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2013 BAMO
2013 BAMO
Part of
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Subcontests
(5)
5
1
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2013 BAMO12 p5 distinct Fibonacci numbers, D_n = P_{n+1}-P_n,
Let
F
1
,
F
2
,
F
3
,
.
.
.
F_1,F_2,F_3,...
F
1
,
F
2
,
F
3
,
...
be the Fibonacci sequence, the sequence of positive integers with
F
1
=
F
2
=
1
F_1 =F_2 =1
F
1
=
F
2
=
1
and
F
n
+
2
=
F
n
+
1
+
F
n
F_{n+2}=F_{n+1}+F_n
F
n
+
2
=
F
n
+
1
+
F
n
for all
n
≥
1
n \ge 1
n
≥
1
. A Fibonacci number is by definition a number appearing in this sequence. Let
P
1
,
P
2
,
P
3
,
.
.
.
P_1,P_2,P_3,...
P
1
,
P
2
,
P
3
,
...
be the sequence consisting of all the integers that are products of two Fibonacci numbers (not necessarily distinct) in increasing order. The first few terms are
1
,
2
,
3
,
4
,
5
,
6
,
8
,
9
,
10
,
13
,
.
.
.
1,2,3,4,5,6,8,9,10,13,...
1
,
2
,
3
,
4
,
5
,
6
,
8
,
9
,
10
,
13
,
...
since, for example
3
=
1
⋅
3
,
4
=
2
⋅
2
3 = 1 \cdot 3, 4 = 2 \cdot 2
3
=
1
⋅
3
,
4
=
2
⋅
2
, and
10
=
2
⋅
5
10 = 2 \cdot 5
10
=
2
⋅
5
. Consider the sequence
D
n
D_n
D
n
of successive differences of the
P
n
P_n
P
n
sequence, where
D
n
=
P
n
+
1
−
P
n
D_n = P_{n+1}-P_n
D
n
=
P
n
+
1
−
P
n
for
n
≥
1
n \ge 1
n
≥
1
. The first few terms of D_n are
1
,
1
,
1
,
1
,
1
,
2
,
1
,
1
,
3
,
.
.
.
1,1,1,1,1,2,1,1,3, ...
1
,
1
,
1
,
1
,
1
,
2
,
1
,
1
,
3
,
...
. Prove that every number in
D
n
D_n
D
n
is a Fibonacci number.
2
1
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Right triangle geo
Let triangle
△
A
B
C
\triangle{ABC}
△
A
BC
have a right angle at
C
C
C
, and let
M
M
M
be the midpoint of the hypotenuse
A
B
AB
A
B
. Choose a point
D
D
D
on line
B
C
BC
BC
so that angle
∠
C
D
M
\angle{CDM}
∠
C
D
M
measures
30
30
30
degrees. Prove that the segments
A
C
AC
A
C
and
M
D
MD
M
D
have equal lengths.
4
2
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Fractions and reciprocals
For a positive integer
n
>
2
n>2
n
>
2
, consider the
n
−
1
n-1
n
−
1
fractions
2
1
,
3
2
,
⋯
,
n
n
−
1
\dfrac21, \dfrac32, \cdots, \dfrac{n}{n-1}
1
2
,
2
3
,
⋯
,
n
−
1
n
The product of these fractions equals
n
n
n
, but if you reciprocate (i.e. turn upside down) some of the fractions, the product will change. Can you make the product equal 1? Find all values of
n
n
n
for which this is possible and prove that you have found them all.
2013 BAMO 12p4 min and max of median in array 10x7
Consider a rectangular array of single digits
d
i
,
j
d_{i,j}
d
i
,
j
with 10 rows and 7 columns, such that
d
i
+
1
,
j
−
d
i
,
j
d_{i+1,j}-d_{i,j}
d
i
+
1
,
j
−
d
i
,
j
is always 1 or -9 for all
1
≤
i
≤
9
1 \leq i \leq 9
1
≤
i
≤
9
and all
1
≤
j
≤
7
1 \leq j \leq 7
1
≤
j
≤
7
, as in the example below. For
1
≤
i
≤
10
1 \leq i \leq 10
1
≤
i
≤
10
, let
m
i
m_i
m
i
be the median of
d
i
,
1
d_{i,1}
d
i
,
1
, ...,
d
i
,
7
d_{i,7}
d
i
,
7
. Determine the least and greatest possible values of the mean of
m
1
m_1
m
1
,
m
2
m_2
m
2
, ...,
m
10
m_{10}
m
10
. Example: https://cdn.artofproblemsolving.com/attachments/8/a/b77c0c3aeef14f0f48d02dde830f979eca1afb.png
3
2
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Tromino tiling
Define a size-
n
n
n
tromino to be the shape you get when you remove one quadrant from a
2
n
×
2
n
2n \times 2n
2
n
×
2
n
square. In the figure below, a size-
1
1
1
tromino is on the left and a size-
2
2
2
tromino is on the right. http://i.imgur.com/2065v7Y.png We say that a shape can be tiled with size-
1
1
1
trominos if we can cover the entire area of the shape—and no excess area—with non-overlapping size-
1
1
1
trominos. For example, a
23
23
23
rectangle can be tiled with size-
1
1
1
trominos as shown below, but a
33
33
33
square cannot be tiled with size-
1
1
1
trominos. http://i.imgur.com/UBPeeRw.pnga) Can a size-
5
5
5
tromino be tiled by size-
1
1
1
trominos?b) Can a size-
2013
2013
2013
tromino be tiled by size-
1
1
1
trominos? Justify your answers.
2013 BAMO12 p3 triangle with vertices circumcenters congruent to ABC
Let
H
H
H
be the orthocenter of an acute triangle
A
B
C
ABC
A
BC
. (The orthocenter is the point at the intersection of the three altitudes. An acute triangle has all angles less than
9
0
o
90^o
9
0
o
.) Draw three circles: one passing through
A
,
B
A, B
A
,
B
, and
H
H
H
, another passing through
B
,
C
B, C
B
,
C
, and
H
H
H
, and finally, one passing through
C
,
A
C, A
C
,
A
, and
H
H
H
. Prove that the triangle whose vertices are the centers of those three circles is congruent to triangle
A
B
C
ABC
A
BC
.
1
1
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Equilateral triangular grid
How many different sets of three points in this equilateral triangular grid are the vertices of an equilateral triangle? Justify your answer.http://i.imgur.com/S6RXkYY.png