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SDML (Middle School)
2014-2015 SDML (Middle School)
2014-2015 SDML (Middle School)
Part of
SDML (Middle School)
Subcontests
(15)
15
1
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Obtuse triangles in a regular 17-gon
How many triangles formed by three vertices of a regular
17
17
17
-gon are obtuse?
(A)
156
(B)
204
(C)
357
(D)
476
(E)
524
\text{(A) }156\qquad\text{(B) }204\qquad\text{(C) }357\qquad\text{(D) }476\qquad\text{(E) }524
(A)
156
(B)
204
(C)
357
(D)
476
(E)
524
14
1
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Driving down Hull Street
Hull Street is nine miles long. The first two miles of Hull Street are in Richmond, and the last two miles of the street are in Midlothian. If Kathy starts driving along Hull Street from a random point in Richmond and stops on the street at a random point in Midlothian, what is the probability that Kathy drove less than six miles?
(A)
1
16
(B)
1
9
(C)
1
8
(D)
1
6
(E)
1
4
\text{(A) }\frac{1}{16}\qquad\text{(B) }\frac{1}{9}\qquad\text{(C) }\frac{1}{8}\qquad\text{(D) }\frac{1}{6}\qquad\text{(E) }\frac{1}{4}
(A)
16
1
(B)
9
1
(C)
8
1
(D)
6
1
(E)
4
1
13
1
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Subset of {100} with no member thrice another
Let
S
S
S
be a subset of the integers
1
,
2
,
…
,
100
1,2,\ldots,100
1
,
2
,
…
,
100
that has the property that none of its members is
3
3
3
times another. What is the largest number of members
S
S
S
can have?
(A)
67
(B)
71
(C)
72
(D)
76
(E)
77
\text{(A) }67\qquad\text{(B) }71\qquad\text{(C) }72\qquad\text{(D) }76\qquad\text{(E) }77
(A)
67
(B)
71
(C)
72
(D)
76
(E)
77
12
1
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Function convolution
Let
f
(
x
)
=
x
2
−
14
x
+
52
f\left(x\right)=x^2-14x+52
f
(
x
)
=
x
2
−
14
x
+
52
and
g
(
x
)
=
a
x
+
b
g\left(x\right)=ax+b
g
(
x
)
=
a
x
+
b
, where
a
a
a
and
b
b
b
are positive. Find
a
a
a
, given that
f
(
g
(
−
5
)
)
=
3
f\left(g\left(-5\right)\right)=3
f
(
g
(
−
5
)
)
=
3
and
f
(
g
(
0
)
)
=
103
f\left(g\left(0\right)\right)=103
f
(
g
(
0
)
)
=
103
.
(A)
2
(B)
5
(C)
7
(D)
10
(E)
17
\text{(A) }2\qquad\text{(B) }5\qquad\text{(C) }7\qquad\text{(D) }10\qquad\text{(E) }17
(A)
2
(B)
5
(C)
7
(D)
10
(E)
17
11
1
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Rational number properties
Phillip and Paula both pick a rational number, and they notice that Phillip's number is greater than Paula's number by
12
12
12
. They each square their numbers to get a new number, and see that the sum of these new numbers is half of
169
169
169
. Finally, they each square their new numbers and note that Phillip's latest number is now greater than Paula's by
5070
5070
5070
. What was the sum of their original numbers?
(A)
−
4
(B)
−
3
(C)
1
(D)
2
(E)
5
\text{(A) }-4\qquad\text{(B) }-3\qquad\text{(C) }1\qquad\text{(D) }2\qquad\text{(E) }5
(A)
−
4
(B)
−
3
(C)
1
(D)
2
(E)
5
10
1
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Repeating decimal denominator
Suppose
a
a
a
and
b
b
b
are digits which are not both nine and not both zero. If the repeating decimal
.
A
B
‾
.\overline{AB}
.
A
B
is expressed as a fraction in lowest terms, how many possible denominators are there?
(A)
2
(B)
3
(C)
4
(D)
5
(E)
6
\text{(A) }2\qquad\text{(B) }3\qquad\text{(C) }4\qquad\text{(D) }5\qquad\text{(E) }6
(A)
2
(B)
3
(C)
4
(D)
5
(E)
6
9
1
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Summer camp classes
At summer camp, there are
20
20
20
campers in each of the swimming class, the archery class, and the rock climbing class. Each camper is in at least one of these classes. If
4
4
4
campers are in all three classes, and
24
24
24
campers are in exactly one of the classes, how many campers are in exactly two classes?
