MathDB
Problems
Contests
National and Regional Contests
USA Contests
USA - Other Middle and High School Contests
SDML/SDMO Contests
SDML (Middle School)
2012-2013 SDML (Middle School)
2012-2013 SDML (Middle School)
Part of
SDML (Middle School)
Subcontests
(15)
15
2
Hide problems
Diagonals of a prism and its volume
Three faces of a rectangular prism have diagonal lengths of
7
7
7
,
8
8
8
, and
9
9
9
inches. How many cubic inches are in the volume of the rectangular prism?
(A)
48
11
(B)
160
(C)
14
95
(D)
35
15
(E)
504
\text{(A) }48\sqrt{11}\qquad\text{(B) }160\qquad\text{(C) }14\sqrt{95}\qquad\text{(D) }35\sqrt{15}\qquad\text{(E) }504
(A)
48
11
(B)
160
(C)
14
95
(D)
35
15
(E)
504
Pentagon area
Pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
is inscribed in a circle such that
A
C
D
E
ACDE
A
C
D
E
is a square with area
12
12
12
. What is the largest possible area of pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
?
(A)
9
+
3
2
(B)
13
(C)
12
+
2
(D)
14
(E)
12
+
6
−
3
\text{(A) }9+3\sqrt{2}\qquad\text{(B) }13\qquad\text{(C) }12+\sqrt{2}\qquad\text{(D) }14\qquad\text{(E) }12+\sqrt{6}-\sqrt{3}
(A)
9
+
3
2
(B)
13
(C)
12
+
2
(D)
14
(E)
12
+
6
−
3
14
2
Hide problems
Kacey's Halloween candy distribution
Kacey is handing out candy for Halloween. She has only
15
15
15
candies left when a ghost, a goblin, and a vampire arrive at her door. She wants to give each trick-or-treater at least one candy, but she does not want to give any two the same number of candies. How many ways can she distribute all
15
15
15
identical candies to the three trick-or-treaters given these restrictions?
(A)
91
(B)
90
(C)
81
(D)
72
(E)
70
\text{(A) }91\qquad\text{(B) }90\qquad\text{(C) }81\qquad\text{(D) }72\qquad\text{(E) }70
(A)
91
(B)
90
(C)
81
(D)
72
(E)
70
Laps around a circular track
Sammy and Tammy run laps around a circular track that has a radius of
1
1
1
kilometer. They begin and end at the same point and at the same time. Sammy runs
3
3
3
laps clockwise while Tammy runs
4
4
4
laps counterclockwise. How many times during their run is the straight-line distance between Sammy and Tammy exactly
1
1
1
kilometer?
(A)
7
(B)
8
(C)
13
(D)
14
(E)
21
\text{(A) }7\qquad\text{(B) }8\qquad\text{(C) }13\qquad\text{(D) }14\qquad\text{(E) }21
(A)
7
(B)
8
(C)
13
(D)
14
(E)
21
13
2
Hide problems
Cubes with colored faces
How many distinct cubes have two red faces, two white faces, and two blue faces? (Two cubes are considered distinct if they cannot be rotated to look the same.)
(A)
5
(B)
6
(C)
7
(D)
8
(E)
9
\text{(A) }5\qquad\text{(B) }6\qquad\text{(C) }7\qquad\text{(D) }8\qquad\text{(E) }9
(A)
5
(B)
6
(C)
7
(D)
8
(E)
9
Two variables and two equations
Let
a
+
1
b
=
8
a+\frac{1}{b}=8
a
+
b
1
=
8
and
b
+
1
a
=
3
b+\frac{1}{a}=3
b
+
a
1
=
3
. Given that there are two possible real values for
a
a
a
, find their sum.
(A)
3
8
(B)
8
3
(C)
3
(D)
5
(E)
8
\text{(A) }\frac{3}{8}\qquad\text{(B) }\frac{8}{3}\qquad\text{(C) }3\qquad\text{(D) }5\qquad\text{(E) }8
(A)
8
3
(B)
3
8
(C)
3
(D)
5
(E)
8
12
2
Hide problems
Odd and even divisors
How many
2
2
2
-digit integers have an equal number of odd and even positive divisors?
(A)
11
(B)
12
(C)
22
(D)
23
(E)
45
\text{(A) }11\qquad\text{(B) }12\qquad\text{(C) }22\qquad\text{(D) }23\qquad\text{(E) }45
(A)
11
(B)
12
(C)
22
(D)
23
(E)
45
Perfect square and perfect cube
For what digit
A
A
A
is the numeral
1
A
A
1AA
1
AA
a perfect square in base-
5
5
5
and a perfect cube in base-
6
6
6
?
