MathDB
Problems
Contests
International Contests
IberoAmerican
2005 IberoAmerican
2005 IberoAmerican
Part of
IberoAmerican
Subcontests
(6)
6
1
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Counting: colorings, circles and 2n points on a line
Let
n
n
n
be a fixed positive integer. The points
A
1
A_1
A
1
,
A
2
A_2
A
2
,
…
\ldots
…
,
A
2
n
A_{2n}
A
2
n
are on a straight line. Color each point blue or red according to the following procedure: draw
n
n
n
pairwise disjoint circumferences, each with diameter
A
i
A
j
A_iA_j
A
i
A
j
for some
i
≠
j
i \neq j
i
=
j
and such that every point
A
k
A_k
A
k
belongs to exactly one circumference. Points in the same circumference must be of the same color. Determine the number of ways of coloring these
2
n
2n
2
n
points when we vary the
n
n
n
circumferences and the distribution of the colors.
5
1
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Prove that a line passes through the orthocenter
Let
O
O
O
be the circumcenter of acutangle triangle
A
B
C
ABC
A
BC
and let
A
1
A_1
A
1
be some point in the smallest arc
B
C
BC
BC
of the circumcircle of
A
B
C
ABC
A
BC
. Let
A
2
A_2
A
2
and
A
3
A_3
A
3
points on sides
A
B
AB
A
B
and
A
C
AC
A
C
, respectively, such that
∠
B
A
1
A
2
=
∠
O
A
C
\angle BA_1A_2 = \angle OAC
∠
B
A
1
A
2
=
∠
O
A
C
and
∠
C
A
1
A
3
=
∠
O
A
B
\angle CA_1A_3 = \angle OAB
∠
C
A
1
A
3
=
∠
O
A
B
. Prove that the line
A
2
A
3
A_2A_3
A
2
A
3
passes through the orthocenter of
A
B
C
ABC
A
BC
.
4
1
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Sum of remainders
Denote by
a
m
o
d
b
a \bmod b
a
mod
b
the remainder of the euclidean division of
a
a
a
by
b
b
b
. Determine all pairs of positive integers
(
a
,
p
)
(a,p)
(
a
,
p
)
such that
p
p
p
is prime and
a
m
o
d
p
+
a
m
o
d
2
p
+
a
m
o
d
3
p
+
a
m
o
d
4
p
=
a
+
p
.
a \bmod p + a\bmod 2p + a\bmod 3p + a\bmod 4p = a + p.
a
mod
p
+
a
mod
2
p
+
a
mod
3
p
+
a
mod
4
p
=
a
+
p
.
3
1
Hide problems
\sum{i^{-p}} = 0 mod p^3
Let
p
>
3
p > 3
p
>
3
be a prime. Prove that if
∑
i
=
1
p
−
1
1
i
p
=
n
m
,
\sum_{i=1 }^{p-1}{1\over i^p} = {n\over m},
i
=
1
∑
p
−
1
i
p
1
=
m
n
,
with \gdc(n,m) = 1, then
p
3
p^3
p
3
divides
n
n
n
.
2
1
Hide problems
Jumping flea
A flea jumps in a straight numbered line. It jumps first from point
0
0
0
to point
1
1
1
. Afterwards, if its last jump was from
A
A
A
to
B
B
B
, then the next jump is from
B
B
B
to one of the points
B
+
(
B
−
A
)
−
1
B + (B - A) - 1
B
+
(
B
−
A
)
−
1
,
B
+
(
B
−
A
)
B + (B - A)
B
+
(
B
−
A
)
,
B
+
(
B
−
A
)
+
1
B + (B-A) + 1
B
+
(
B
−
A
)
+
1
. Prove that if the flea arrived twice at the point
n
n
n
,
n
n
n
positive integer, then it performed at least
⌈
2
n
⌉
\lceil 2\sqrt n\rceil
⌈
2
n
⌉
jumps.
1
1
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Three-variable system
Determine all triples of real numbers
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
such that \begin{eqnarray*} xyz &=& 8 \\ x^2y + y^2z + z^2x &=& 73 \\ x(y-z)^2 + y(z-x)^2 + z(x-y)^2 &=& 98 . \end{eqnarray*}