(A)
12
(B)
13
(C)
14
(D)
15
(E)
16
\text{(A) }12\qquad\text{(B) }13\qquad\text{(C) }14\qquad\text{(D) }15\qquad\text{(E) }16
(A)
12
(B)
13
(C)
14
(D)
15
(E)
16
8
2
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Five digit number with divisibility properties
If the five-digit number
3
A
B
7
C
3AB7C
3
A
B
7
C
is divisible by
4
4
4
and
9
9
9
and
A
<
B
<
C
A<B<C
A
<
B
<
C
, what is
A
+
B
+
C
A+B+C
A
+
B
+
C
?
(A)
3
(B)
8
(C)
9
(D)
17
(E)
26
\text{(A) }3\qquad\text{(B) }8\qquad\text{(C) }9\qquad\text{(D) }17\qquad\text{(E) }26
(A)
3
(B)
8
(C)
9
(D)
17
(E)
26
Regular square pyramids
Two regular square pyramids have all edges
12
12
12
cm in length. The pyramids have parallel bases and those bases have parallel edges, and each pyramid has its apex at the center of the other pyramid's base. What is the total number of cubic centimeters in the volume of the solid of intersection of the two pyramids?
7
2
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Gizmo's geometric sequence
Gizmo is thinking of a geometric sequence in which the third term is
1215
1215
1215
and the fifth is
540
540
540
. Which of the following could be the eighth term of Gizmo's sequence?
(A)
−
160
(B)
−
135.5
(C)
216
(D)
240
(E)
472.5
\text{(A) }-160\qquad\text{(B) }-135.5\qquad\text{(C) }216\qquad\text{(D) }240\qquad\text{(E) }472.5
(A)
−
160
(B)
−
135.5
(C)
216
(D)
240
(E)
472.5
Distinct digits of 2^29
Nine distinct digits appear in the decimal expansion of
2
29
2^{29}
2
29
. Which digit is missing?
6
2
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Triangle angles
In
△
A
B
C
\triangle{ABC}
△
A
BC
,
A
X
=
X
Y
=
Y
B
=
B
C
AX=XY=YB=BC
A
X
=
X
Y
=
Y
B
=
BC
, and
m
∠
A
B
C
=
12
0
∘
m\angle{ABC}=120^{\circ}
m
∠
A
BC
=
12
0
∘
. What is
m
∠
B
A
C
m\angle{BAC}
m
∠
B
A
C
?[asy] pair A, B, C, X, Y; A = origin; X = dir(30); Y = X + dir(0); B = Y + dir(60); C = B + dir(330); draw(A--B--C--cycle); draw(X--Y--B); label("
A
A
A
",A,W); label("
B
B
B
",B,N); label("
C
C
C
",C,E); label("
X
X
X
",X,NW); label("
Y
Y
Y
",Y,SE); [/asy]
(A)
15
(B)
20
(C)
25
(D)
30
(E)
35
\text{(A) }15\qquad\text{(B) }20\qquad\text{(C) }25\qquad\text{(D) }30\qquad\text{(E) }35
(A)
15
(B)
20
(C)
25
(D)
30
(E)
35
Alex, Beth, and Carl rake a lawn
Yesterday, Alex, Beth, and Carl raked their lawn. First, Alex and Beth raked half of the lawn together in
30
30
30
minutes. While they took a break, Carl raked a third of the remaining lawn in
60
60
60
minutes. Finally, Beth joined Carl and together they finished raking the lawn in
24
24
24
minutes. If they each rake at a constant rate, how many hours would it have taken Alex to rake the entire lawn by himself?
5
2
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Trailing zeros of a factor of a factorial
In how many consecutive zeros does the decimal expansion of
26
!
3
5
3
\frac{26!}{35^3}
3
5
3
26
!
end?
(A)
1
(B)
2
(C)
3
(D)
4
(E)
5
\text{(A) }1\qquad\text{(B) }2\qquad\text{(C) }3\qquad\text{(D) }4\qquad\text{(E) }5
(A)
1
(B)
2
(C)
3
(D)
4
(E)
5
Circles and right triangles
A circle of radius
5
5
5
is inscribed in an isosceles right triangle,
A
B
C
ABC
A
BC
. The length of the hypotenuse of
A
B
C
ABC
A
BC
can be expressed as
a
+
a
2
a+a\sqrt{2}
a
+
a
2
for some
a
a
a
. What is
a
a
a
?
