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Rumson Fair Haven Reg H *
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Subject
Mathematics
Date
Nov 24, 2024
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Uploaded by CoachRiverTiger30
David Essner Mathematics Competition XXXI, 2011-2012
Preliminary Exam
1.
Tom, Alice and John took an exam. Alice scored 80. Tom scored 10 more than the average of
the three, while John scored 16 less than the average of the three. The average of the three was
then
(a) 72
(b) 74
(c) 75
2
3
(d) 76
1
3
(e) 78
2.
Two sides of an isosceles triangle have length 2 and 5. What is the area of the triangle?
(a) 5
(b) 2
√
6
(c)
√
21
(d) 2
√
5
(e) There is more than one possible value.
3.
What is the sum 1
-
3+5
-
7+9
-
. . .
+81? (The terms increase in magnitude by 2 and alternate
in sign.)
(a)
-
15
(b) 1
(c) 27
(d) 39
(e) 41
4.
What is the area of a rectangle if the diagonals have length 1 and 60
◦
is an angle of their
intersection?
(a) 1
/
2
(b)
√
2
/
2
(c)
√
3
/
2
(d)
√
3
/
4
(e) (
√
3 + 1)
/
2
5.
A multiple choice test has 30 questions and 5 choices for each question. If a student answers all
30 questions and the score is [number right - (number wrong/4)] then which of the following is
a possible score?
(a)
-
10
(b) 5
.
25
(c) 7
.
75
(d) 8
.
75
(e) 9
.
25
6.
Line
L
1
has slope 1
/
2 and Line
L
2
has slope 1
/
3. If
L
1
and
L
2
have the same
y
–intercept
b
and
the sum of the
x
–intercepts of
L
1
and
L
2
is 10, then
b
equals
(a) 5
/
6
(b)
-
6
/
5
(c)
-
2
(d) 3
/
2
(e) 2
7.
What is the value of (log
2
3)(log
3
4)?
(a) 3
/
4
(b) 4
/
3
(c) 3
/
2
(d) 2
(e) 8
/
3
8.
If 0
< x < π/
2 and sin
x
= 2 cos
x
then (sin
x
)(cos
x
) equals
(a) 1
/
3
(b) 2
/
5
(c) 1
/
5
(d) 3
/
8
(e)
√
3
/
4
9.
How many positive integer pairs (
m, n
) satisfy the equation 2
m
+ 7
n
= 835?
(a) 44
(b) 51
(c) 60
(d) 71
(e) 119
10.
From a point
P
two tangent lines are drawn to a circle
C
. If
A, B
are the tangent points,
O
is
the center of
C
, and
negationslash
APB
= 30
◦
then
negationslash
AOB
equals
(a) 60
◦
(b) 90
◦
(c) 120
◦
(d) 150
◦
(e) 180
◦
11.
Let the function
f
satisfy
f
(
xy
) =
f
(
x
)
/y
for all positive numbers
x, y
. If
f
(5) = 10 then what
is the value of
f
(8)?
(a) 4
(b) 16
/
5
(c) 32
/
5
(d) 25
/
4
(e) There are many possible answers.
12.
A full glass contains a mixture of 4
/
5 water and 1
/
5 alcohol. First 1
/
4 of the content is removed
and replaced with alcohol.
Then 1
/
3 of the resulting mixture is removed and replaced with
alcohol. What fraction of the final mixture is alcohol?
(a) 13
/
20
(b) 11
/
20
(c) 2
/
5
(d) 3
/
5
(e) 47
/
60
13.
If
a, b
are real numbers,
a
+
b
= 3 and
a
2
+
b
2
= 45 then the value of
a
3
+
b
3
is
(a) 27
(b) 87
(c) 189
(d) 135
(e) not uniquely determined.
14.
Which of the following sets of three numbers form the lengths of the sides of an obtuse triangle?
(a) 5
,
6
,
12
(b) 4
,
6
,
7
(c) 4
,
5
,
6
(d) 3
,
4
,
5
(e) 2
,
3
,
4
15.
