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Rioplatense Mathematical Olympiad, Level 3
1990 Rioplatense Mathematical Olympiad, Level 3
1990 Rioplatense Mathematical Olympiad, Level 3
Part of
Rioplatense Mathematical Olympiad, Level 3
Subcontests
(3)
1
1
Hide problems
[x/10]=[x/11]+1
How many positive integer solutions does the equation have
⌊
x
10
⌋
=
⌊
x
11
⌋
+
1
?
\left\lfloor\frac{x}{10}\right\rfloor= \left\lfloor\frac{x}{11}\right\rfloor + 1?
⌊
10
x
⌋
=
⌊
11
x
⌋
+
1
?
(
⌊
x
⌋
\lfloor x \rfloor
⌊
x
⌋
denotes the integer part of
x
x
x
, for example
⌊
2
⌋
=
2
\lfloor 2\rfloor = 2
⌊
2
⌋
=
2
,
⌊
π
⌋
=
3
\lfloor \pi\rfloor = 3
⌊
π
⌋
=
3
,
⌊
2
⌋
=
1
\lfloor \sqrt2 \rfloor =1
⌊
2
⌋
=
1
)
2
1
Hide problems
no of people meeting odd number of others is even
Some of the people attending a meeting greet each other. Let
n
n
n
be the number of people who greet an odd number of people. Prove that
n
n
n
is even.
3
1
Hide problems
trapezium with one base double of the other, 4 perpendiculars, collinear wanted
Let
A
B
C
D
ABCD
A
BC
D
be a trapezium with bases
A
B
AB
A
B
and
C
D
CD
C
D
such that
A
B
=
2
C
D
AB = 2 CD
A
B
=
2
C
D
. From
A
A
A
the line
r
r
r
is drawn perpendicular to
B
C
BC
BC
and from
B
B
B
the line
t
t
t
is drawn perpendicular to
A
D
AD
A
D
. Let
P
P
P
be the intersection point of
r
r
r
and
t
t
t
. From
C
C
C
the line
s
s
s
is drawn perpendicular to
B
C
BC
BC
and from
D
D
D
the line
u
u
u
perpendicular to
A
D
AD
A
D
. Let
Q
Q
Q
be the intersection point of
s
s
s
and
u
u
u
. If
R
R
R
is the intersection point of the diagonals of the trapezium, prove that points
P
,
Q
P, Q
P
,
Q
and
R
R
R
are collinear.