MathDB
Problems
Contests
International Contests
Rioplatense Mathematical Olympiad, Level 3
1993 Rioplatense Mathematical Olympiad, Level 3
1993 Rioplatense Mathematical Olympiad, Level 3
Part of
Rioplatense Mathematical Olympiad, Level 3
Subcontests
(6)
5
1
Hide problems
sum 1/n_k =3/17
Prove that for every integer
k
≥
2
k \ge 2
k
≥
2
there are
k
k
k
different natural numbers
n
1
n_1
n
1
,
n
2
n_2
n
2
,
.
.
.
...
...
,
n
k
n_k
n
k
such that:
1
n
1
+
1
n
2
+
.
.
.
+
1
n
k
=
3
17
\frac{1}{n_1}+\frac{1}{n_2}+...+\frac{1}{n_k}=\frac{3}{17}
n
1
1
+
n
2
1
+
...
+
n
k
1
=
17
3
4
1
Hide problems
max of f(x,y) = \sqrt{(6 -x)(3 + y^2)} + \sqrt{(11 + x)(14 - y^2)}
x
x
x
and
y
y
y
are real numbers such that
6
−
x
6 -x
6
−
x
,
3
+
y
2
3 + y^2
3
+
y
2
,
11
+
x
11 + x
11
+
x
,
14
−
y
2
14 - y^2
14
−
y
2
are greater than zero. Find the maximum of the function
f
(
x
,
y
)
=
(
6
−
x
)
(
3
+
y
2
)
+
(
11
+
x
)
(
14
−
y
2
)
.
f(x,y) = \sqrt{(6 -x)(3 + y^2)} + \sqrt{(11 + x)(14 - y^2)}.
f
(
x
,
y
)
=
(
6
−
x
)
(
3
+
y
2
)
+
(
11
+
x
)
(
14
−
y
2
)
.
2
1
Hide problems
integer on each cell of N x N+1
An integer is written in each cell of a board of
N
N
N
rows and
N
+
1
N + 1
N
+
1
columns. Prove that some columns (possibly none) can be deleted so that in each row the sum of the numbers left uncrossed out is even.
1
1
Hide problems
f(n + m) =f(n)f(m) from m,n integers >=1
Find all functions
f
f
f
defined on the integers greater than or equal to
1
1
1
that satisfy: (a) for all
n
,
f
(
n
)
n,f(n)
n
,
f
(
n
)
is a positive integer. (b)
f
(
n
+
m
)
=
f
(
n
)
f
(
m
)
f(n + m) =f(n)f(m)
f
(
n
+
m
)
=
f
(
n
)
f
(
m
)
for all
m
m
m
and
n
n
n
. (c) There exists
n
0
n_0
n
0
such that
f
(
f
(
n
0
)
)
=
[
f
(
n
0
)
]
2
f(f(n_0)) = [f(n_0)]^2
f
(
f
(
n
0
))
=
[
f
(
n
0
)
]
2
.
6
1
Hide problems
right triangle wanted, related to pentagon ABCDE with AE = ED, BC = CD
Let
A
B
C
D
E
ABCDE
A
BC
D
E
be pentagon such that
A
E
=
E
D
AE = ED
A
E
=
E
D
and
B
C
=
C
D
BC = CD
BC
=
C
D
. It is known that
∠
B
A
E
+
∠
E
D
C
+
∠
C
B
A
=
36
0
o
\angle BAE + \angle EDC + \angle CB A = 360^o
∠
B
A
E
+
∠
E
D
C
+
∠
CB
A
=
36
0
o
and that
P
P
P
is the midpoint of
A
B
AB
A
B
. Show that the triangle
E
C
P
ECP
ECP
is right.
3
1
Hide problems
equilateral with max perimeter that sides pass through 3 given points
Given three points
A
,
B
A, B
A
,
B
and
C
C
C
(not collinear) construct the equilateral triangle of greater perimeter such that each of its sides passes through one of the given points.