4
2
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Colored shirts at work
Shannon, Laura, and Tasha found a shirt which came in five colors at their favorite store, and they each bought one of each color of that shirt. On Monday, they all wear one of their new shirts to work. What is the probability that Shannon, Laura, and Tasha will not all be wearing the same color shirt that day?
(A)
12
25
(B)
16
25
(C)
21
25
(D)
22
25
(E)
24
25
\text{(A) }\frac{12}{25}\qquad\text{(B) }\frac{16}{25}\qquad\text{(C) }\frac{21}{25}\qquad\text{(D) }\frac{22}{25}\qquad\text{(E) }\frac{24}{25}
(A)
25
12
(B)
25
16
(C)
25
21
(D)
25
22
(E)
25
24
Hundreds digit is thrice the ones digit
If you pick a random
3
3
3
-digit number, what is the probability that its hundreds digit is triple the ones digit?
3
2
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Box of marbles
In a box of
100
100
100
marbles, just
3
%
3\%
3%
of the marbles are purple and the rest are green. How many green marbles must be removed from the box so that
95
%
95\%
95%
of the remaining marbles are green?
(A)
2
(B)
15
(C)
37
(D)
40
(E)
57
\text{(A) }2\qquad\text{(B) }15\qquad\text{(C) }37\qquad\text{(D) }40\qquad\text{(E) }57
(A)
2
(B)
15
(C)
37
(D)
40
(E)
57
Layna's rectangular green wall
Layna wants to paint a rectangular wall green, but she only has blue and yellow paint. She finds that a
2
:
1
2:1
2
:
1
mix of blue paint to yellow paint produces the color green she wants, and she knows that one gallon of paint will cover
80
80
80
square feet of wall. If the wall is
8
8
8
feet tall and
21
21
21
feet long, how many gallons of blue paint does Layna need? Express your answer as a fraction in simplest form.
2
2
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x-intercept of a line
A line passes through the points
(
−
1
,
3
)
\left(-1,3\right)
(
−
1
,
3
)
and
(
7
,
−
2
)
\left(7,-2\right)
(
7
,
−
2
)
. At what value of
x
x
x
does this line intercept the
x
x
x
-axis?
(A)
7
5
(B)
19
8
(C)
19
5
(D)
27
5
(E)
23
4
\text{(A) }\frac{7}{5}\qquad\text{(B) }\frac{19}{8}\qquad\text{(C) }\frac{19}{5}\qquad\text{(D) }\frac{27}{5}\qquad\text{(E) }\frac{23}{4}
(A)
5
7
(B)
8
19
(C)
5
19
(D)
5
27
(E)
4
23
Fractions and powers
Suppose
a
=
1332
a=1332
a
=
1332
and
b
=
−
222
b=-222
b
=
−
222
. Find
c
c
c
such that
(
a
c
)
3
=
b
6
\left(\frac{a}{c}\right)^3=\sqrt{b^6}
(
c
a
)
3
=
b
6
.
1
2
Hide problems
Shaded figure in grid
Given that each unit square in the grid below is a
1
×
1
1\times1
1
×
1
square, find the area of the shaded region in square units.[asy] fill((3,0)--(4,0)--(6,3)--(4,4)--(4,3)--(0,2)--(2,2)--cycle, grey); draw((0,0)--(6,0)); draw((0,1)--(6,1)); draw((0,2)--(6,2)); draw((0,3)--(6,3)); draw((0,4)--(6,4)); draw((0,0)--(0,4)); draw((1,0)--(1,4)); draw((2,0)--(2,4)); draw((3,0)--(3,4)); draw((4,0)--(4,4)); draw((5,0)--(5,4)); draw((6,0)--(6,4)); [/asy]
(A)
8
(B)
9
(C)
10
(D)
11
(E)
12
\text{(A) }8\qquad\text{(B) }9\qquad\text{(C) }10\qquad\text{(D) }11\qquad\text{(E) }12
(A)
8
(B)
9
(C)
10
(D)
11
(E)
12
Sum of 10 consecutive integers
The sum of
10
10
10
consecutive integers is
75
75
75
. What is the smallest of these
10
10
10
integers?