MathDB
Problems
Contests
National and Regional Contests
Brazil Contests
Brazil National Olympiad
2006 Brazil National Olympiad
2006 Brazil National Olympiad
Part of
Brazil National Olympiad
Subcontests
(6)
6
1
Hide problems
Piraldo's soccer matches
Professor Piraldo takes part in soccer matches with a lot of goals and judges a match in his own peculiar way. A match with score of
m
m
m
goals to
n
n
n
goals,
m
≥
n
m\geq n
m
≥
n
, is tough when
m
≤
f
(
n
)
m\leq f(n)
m
≤
f
(
n
)
, where
f
(
n
)
f(n)
f
(
n
)
is defined by
f
(
0
)
=
0
f(0) = 0
f
(
0
)
=
0
and, for
n
≥
1
n \geq 1
n
≥
1
,
f
(
n
)
=
2
n
−
f
(
r
)
+
r
f(n) = 2n-f(r)+r
f
(
n
)
=
2
n
−
f
(
r
)
+
r
, where
r
r
r
is the largest integer such that
r
<
n
r < n
r
<
n
and
f
(
r
)
≤
n
f(r) \leq n
f
(
r
)
≤
n
. Let
ϕ
=
1
+
5
2
\phi ={1+\sqrt 5\over 2}
ϕ
=
2
1
+
5
. Prove that a match with score of
m
m
m
goals to
n
n
n
,
m
≥
n
m\geq n
m
≥
n
, is tough if
m
≤
ϕ
n
m\leq \phi n
m
≤
ϕ
n
and is not tough if
m
≥
ϕ
n
+
1
m \geq \phi n+1
m
≥
ϕ
n
+
1
.
5
1
Hide problems
2006 concurrent lines
Let
P
P
P
be a convex
2006
2006
2006
-gon. The
1003
1003
1003
diagonals connecting opposite vertices and the
1003
1003
1003
lines connecting the midpoints of opposite sides are concurrent, that is, all
2006
2006
2006
lines have a common point. Prove that the opposite sides of
P
P
P
are parallel and congruent.
4
1
Hide problems
Bold numbers
A positive integer is bold iff it has
8
8
8
positive divisors that sum up to
3240
3240
3240
. For example,
2006
2006
2006
is bold because its
8
8
8
positive divisors,
1
1
1
,
2
2
2
,
17
17
17
,
34
34
34
,
59
59
59
,
118
118
118
,
1003
1003
1003
and
2006
2006
2006
, sum up to
3240
3240
3240
. Find the smallest positive bold number.
3
1
Hide problems
Yet another (but nice!) functional equation
Find all functions
f
:
R
→
R
f\colon \mathbb{R}\to \mathbb{R}
f
:
R
→
R
such that
f
(
x
f
(
y
)
+
f
(
x
)
)
=
2
f
(
x
)
+
x
y
f(xf(y)+f(x)) = 2f(x)+xy
f
(
x
f
(
y
)
+
f
(
x
))
=
2
f
(
x
)
+
x
y
for every reals
x
,
y
x,y
x
,
y
.
2
1
Hide problems
Lots of isosceles triangles
Let
n
n
n
be an integer,
n
≥
3
n \geq 3
n
≥
3
. Let
f
(
n
)
f(n)
f
(
n
)
be the largest number of isosceles triangles whose vertices belong to some set of
n
n
n
points in the plane without three colinear points. Prove that there exists positive real constants
a
a
a
and
b
b
b
such that
a
n
2
<
f
(
n
)
<
b
n
2
an^{2}< f(n) < bn^{2}
a
n
2
<
f
(
n
)
<
b
n
2
for every integer
n
n
n
,
n
≥
3
n \geq 3
n
≥
3
.
1
1
Hide problems
Isosceles triangles
Let
A
B
C
ABC
A
BC
be a triangle. The internal bisector of
∠
B
\angle B
∠
B
meets
A
C
AC
A
C
in
P
P
P
and
I
I
I
is the incenter of
A
B
C
ABC
A
BC
. Prove that if
A
P
+
A
B
=
C
B
AP+AB = CB
A
P
+
A
B
=
CB
, then
A
P
I
API
A
P
I
is an isosceles triangle.