MathDB
Problems
Contests
National and Regional Contests
Brazil Contests
Brazil National Olympiad
2011 Brazil National Olympiad
2011 Brazil National Olympiad
Part of
Brazil National Olympiad
Subcontests
(6)
4
1
Hide problems
Does there exist 2011 numbers?
Do there exist
2011
2011
2011
positive integers
a
1
<
a
2
<
…
<
a
2011
a_1 < a_2 < \ldots < a_{2011}
a
1
<
a
2
<
…
<
a
2011
such that
gcd
(
a
i
,
a
j
)
=
a
j
−
a
i
\gcd(a_i,a_j) = a_j - a_i
g
cd
(
a
i
,
a
j
)
=
a
j
−
a
i
for any
i
i
i
,
j
j
j
such that
1
≤
i
<
j
≤
2011
1 \le i < j \le 2011
1
≤
i
<
j
≤
2011
?
1
1
Hide problems
Sum of squares of digits
We call a number pal if it doesn't have a zero digit and the sum of the squares of the digits is a perfect square. For example,
122
122
122
and
34
34
34
are pal but
304
304
304
and
12
12
12
are not pal. Prove that there exists a pal number with
n
n
n
digits,
n
>
1
n > 1
n
>
1
.
3
1
Hide problems
Inequality with convex pentagons
Prove that, for all convex pentagons
P
1
P
2
P
3
P
4
P
5
P_1 P_2 P_3 P_4 P_5
P
1
P
2
P
3
P
4
P
5
with area 1, there are indices
i
i
i
and
j
j
j
(assume
P
7
=
P
2
P_7 = P_2
P
7
=
P
2
and
P
6
=
P
1
P_6 = P_1
P
6
=
P
1
) such that:
Area of
△
P
i
P
i
+
1
P
i
+
2
≤
5
−
5
10
≤
Area of
△
P
j
P
j
+
1
P
j
+
2
\text{Area of} \ \triangle P_i P_{i+1} P_{i+2} \le \frac{5 - \sqrt 5}{10} \le \text{Area of} \ \triangle P_j P_{j+1} P_{j+2}
Area of
△
P
i
P
i
+
1
P
i
+
2
≤
10
5
−
5
≤
Area of
△
P
j
P
j
+
1
P
j
+
2
5
1
Hide problems
Concurrence of angle bisectors
Let
A
B
C
ABC
A
BC
be an acute triangle and
H
H
H
is orthocenter. Let
D
D
D
be the intersection of
B
H
BH
B
H
and
A
C
AC
A
C
and
E
E
E
be the intersection of
C
H
CH
C
H
and
A
B
AB
A
B
. The circumcircle of
A
D
E
ADE
A
D
E
cuts the circumcircle of
A
B
C
ABC
A
BC
at
F
≠
A
F \neq A
F
=
A
. Prove that the angle bisectors of
∠
B
F
C
\angle BFC
∠
BFC
and
∠
B
H
C
\angle BHC
∠
B
H
C
concur at a point on
B
C
.
BC.
BC
.
2
1
Hide problems
Distributions of stickers
33 friends are collecting stickers for a 2011-sticker album. A distribution of stickers among the 33 friends is incomplete when there is a sticker that no friend has. Determine the least
m
m
m
with the following property: every distribution of stickers among the 33 friends such that, for any two friends, there are at least
m
m
m
stickers both don't have, is incomplete.
6
1
Hide problems
Hard inequality
Let
a
1
,
a
2
,
a
3
,
.
.
.
a
2011
a_{1}, a_{2}, a_{3}, ... a_{2011}
a
1
,
a
2
,
a
3
,
...
a
2011
be nonnegative reals with sum
2011
2
\frac{2011}{2}
2
2011
, prove :
∣
∏
c
y
c
(
a
n
−
a
n
+
1
)
∣
=
∣
(
a
1
−
a
2
)
(
a
2
−
a
3
)
.
.
.
(
a
2011
−
a
1
)
∣
≤
3
3
16
.
|\prod_{cyc} (a_{n} - a_{n+1})| = |(a_{1} - a_{2})(a_{2} - a_{3})...(a_{2011}-a_{1})| \le \frac{3 \sqrt3}{16}.
∣
∏
cyc
(
a
n
−
a
n
+
1
)
∣
=
∣
(
a
1
−
a
2
)
(
a
2
−
a
3
)
...
(
a
2011
−
a
1
)
∣
≤
16
3
3
.