St. Louis 2014
PAGE I
Problem 1. If 2 x + 3 = 4 x + 6 then x is
a) 1
b) -1
c) 3/2
d) -3/2
e) none of these
Problem 2. If A and B are sets with A = {1,2,4,6,8} and B = {2,3,4,5,6} then ( A ∩ B) ∪ B is
b) empty
a) {2,4,6}
c) {1,2,3,4,5,6,8}
d) {2,3,4,5,6}
e) none of these
Problem 3. The number of cents in d dimes and q quarters is
a) 10d + q / 4
b)
c)
d / 10 + q / 4
d)
d / 10 + q / 25
d / 10 + 25q
e) none of these
1
1
−
Problem 4. A simplified form of a + b a − b is
2
a)
b
a
b)
b2 − a2
a2 − b2
c)
b
a2 − b2
d)
a
b − a2
2
e) none of these
Problem 5. A boy and a girl start at noon on a trip from P to Q and they travel at constant speed (not
the same speed). If the boy takes 8 hours for the trip and the girl 6 hours, then how many hours after
noon will the boy be twice as far from Q as the girl?
a) 4.5
b) 4.8
c) 5.0
d) 5.2
e) none of these
Problem 6. The rectangle has length 2 and width 1. The two circles are
tangent to each other and to three sides of the rectangle as shown. The
area of the shaded region outside the circles and inside the rectangle is
a) 2 − π
b) 2 − 2π
c) 2(1 − π / 2)
d) 1 − π / 2
e) none of these
PAGE II
Problem 7. A dozen tennis players set up a series of matches so that each player plays each of the
other players one time. How many matches are required?
a) 24
b) 66
c) 132
d) 12 ! / 2
e) none of these
Problem 8. The measure of the central angle of a circular sector is 30 degrees. If
the radius is 10 inches then the area of the sector in square inches is
a)
100 π
b) 25π
c) 25π / 4
d) 25π / 3
e) none of these
Problem 9. How many different combinations of pennies, nickels and dimes are there such that
their sum is 25 cents?
a) 10
b) 12
c) 14
d) 16
e) none of these
Problem 10. The prices of all winter coats have been reduced by 20%. If the sale price was P, then
the original price was
a) 1.2 P b) 0.8 P
c) 1.25 P
d) 1.375 P
e) none of these
Problem 11. If the length of a rectangle is twice its width and the area is 50 sq. in., then its perimeter
is
a) 20 in.
b) 15 in.
c) 35 in.
d) 30 in.
e) none of these
Problem 12. John took a trip and traveled 50 mph (miles per hour) for the first two hours and then
60 mph for the next three hours. So his average speed for the trip in mph was
a) 55
b) 57
c) 59
d) 60
e) none of these
PAGE III
Problem 13. A flag in the shape of an equilateral triangle is suspended by
two of its corners from the tops of two vertical poles of lengths 8 and 11 as
shown. If the third corner just touches the ground then the area of the flag is
a)
97
b)
3
95
98
c)
3
3
d)
96
e) none of these
3
Problem 14. A multiple choice test has 5 possible answers for each question. A correct answer is
worth 5 points. An unanswered question is worth 0 points. How many points should be subtracted for
an incorrect answer to eliminate any expected advantage or disadvantage of random guessing?
a) 1
b) 2
c) 1.2
d) 1.25
e) none of these
Problem 15. The absolute value of x is denoted by |x|. How many distinct integer values of m satisfy
m
the inequality
+ 2 ≤ 1?
3
a) 6
b) 7
c) 8
d) 9
e) none of these
Problem 16.
Let c be the hypotenuse, h the altitude to c, and P the
perimeter of a right triangle. If h = 2 and P = 10 then c is
a)
25
8
b)
25
6
c)
16
5
d)
16
9
e) none of these
Problem 17. How many three digit positive integers are there such that the sum of the units digit
and tens digit is 10?
a) 99
b) 90
c) 81
d) 72
e) none of these
PAGE IV
Problem 18.
The graph of the function y =
4a − x 2
a2
value of y when x = 5 ?
a)
− 19
3
− 17
3
b)
c)
− 17
5
d)
passes through the point (2, 1). What is the
− 19
5
e) none of these
Problem 19. Consider a square sheet of paper ABCD with sides of length one. The
paper is folded so that the upper left corner D goes to a point D' on side AB as
shown. The fold line is FE. Let Q = 2( AF ) 2 + ( FD' ) 2 . If D' is chosen so that Q is
minimized then the area of the shaded triangle GBD' is
a)
2 3 −3
3
b)
2− 3
2
c)
2− 3
3
d)
2 3 −3
2
e) none of these
Problem 20. A large rectangle is divided into a number of smaller
rectangles as shown in the figure. The only lengths that are given are
the three shown. The sum of the perimeters of all the rectangles in
the figure is
a) 282
b) 260
c) 276
d) 300 e) none of these
ANSWERS St. Louis 2014
1
2
3
4
5
d
d
e
a
b
6 e
7 b
8 d
9 b
10 c
11 d
12 e
13 a
14 d
15 b
16
17
18
19
20
b
c
e
a
d
Some solutions:
13.
c 2 − a 2 + c 2 − b 2 = c 2 − (b − a ) 2
4(a 2 + b 2 − ab)
and thus the area of
Squaring twice gives c 2 =
3
3 4(a 2 + b 2 − ab)
. For a=8 and b=11,
the equilateral triangle =
4
3
area= 97 / 3
16. If the sides are a and b, then ab = 2c . Also a 2 + b 2 = c 2 . Thus (a + b) 2 = c 2 + 4c . Using
a + b + c = 10 gives
c 2 + 4c + c = 10 . Solving for c gives c=25/6.
19. Let FA = x , then FD' = 1 − x and Q = 3 x 2 − 2 x + 1 = 3( x − 1 / 3) 2 + 2 / 3 .
Thus Q is minimal at x = 1 / 3 . Hence AD' = 3 / 3 and D' B = 1 − 3 / 3 . Since
DFAD' ≅ DD' BG then ∠ BD' G = ∠ AFD' = 60 degrees and thus area
( BD' G ) =
2 3 −3
.
3
20. We find the total perimeter for the 6 kinds of rectangles.
large: 2a+2b,
singles: 4b+6a ,
blocks of two vertical: 6a+2b,
blocks of three horizontal: 4b+2a,
blocks of two horizontal: 4a+4b+4c
blocks of 4- two horizontal and two vertical: 4a+2b+2c.
Sum=24a+18b+6c. If a=5, b=9, and c=3 then sum=300.