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2016 SDMO (Middle School)
2016 SDMO (Middle School)
Part of
SDMO (Middle School)
Subcontests
(5)
5
1
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Quadratic and a not-prime
Suppose
a
a
a
and
b
b
b
are integers such that
x
2
+
a
x
+
b
+
1
=
0
x^2+ax+b+1=0
x
2
+
a
x
+
b
+
1
=
0
has
2
2
2
positive integer solutions. Show that
a
2
+
b
2
a^2+b^2
a
2
+
b
2
is not prime.
4
1
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Pyramid of spheres
There is an infinitely tall tetrahedral stack of spheres where each row of the tetrahedron consists of a triangular arrangement of spheres, as shown below. There is
1
1
1
sphere in the top row (which we will call row
0
0
0
),
3
3
3
spheres in row
1
1
1
,
6
6
6
spheres in row
2
2
2
,
10
10
10
spheres in row
3
3
3
, etc. The top-most sphere in row
0
0
0
is assigned the number
1
1
1
. We then assign each other sphere the sum of the number(s) assigned to the sphere(s) which touch it in the row directly above it. Find a simplified expression in terms of
n
n
n
for the sum of the numbers assigned to each sphere from row
0
0
0
to row
n
n
n
.[asy] import three; import solids; size(8cm);//currentprojection = perspective(1, 1, 10);triple backright = (-2, 0, 0), backleft = (-1, -sqrt(3), 0), backup = (-1, -sqrt(3) / 3, 2 * sqrt(6) / 3);draw(shift(2 * backleft) * surface(sphere(1,20)), white); //2 draw(shift(backleft + backright) * surface(sphere(1,20)), white); //2 draw(shift(2 * backright) * surface(sphere(1,20)), white); //3 draw(shift(backup + backleft) * surface(sphere(1,20)), white); draw(shift(backup + backright) * surface(sphere(1,20)), white); draw(shift(2 * backup) * surface(sphere(1,20)), white);draw(shift(backleft) * surface(sphere(1,20)), white); draw(shift(backright) * surface(sphere(1,20)), white); draw(shift(backup) * surface(sphere(1,20)), white);draw(surface(sphere(1,20)), white);label("Row 0", 2 * backup, 15 * dir(20)); label("Row 1", backup, 25 * dir(20)); label("Row 2", O, 35 * dir(20));dot(-backup); dot(-7 * backup / 8); dot(-6 * backup / 8);dot((backleft - backup) + backleft * 2); dot(5 * (backleft - backup) / 4 + backleft * 2); dot(6 * (backleft - backup) / 4 + backleft * 2);dot((backright - backup) + backright * 2); dot(5 * (backright - backup) / 4 + backright * 2); dot(6 * (backright - backup) / 4 + backright * 2); [/asy]
3
1
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Flip heads to win!
Gwen, Eli, and Kat take turns flipping a coin in their respective order. The first one to flip heads wins. What is the probability that Kat will win?
2
1
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Circle area ratio
Let
A
B
AB
A
B
be a diameter of a circle and let
C
C
C
be a point on
A
B
AB
A
B
with
2
⋅
A
C
=
B
C
2\cdot AC=BC
2
⋅
A
C
=
BC
. Let
D
D
D
and
E
E
E
be points on the circle such that
D
C
⊥
A
B
DC\perp AB
D
C
⊥
A
B
and
D
E
DE
D
E
is a second diameter. What is the ratio of the area of
△
D
C
E
\triangle{DCE}
△
D
CE
to the area of
△
A
B
D
\triangle{ABD}
△
A
B
D
?
1
1
Hide problems
Clover convolution
Let
♣
(
x
)
\clubsuit\left(x\right)
♣
(
x
)
denote the sum of the digits of the positive integer
x
x
x
. For example,
♣
(
8
)
=
8
\clubsuit\left(8\right)=8
♣
(
8
)
=
8
and
♣
(
123
)
=
1
+
2
+
3
=
6
\clubsuit\left(123\right)=1+2+3=6
♣
(
123
)
=
1
+
2
+
3
=
6
. For how many two-digit values of
x
x
x
is
♣
(
♣
(
x
)
)
=
3
\clubsuit\left(\clubsuit\left(x\right)\right)=3
♣
(
♣
(
x
)
)
=
3
?