MathDB
Problems
Contests
National and Regional Contests
Brazil Contests
Olympic Revenge
2003 Olympic Revenge
2003 Olympic Revenge
Part of
Olympic Revenge
Subcontests
(7)
6
1
Hide problems
f(y(f(x)-x))= f(x)/y - f(y)/x for x,y \ne 0, f(x) \ne x
Find all functions
f
:
R
∗
→
R
f:R^{*} \rightarrow R
f
:
R
∗
→
R
such that
f
(
x
)
≠
x
f(x)\not = x
f
(
x
)
=
x
and
f
(
y
(
f
(
x
)
−
x
)
)
=
f
(
x
)
y
−
f
(
y
)
x
f(y(f(x)-x))=\frac{f(x)}{y}-\frac{f(y)}{x}
f
(
y
(
f
(
x
)
−
x
))
=
y
f
(
x
)
−
x
f
(
y
)
for any
x
,
y
≠
0
x,y \not = 0
x
,
y
=
0
.
7
1
Hide problems
subset of R_{+}^{*} with m elements such nο of subsets with same sum is max
Let
X
X
X
be a subset of
R
+
∗
R_{+}^{*}
R
+
∗
with
m
m
m
elements. Find
X
X
X
such that the number of subsets with the same sum is maximum.
5
1
Hide problems
p | \sum_{(x,y) \in S}{x},p | \sum_{(x,y) \in S}{y}
Let
[
n
]
=
{
1
,
2
,
.
.
.
,
n
}
[n]=\{1,2,...,n\}
[
n
]
=
{
1
,
2
,
...
,
n
}
.Let
p
p
p
be any prime number. Find how many finite non-empty sets
S
∈
[
p
]
×
[
p
]
S\in [p] \times [p]
S
∈
[
p
]
×
[
p
]
are such that
p
∣
∑
(
x
,
y
)
∈
S
x
,
p
∣
∑
(
x
,
y
)
∈
S
y
\displaystyle \large p | \sum_{(x,y) \in S}{x},p | \sum_{(x,y) \in S}{y}
p
∣
(
x
,
y
)
∈
S
∑
x
,
p
∣
(
x
,
y
)
∈
S
∑
y
3
1
Hide problems
equal segments wanted, symmetric point , <A=60 given
Let
A
B
C
ABC
A
BC
be a triangle with
∠
B
A
C
=
6
0
∘
\angle BAC =60^\circ
∠
B
A
C
=
6
0
∘
.
A
′
A'
A
′
is the symmetric point of
A
A
A
wrt
B
C
‾
\overline{BC}
BC
.
D
D
D
is the point in
A
C
‾
\overline{AC}
A
C
such that
A
B
‾
=
A
D
‾
\overline{AB}=\overline{AD}
A
B
=
A
D
.
H
H
H
is the orthocenter of triangle
A
B
C
ABC
A
BC
.
l
l
l
is the external angle bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
.
{
M
}
=
A
′
D
‾
∩
l
\{M\}=\overline{A'D}\cap l
{
M
}
=
A
′
D
∩
l
,
{
N
}
=
C
H
‾
∩
l
\{N\}=\overline{CH} \cap l
{
N
}
=
C
H
∩
l
. Show that
A
M
‾
=
A
N
‾
\overline{AM}=\overline{AN}
A
M
=
A
N
.
2
1
Hide problems
x_{n+1}=1+\prod_{0 \leq i \leq n}{x_i}, exists prime p such p doesn't divide x_i
Let
x
n
x_n
x
n
the sequence defined by any nonnegatine integer
x
0
x_0
x
0
and
x
n
+
1
=
1
+
∏
0
≤
i
≤
n
x
i
x_{n+1}=1+\prod_{0 \leq i \leq n}{x_i}
x
n
+
1
=
1
+
∏
0
≤
i
≤
n
x
i
Show that there exists prime
p
p
p
such that
p
∤
x
n
p\not|x_n
p
∣
x
n
for any
n
n
n
.
4
1
Hide problems
max area of water resevoir in function of k bricks in Mobius Planet
In the Mobius Planet (a plane and infinite planet!, in a similar manner to the
N
×
N
N \times N
N
×
N
lattice), the Supreme King Mobius is planning to construct a water reservoir. There are some restrictions to this project: 1. There exists only
k
<
∞
k < \infty
k
<
∞
bricks. 2. These bricks will delimit a closed finite area. What is the maximum area of this resevoir in function of
k
k
k
?
1
1
Hide problems
isosceles wanted, arc midpoint, circumcircle, perp. bisector related
Let
A
B
C
ABC
A
BC
be a triangle with circumcircle
Γ
\Gamma
Γ
.
D
D
D
is the midpoint of arc
B
C
BC
BC
(this arc does not contain
A
A
A
).
E
E
E
is the common point of
B
C
BC
BC
and the perpendicular bisector of
B
D
BD
B
D
.
F
F
F
is the common point of
A
C
AC
A
C
and the parallel to
A
B
AB
A
B
containing
D
D
D
.
G
G
G
is the common point of
E
F
EF
EF
and
A
B
AB
A
B
.
H
H
H
is the common point of
G
D
GD
G
D
and
A
C
AC
A
C
. Show that
G
A
H
GAH
G
A
H
is isosceles.