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2015 SDMO (High School)
2015 SDMO (High School)
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SDMO (High School)
Subcontests
(5)
5
1
Hide problems
Always a 0.125-good point
Let
A
A
A
be a finite set of points in the coordinate plane. Suppose that
A
A
A
has
n
≥
3
n\geq3
n
≥
3
points. Given any
a
a
a
in
A
A
A
, the horizontal and vertical lines through
a
a
a
define four closed quadrants centered at
a
a
a
. For any real number
α
\alpha
α
, call a point
a
a
a
in
A
A
A
α
\alpha
α
-good if there are two diagonally opposite closed quadrants centered at
a
a
a
that each contain at least
α
n
\alpha n
α
n
points from
A
A
A
. Show that there is some
a
a
a
in
A
A
A
that is
1
8
\frac{1}{8}
8
1
-good.
4
1
Hide problems
Power to the three-variable Diophantine
Find all triples
(
x
,
y
,
z
)
\left(x,y,z\right)
(
x
,
y
,
z
)
of nonnegative integers such that
5
x
7
y
+
4
=
3
z
.
5^x7^y+4=3^z.
5
x
7
y
+
4
=
3
z
.
3
1
Hide problems
1/p is not the cosine of a rational degree marking
Let
p
p
p
be an odd prime. Show that
1
π
⋅
cos
−
1
(
1
p
)
\frac{1}{\pi}\cdot\cos^{-1}\left(\frac{1}{p}\right)
π
1
⋅
cos
−
1
(
p
1
)
is irrational. (Note:
cos
−
1
(
x
)
\cos^{-1}\left(x\right)
cos
−
1
(
x
)
is defined to be the unique
y
y
y
with
0
≤
y
≤
π
0\leq y\leq\pi
0
≤
y
≤
π
such that
cos
(
y
)
=
x
\cos\left(y\right)=x
cos
(
y
)
=
x
.)
2
1
Hide problems
Card flipping to get all the same
N
N
N
cards are arranged in a circle, with exactly one card face up and the rest face-down. In a turn, choose a proper divisor
k
k
k
of
N
N
N
. You may begin at any card on the circle and flip every
k
k
k
-th card, counting clockwise, if and only if every
k
k
k
-th card begins the turn in the same orientation (either all face-up or all face-down).For example, with
15
15
15
cards, you may start at any position and flip the
3
3
3
rd,
6
6
6
th,
9
9
9
th,
12
12
12
th, and
15
15
15
th cards around the circle if they all begin the turn face up (or all face-down).For what values of
N
N
N
can all of the cards be flipped face-up in a finite number of turns?
1
1
Hide problems
Infinitely many circles in a circle
Let
C
\mathcal{C}
C
be a circle and let
P
P
P
and
Q
Q
Q
be points inside
C
\mathcal{C}
C
. Prove that there are infinitely many circle through
P
P
P
and
Q
Q
Q
that are completely contained inside of
C
\mathcal{C}
C
.