MathDB
Problems
Contests
National and Regional Contests
Brazil Contests
Brazil National Olympiad
1985 Brazil National Olympiad
1985 Brazil National Olympiad
Part of
Brazil National Olympiad
Subcontests
(5)
5
1
Hide problems
Ax + B[x] = Ay + B[y] no solutions except x = y, when A, B reals
A
,
B
A, B
A
,
B
are reals. Find a necessary and sufficient condition for
A
x
+
B
[
x
]
=
A
y
+
B
[
y
]
Ax + B[x] = Ay + B[y]
A
x
+
B
[
x
]
=
A
y
+
B
[
y
]
to have no solutions except
x
=
y
x = y
x
=
y
.
4
1
Hide problems
x^2 +ax+b=y^2+cy+d has infinitely integer solutions iff a^2 -4b = c^2 - 4d
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
are integers. Show that
x
2
+
a
x
+
b
=
y
2
+
c
y
+
d
x^2 + ax + b = y^2 + cy + d
x
2
+
a
x
+
b
=
y
2
+
cy
+
d
has infinitely many integer solutions iff
a
2
−
4
b
=
c
2
−
4
d
a^2 - 4b = c^2 - 4d
a
2
−
4
b
=
c
2
−
4
d
.
3
1
Hide problems
inequality with perimeter and sum of diagonals of inscribed quadrilateral
A convex quadrilateral is inscribed in a circle of radius
1
1
1
. Show that the its perimeter less the sum of its two diagonals lies between
0
0
0
and
2
2
2
.
2
1
Hide problems
3 out of n points exist always which an angle < = \pi / n
Given
n
n
n
points in the plane, show that we can always find three which give an angle
≤
π
/
n
\le \pi / n
≤
π
/
n
.
1
1
Hide problems
1/1 \cdot 4 + 1/4\cdot 7 + 1/7\cdot 10+ ...+1/2998 \cdot 3001, sum with hint
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
are integers with
a
d
≠
b
c
ad \ne bc
a
d
=
b
c
. Show that
1
/
(
(
a
x
+
b
)
(
c
x
+
d
)
)
1/((ax+b)(cx+d))
1/
((
a
x
+
b
)
(
c
x
+
d
))
can be written in the form
r
/
(
a
x
+
b
)
+
s
/
(
c
x
+
d
)
r/(ax+b) + s/(cx+d)
r
/
(
a
x
+
b
)
+
s
/
(
c
x
+
d
)
. Find the sum
1
/
1
⋅
4
+
1
/
4
⋅
7
+
1
/
7
⋅
10
+
.
.
.
+
1
/
2998
⋅
3001
1/1\cdot 4 + 1/4\cdot 7 + 1/7\cdot 10 + ... + 1/2998 \cdot 3001
1/1
⋅
4
+
1/4
⋅
7
+
1/7
⋅
10
+
...
+
1/2998
⋅
3001
.