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2014 BAMO
2014 BAMO
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BAMO Contests
Subcontests
(6)
1
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BAMO Collection
I'd like to note that the 2016 BAMO collection is up now! You can visit the Bay Area Mathematical Olympiad 2016 collection at the below link: [url=http://artofproblemsolving.com/community/c235604_2016_bamo]2016 BAMO Collection And you can visit the collection of BAMO collections (based by year) at the below link: [url=http://artofproblemsolving.com/community/c233906_bamo_contests]BAMO Olympiad Collection Good luck to everyone who participated!
5
1
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Chess game challenge
A chess tournament took place between
2
n
+
1
2n+1
2
n
+
1
players. Every player played every other player once, with no draws. In addition, each player had a numerical rating before the tournament began, with no two players having equal ratings. It turns out there were exactly
k
k
k
games in which the lower-rated player beat the higher-rated player. Prove that there is some player who won no less than
n
−
2
k
n-\sqrt{2k}
n
−
2
k
and no more than
n
+
2
k
n+\sqrt{2k}
n
+
2
k
games.
4
2
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Scalene triangles with circles
Let
△
A
B
C
\triangle{ABC}
△
A
BC
be a scalene triangle with the longest side
A
C
AC
A
C
. (A
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
s
c
a
l
e
n
e
t
r
i
a
n
g
l
e
<
/
s
p
a
n
>
{<span class='latex-italic'>scalene triangle</span>}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
sc
a
l
e
n
e
t
r
ian
g
l
e
<
/
s
p
an
>
has sides of different lengths.) Let
P
P
P
and
Q
Q
Q
be the points on the side
A
C
AC
A
C
such that
A
P
=
A
B
AP=AB
A
P
=
A
B
and
C
Q
=
C
B
CQ=CB
CQ
=
CB
. Thus we have a new triangle
△
B
P
Q
\triangle{BPQ}
△
BPQ
inside
△
A
B
C
\triangle{ABC}
△
A
BC
. Let
k
1
k_1
k
1
be the circle circumscribed around the triangle
△
B
P
Q
\triangle{BPQ}
△
BPQ
(that is, the circle passing through the vertices
B
,
P
,
B,P,
B
,
P
,
and
Q
Q
Q
of the triangle
△
B
P
Q
\triangle{BPQ}
△
BPQ
); and let
k
2
k_2
k
2
be the circle inscribed in triangle
△
A
B
C
\triangle{ABC}
△
A
BC
(that is, the circle inside triangle
△
A
B
C
\triangle{ABC}
△
A
BC
that is tangent to the three sides
A
B
,
B
C
AB,BC
A
B
,
BC
, and
C
A
CA
C
A
). Prove that the two circles
k
1
k_1
k
1
and
k
2
k_2
k
2
are concentric, that is, they have the same center.
Fibonacci divisibilities
Let
F
1
,
F
2
,
F
3
⋯
F_1, F_2, F_3 \cdots
F
1
,
F
2
,
F
3
⋯
be the Fibonacci sequence, the sequence of positive integers satisfying
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
.Does there exist an
n
≥
1
n \ge 1
n
≥
1
such that
F
n
F_n
F
n
is divisible by
2014
2014
2014
? Prove your answer.
3
2
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Symmetric square root equality
Suppose that for two real numbers
x
x
x
and
y
y
y
the following equality is true:
(
x
+
1
+
x
2
)
(
y
+
1
+
y
2
)
=
1
(x+ \sqrt{1+ x^2})(y+\sqrt{1+y^2})=1
(
x
+
1
+
x
2
)
(
y
+
1
+
y
2
)
=
1
Find (with proof) the value of
x
+
y
x+y
x
+
y
.
Cookie game
Amy and Bob play a game. They alternate turns, with Amy going first. At the start of the game, there are
20
20
20
cookies on a red plate and
14
14
14
on a blue plate. A legal move consists of eating two cookies taken from one plate, or moving one cookie from the red plate to the blue plate (but never from the blue plate to the red plate). The last player to make a legal move wins; in other words, if it is your turn and you cannot make a legal move, you lose, and the other player has won. Which player can guarantee that they win no matter what strategy their opponent chooses? Prove that your answer is correct.
2
1
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Pegs jumping in holes
There are
n
n
n
holes in a circle. The holes are numbered
1
,
2
,
3
1,2,3
1
,
2
,
3
and so on to
n
n
n
. In the beginning, there is a peg in every hole except for hole
1
1
1
. A peg can jump in either direction over one adjacent peg to an empty hole immediately on the other side. After a peg moves, the peg it jumped over is removed. The puzzle will be solved if all pegs disappear except for one. For example, if
n
=
4
n=4
n
=
4
the puzzle can be solved in two jumps: peg
3
3
3
jumps peg
4
4
4
to hole
1
1
1
, then peg
2
2
2
jumps the peg in
1
1
1
to hole
4
4
4
. (See illustration below, in which black circles indicate pegs and white circles are holes.) http://i.imgur.com/4ggOa8m.png[*]Can the puzzle be solved for
n
=
5
n=5
n
=
5
? [*]Can the puzzle be solved for
n
=
2014
n=2014
n
=
2014
?In each part (a) and (b) either describe a sequence of moves to solve the puzzle or explain why it is impossible to solve the puzzle.
1
1
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Cube colorings
The four bottom corners of a cube are colored red, green, blue, and purple. How many ways are there to color the top four corners of the cube so that every face has four different colored corners? Prove that your answer is correct.