(A)
0
(B)
1
(C)
2
(D)
3
(E)
4
\text{(A) }0\qquad\text{(B) }1\qquad\text{(C) }2\qquad\text{(D) }3\qquad\text{(E) }4
(A)
0
(B)
1
(C)
2
(D)
3
(E)
4
11
2
Hide problems
Circumscribing a rectangle
What is the smallest possible area of a rectangle that can completely contain the shape formed by joining six squares of side length
8
8
8
cm as shown below?[asy] size(5cm,0); draw((0,2)--(0,3)); draw((1,1)--(1,3)); draw((2,0)--(2,3)); draw((3,0)--(3,2)); draw((4,0)--(4,1)); draw((2,0)--(4,0)); draw((1,1)--(4,1)); draw((0,2)--(3,2)); draw((0,3)--(2,3)); [/asy]
(A)
384
cm
2
(B)
576
cm
2
(C)
672
cm
2
(D)
768
cm
2
(E)
832
cm
2
\text{(A) }384\text{ cm}^2\qquad\text{(B) }576\text{ cm}^2\qquad\text{(C) }672\text{ cm}^2\qquad\text{(D) }768\text{ cm}^2\qquad\text{(E) }832\text{ cm}^2
(A)
384
cm
2
(B)
576
cm
2
(C)
672
cm
2
(D)
768
cm
2
(E)
832
cm
2
Gluing cubes together
Six different-sized cubes are glued together, one on top of the other. The bottom cube has edge length
8
8
8
. Each of the other cubes has four vertices at the midpoints of the edges of the cube below it as shown. The entire solid is then dipped in red paint. What is the total area of the red-painted surface on the solid?(will insert image here later)
(A)
630
(B)
632
(C)
648
(D)
694
(E)
756
\text{(A) }630\qquad\text{(B) }632\qquad\text{(C) }648\qquad\text{(D) }694\qquad\text{(E) }756
(A)
630
(B)
632
(C)
648
(D)
694
(E)
756
10
2
Hide problems
Diagonals of a pentagon
Two of the diagonals of a regular pentagon are selected at random. What is the probability that the two selected diagonals intersect inside the pentagon?
(A)
2
5
(B)
1
5
(C)
7
10
(D)
3
5
(E)
1
2
\text{(A) }\frac{2}{5}\qquad\text{(B) }\frac{1}{5}\qquad\text{(C) }\frac{7}{10}\qquad\text{(D) }\frac{3}{5}\qquad\text{(E) }\frac{1}{2}
(A)
5
2
(B)
5
1
(C)
10
7
(D)
5
3
(E)
2
1
Palmer's prime product
Palmer correctly computes the product of the first
1
,
001
1,001
1
,
001
prime numbers. Which of the following is NOT a factor of Palmer's product?
(A)
2
,
002
(B)
3
,
003
(C)
5
,
005
(D)
6
,
006
(E)
7
,
007
\text{(A) }2,002\qquad\text{(B) }3,003\qquad\text{(C) }5,005\qquad\text{(D) }6,006\qquad\text{(E) }7,007
(A)
2
,
002
(B)
3
,
003
(C)
5
,
005
(D)
6
,
006
(E)
7
,
007
9
2
Hide problems
Equilateral triangle on plane
Find the area of the equilateral triangle that includes vertices at
(
−
3
,
5
)
\left(-3,5\right)
(
−
3
,
5
)
and
(
−
5
,
9
)
\left(-5,9\right)
(
−
5
,
9
)
.
(A)
3
3
(B)
10
3
(C)
30
(D)
2
15
(E)
5
3
\text{(A) }3\sqrt{3}\qquad\text{(B) }10\sqrt{3}\qquad\text{(C) }\sqrt{30}\qquad\text{(D) }2\sqrt{15}\qquad\text{(E) }5\sqrt{3}
(A)
3
3
(B)
10
3
(C)
30
(D)
2
15
(E)
5
3
Gender division
If five boys and three girls are randomly divided into two four-person teams, what is the probability that all three girls will end up on the same team?
(A)
1
7
(B)
2
7
(C)
1
10
(D)
1
14
(E)
1
28
\text{(A) }\frac{1}{7}\qquad\text{(B) }\frac{2}{7}\qquad\text{(C) }\frac{1}{10}\qquad\text{(D) }\frac{1}{14}\qquad\text{(E) }\frac{1}{28}
(A)
7
1
(B)
7
2
(C)
10
1
(D)
14
1
(E)
28
1
8
4
Show problems
7
4
Show problems
6
4
Show problems
5
4
Show problems
4
4
Show problems
3
4
Show problems
2
4
Show problems
1
4
Show problems