MathDB
Problems
Contests
National and Regional Contests
Brazil Contests
Brazil National Olympiad
2000 Brazil National Olympiad
2000 Brazil National Olympiad
Part of
Brazil National Olympiad
Subcontests
(6)
3
1
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Find the smallest n such that f(1)+...+f(n) >=12345
Define
f
f
f
on the positive integers by
f
(
n
)
=
k
2
+
k
+
1
f(n) = k^2 + k + 1
f
(
n
)
=
k
2
+
k
+
1
, where
n
=
2
k
(
2
l
+
1
)
n=2^k(2l+1)
n
=
2
k
(
2
l
+
1
)
for some
k
,
l
k,l
k
,
l
nonnegative integers. Find the smallest
n
n
n
such that
f
(
1
)
+
f
(
2
)
+
.
.
.
+
f
(
n
)
≥
123456
f(1) + f(2) + ... + f(n) \geq 123456
f
(
1
)
+
f
(
2
)
+
...
+
f
(
n
)
≥
123456
.
6
1
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Cutting the cube
Let it be is a wooden unit cube. We cut along every plane which is perpendicular to the segment joining two distinct vertices and bisects it. How many pieces do we get?
4
1
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Stop-and-go problem
An infinite road has traffic lights at intervals of 1500m. The lights are all synchronised and are alternately green for
3
2
\frac 32
2
3
minutes and red for 1 minute. For which
v
v
v
can a car travel at a constant speed of
v
v
v
m/s without ever going through a red light?
2
1
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A caracterization of almost-perfect numbers
Let
s
(
n
)
s(n)
s
(
n
)
be the sum of all positive divisors of
n
n
n
, so
s
(
6
)
=
12
s(6) = 12
s
(
6
)
=
12
. We say
n
n
n
is almost perfect if
s
(
n
)
=
2
n
−
1
s(n) = 2n - 1
s
(
n
)
=
2
n
−
1
. Let
m
o
d
(
n
,
k
)
\mod(n, k)
mod
(
n
,
k
)
denote the residue of
n
n
n
modulo
k
k
k
(in other words, the remainder of dividing
n
n
n
by
k
k
k
). Put
t
(
n
)
=
m
o
d
(
n
,
1
)
+
m
o
d
(
n
,
2
)
+
⋯
+
m
o
d
(
n
,
n
)
t(n) = \mod(n, 1) + \mod(n, 2) + \cdots + \mod(n, n)
t
(
n
)
=
mod
(
n
,
1
)
+
mod
(
n
,
2
)
+
⋯
+
mod
(
n
,
n
)
. Show that
n
n
n
is almost perfect if and only if
t
(
n
)
=
t
(
n
−
1
)
t(n) = t(n-1)
t
(
n
)
=
t
(
n
−
1
)
.
1
1
Hide problems
Paper-folding trisectors
A rectangular piece of paper has top edge
A
D
AD
A
D
. A line
L
L
L
from
A
A
A
to the bottom edge makes an angle
x
x
x
with the line
A
D
AD
A
D
. We want to trisect
x
x
x
. We take
B
B
B
and
C
C
C
on the vertical ege through
A
A
A
such that
A
B
=
B
C
AB = BC
A
B
=
BC
. We then fold the paper so that
C
C
C
goes to a point
C
′
C'
C
′
on the line
L
L
L
and
A
A
A
goes to a point
A
′
A'
A
′
on the horizontal line through
B
B
B
. The fold takes
B
B
B
to
B
′
B'
B
′
. Show that
A
A
′
AA'
A
A
′
and
A
B
′
AB'
A
B
′
are the required trisectors.
5
1
Hide problems
Find functions: The dist. betw.a & b of X is defined by.
Let
X
X
X
the set of all sequences
{
a
1
,
a
2
,
…
,
a
2000
}
\{a_1, a_2,\ldots , a_{2000}\}
{
a
1
,
a
2
,
…
,
a
2000
}
, such that each of the first 1000 terms is 0, 1 or 2, and each of the remaining terms is 0 or 1. The distance between two members
a
a
a
and
b
b
b
of
X
X
X
is defined as the number of
i
i
i
for which
a
i
a_i
a
i
and
b
i
b_i
b
i
are different. Find the number of functions
f
:
X
→
X
f : X \to X
f
:
X
→
X
which preserve the distance.