In the Cartesian plane points
P, Q
are respectively in the first and fourth quadrants and each is
a distance of 10 from the origin
O
. If segments
OP
,
OQ
respectively make an angle of 40
◦
, 20
◦
with the positive
x
–axis and
M
is the midpoint of
PQ
, then the distance from
O
to
M
is
(a) 4
√
2
(b) 5
√
3
(c) 5
(d) 5
√
2
(e) 4
√
3
16.
What is the real number value of
x
such that 2
x
+ 4
x
= 12?
(a) log
2
3
(b) log
3
2
(c) log
3
4
(d) log
4
3
(e) log
2
12
17.
The length of a chord of a circle is 14 and the shortest distance from the midpoint of the chord
to the circle is 5. Then the radius of the circle equals
(a) 28
/
5
(b) 37
/
5
(c) 28
/
3
(d) 19
/
2
(e) 2
√
6
18.
In a race
A
,
B
, and
C
run at a constant speed. When
A
finishes,
B
is 20 feet behind and
C
is
29 feet behind. When
B
finishes,
C
is 10 feet behind. What was the distance of the race?
(a) 100 feet
(b) 150 feet
(c) 200 feet
(d) 250 feet
(e) 300 feet
19.
Three standard six-sided dice are rolled. What is the probability that the largest number that
occurs is a five?
(a) 16
/
216
(b) 25
/
216
(c) 48
/
216
(d) 61
/
216
(e) 75
/
216
20.
How many non-congruent rectangles are there which satisfy both (i) the lengths of the sides are
integers and (ii) the perimeter equals the area?
(a) 0
(b) 1
(c) 2
(d) 3
(e) more than 3
21.
If the polynomial
P
(
x
) =
x
3
-
4
x
2
+
ax
+ 30, where
a
is a real number, has roots 2
, r, s
, then
what is
|
r
-
s
|
?
(a) 2
(b) 4
(c) 6
(d) 8
(e) There is more than one possible value.
22.
Mr. Jones invested the amount $
P
. During the first year the value of the investment increased
by
x
% and during the second year it decreased by
y
% of the amount at the end of the first year.
If the value of the investment was exactly $
P
at the end of the second year then
x
equals
(a)
y
(b) 101
y/
100
(c) 100
y/
101
(d) 100
y/
(100
-
y
)
(e) 101
y/
(101
-
y
)
23.
How many numbers between 1 and 1000 are integer multiples of either 6 or 15?
(a) 265
(b) 232
(c) 221
(d) 199
(e) 33
24.
Three men and three women are assigned different numbers selected at random from the integers
1 through 9. What is the probability the three numbers assigned to the men are all greater than
the three numbers assigned to the women?
(a) 1
/
2
(b) 1
/
6
(c) 1
/
9
(d) 1
/
20
(e) 1
/
120
25.
If
y
=
x
2
-
2
x
, what is the sum of all distinct real numbers
x
such that (
y
+ 2)
y
= 1?
(a)
-
1
(b) 1
(c) 2
(d) 3
(e) 3
/
2
26.
The difference
√
10
-
3 is nearest to which of the following fractions?
(a) 1
/
3
(b) 3
/
10
(c) 2
/
9
(d) 1
/
4
(e) 1
/
6
27.
In the Cartesian plane if a line through the point (10
,
0) is tangent to the circle
x
2
+
y
2
= 25 at
the point (
a, b
) then
a
equals
(a) 5
/
2
(b) 5
/
√
2
(c) 5
/
√
3
(d) 5
√
3
/
2
(e) 12
/
5
28.
If 50! =
N
(10
k
), where
N
is not a multiple of 10, then
k
equals
(a) 5
(b) 7
(c) 12
(d) 15
(e) 20
29.
At a high school 2/5 of the students are boys and 1/3 of the seniors are boys. If 1/5 of the boys
are seniors then what fraction of the girls are seniors?
(a) 7
/
25
(b) 3
/
10
(c) 6
/
25
(d) 4
/
15
(e) 1
/
4
30.
The ratio
(1 +
i
)
10
(1
-
i
)
7
, where
i
2
=
-
1, equals
(a) 4
-
i
(b) 2 + 2
i
(c) 1
-
2
i
(d) 3
-
3
i
(e) 4
i
-
1
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