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Contests
National and Regional Contests
Peru Contests
Peru Iberoamerican Team Selection Test
2010 Peru Iberoamerican Team Selection Test
2010 Peru Iberoamerican Team Selection Test
Part of
Peru Iberoamerican Team Selection Test
Subcontests
(6)
P6
1
Hide problems
In how many ways can a 99-ladder be divided?
On an
n
n
n
×
n
n
n
board, the set of all squares that are located on or below the main diagonal of the board is called the
n
−
l
a
d
d
e
r
n-ladder
n
−
l
a
dd
er
. For example, the following figure shows a
3
−
l
a
d
d
e
r
3-ladder
3
−
l
a
dd
er
: [asy] draw((0,0)--(0,3)); draw((0,0)--(3,0)); draw((0,1)--(3,1)); draw((1,0)--(1,3)); draw((0,2)--(2,2)); draw((2,0)--(2,2)); draw((0,3)--(1,3)); draw((3,0)--(3,1)); [/asy] In how many ways can a
99
−
l
a
d
d
e
r
99-ladder
99
−
l
a
dd
er
be divided into some rectangles, which have their sides on grid lines, in such a way that all the rectangles have distinct areas?
P5
1
Hide problems
Show that the measure of the angle AZX does not depend on the choice of X.
The trapeze
A
B
C
D
ABCD
A
BC
D
with bases
A
B
AB
A
B
and
C
D
CD
C
D
is inscribed in a circle
Γ
\Gamma
Γ
. Let
X
X
X
be a variable point of the arc \overarc{AB} that does not contain either
C
C
C
or
D
D
D
. Let
Y
Y
Y
be the point of intersection of
A
B
AB
A
B
and
D
X
DX
D
X
, and let
Z
Z
Z
be the point of the segment
C
X
CX
CX
such that
X
Z
X
C
=
A
Y
A
B
\frac{XZ}{XC}=\frac{AY}{AB}
XC
XZ
=
A
B
A
Y
. Prove that the measure of the angle
∠
A
Z
X
\angle AZX
∠
A
ZX
does not depend on the choice of
X
X
X
.
P4
1
Hide problems
n^k − n is a multiple of 2010
Find the smallest integer
k
>
1
k > 1
k
>
1
for which
n
k
−
n
n^k-n
n
k
−
n
is a multiple of
2010
2010
2010
for every integer positive
n
n
n
.
P3
1
Hide problems
Determine the locus that determines the circumcenter of triangle ABC as L varies
Let
C
1
C_1
C
1
and
C
2
C_2
C
2
be two concentric circles with center
O
O
O
, in such a way that the radius of
C
1
C_1
C
1
is smaller than the radius of
C
2
C_2
C
2
. Let
P
P
P
be a point other than
O
O
O
that is in the interior of
C
1
C_1
C
1
, and
L
L
L
a line through
P
P
P
and intersects
C
1
C_1
C
1
at
A
A
A
and
B
B
B
. Ray
O
B
→
\overrightarrow{OB}
OB
intersects
C
2
C_2
C
2
at
C
C
C
. Determine the locus that determines the circumcenter of triangle
A
B
C
ABC
A
BC
as
L
L
L
varies.
P2
1
Hide problems
S(ab) = S(bc) = S(ca) = N
For each positive integer
k
k
k
, let
S
(
k
)
S(k)
S
(
k
)
be the sum of the digits of
k
k
k
in the decimal system. Find all positive integers N for which there exist positive integers
a
a
a
,
b
b
b
,
c
c
c
, coprime two by two, such that:
S
(
a
b
)
=
S
(
b
c
)
=
S
(
c
a
)
=
N
S(ab) = S(bc) = S(ca) = N
S
(
ab
)
=
S
(
b
c
)
=
S
(
c
a
)
=
N
.
P1
1
Hide problems
Prove that the sum of the 8^n numbers considered is a multiple of n
Let
n
n
n
be a positive integer. We know that the set
I
n
=
{
1
,
2
,
…
,
n
}
I_n = \{ 1, 2,\ldots , n\}
I
n
=
{
1
,
2
,
…
,
n
}
has exactly
2
n
2^n
2
n
subsets, so there are
8
n
8^n
8
n
ordered triples
(
A
,
B
,
C
)
(A, B, C)
(
A
,
B
,
C
)
, where
A
,
B
A, B
A
,
B
, and
C
C
C
are subsets of
I
n
I_n
I
n
. For each of these triples we consider the number
∣
A
∩
B
∩
C
∣
\mid A \cap B \cap C\mid
∣
A
∩
B
∩
C
∣
. Prove that the sum of the
8
n
8^n
8
n
numbers considered is a multiple of
n
n
n
. Clarification:
∣
Y
∣
\mid Y\mid
∣
Y
∣
denotes the number of elements in the set
Y
Y
Y
.