=
0
.
5
1
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inequalities: x^2 y + y^2 z + z^2 x <= 4/27 if x + y + z = 1
Let
x
x
x
,
y
y
y
, and
z
z
z
be non-negative real numbers satisfying x \plus{} y \plus{} z \equal{} 1. Show that x^2 y \plus{} y^2 z \plus{} z^2 x \leq \frac {4}{27} and find when equality occurs.