MathDB
Problems
Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
2002 Canada National Olympiad
2002 Canada National Olympiad
Part of
Canada National Olympiad
Subcontests
(5)
5
1
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Functional equation
Let
N
=
{
0
,
1
,
2
,
…
}
\mathbb N = \{0,1,2,\ldots\}
N
=
{
0
,
1
,
2
,
…
}
. Determine all functions
f
:
N
→
N
f: \mathbb N \to \mathbb N
f
:
N
→
N
such that
x
f
(
y
)
+
y
f
(
x
)
=
(
x
+
y
)
f
(
x
2
+
y
2
)
xf(y) + yf(x) = (x+y) f(x^2+y^2)
x
f
(
y
)
+
y
f
(
x
)
=
(
x
+
y
)
f
(
x
2
+
y
2
)
for all
x
x
x
and
y
y
y
in
N
\mathbb N
N
.
4
1
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Pq=r
Let
Γ
\Gamma
Γ
be a circle with radius
r
r
r
. Let
A
A
A
and
B
B
B
be distinct points on
Γ
\Gamma
Γ
such that
A
B
<
3
r
AB < \sqrt{3}r
A
B
<
3
r
. Let the circle with centre
B
B
B
and radius
A
B
AB
A
B
meet
Γ
\Gamma
Γ
again at
C
C
C
. Let
P
P
P
be the point inside
Γ
\Gamma
Γ
such that triangle
A
B
P
ABP
A
BP
is equilateral. Finally, let the line
C
P
CP
CP
meet
Γ
\Gamma
Γ
again at
Q
Q
Q
. Prove that
P
Q
=
r
PQ = r
PQ
=
r
.
3
1
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Really classical inequatily from canada
Prove that for all positive real numbers
a
a
a
,
b
b
b
, and
c
c
c
,
a
3
b
c
+
b
3
c
a
+
c
3
a
b
≥
a
+
b
+
c
\frac{a^3}{bc} + \frac{b^3}{ca} + \frac{c^3}{ab} \geq a+b+c
b
c
a
3
+
c
a
b
3
+
ab
c
3
≥
a
+
b
+
c
and determine when equality occurs.
2
1
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Sum of distinct divisors
Call a positive integer
n
n
n
practical if every positive integer less than or equal to
n
n
n
can be written as the sum of distinct divisors of
n
n
n
. For example, the divisors of 6 are 1, 2, 3, and 6. Since \centerline{1={\bf 1}, ~~ 2={\bf 2}, ~~ 3={\bf 3}, ~~ 4={\bf 1}+{\bf 3}, ~~ 5={\bf 2}+ {\bf 3}, ~~ 6={\bf 6},} we see that 6 is practical. Prove that the product of two practical numbers is also practical.
1
1
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Within 9 numbers
Let
S
S
S
be a subset of
{
1
,
2
,
…
,
9
}
\{1, 2, \dots, 9\}
{
1
,
2
,
…
,
9
}
, such that the sums formed by adding each unordered pair of distinct numbers from
S
S
S
are all different. For example, the subset
{
1
,
2
,
3
,
5
}
\{1, 2, 3, 5\}
{
1
,
2
,
3
,
5
}
has this property, but
{
1
,
2
,
3
,
4
,
5
}
\{1, 2, 3, 4, 5\}
{
1
,
2
,
3
,
4
,
5
}
does not, since the pairs
{
1
,
4
}
\{1, 4\}
{
1
,
4
}
and
{
2
,
3
}
\{2, 3\}
{
2
,
3
}
have the same sum, namely 5. What is the maximum number of elements that
S
S
S
can contain?