University of South Carolina
January 29, 2005
1. Tim buys apples at three for $1. He resells them at ve for $2. Assuming
he resells every apple that he buys, how many apples must Tim buy in
order to make a prot of $10 ?
is a point on CA suh that BD is perpendiular to
6. What are the last three digits of 2005 ?
(a) 025 (b) 125 () 375 (d) 625 (e) 875
7. How many dierent real-valued pairs (x;y) satisfy the system of two
equations below?
8
<
:
j x yj= 1
x
y
= xy
(a) 2 (b) 3 () 4 (d) 5 (e) 6
8. Suppose aand bare positiveintegers suhthat(a+2b)(a b) = 10. What
is the value of 2a b ?
(a) 1 (b) 2 () 3 (d) 4 (e) 5
9. Suppose m andn arepositive integerswith mn = 40000. Suppose further
that neither m nor n is divisible by 10. What is the value of m +n ?
(a) 650 (b) 660 () 689 (d) 691 (e) 695
10. If log
y
x+log
x
y = 7, then what is the value of log
y x
2
+ (log
x y)
2
?
(a) 40 (b) 43 () 45 (d) 47 (e) 49
11. Whih of the ve numbers below is the largest?
(a) log
2
3 (b) log
3
5 ()
p
2 (d)
3 p
3 (e)
3
The faster partile moves at a onstant speed of R feet per seond, and
the slower partile moves at a onstant speed of r feet per seond. If the
partiles meet every 50 seonds, then what is the value of R r in feet
per seond?
(a) 6 (b) 8 () 10 (d) 12 (e) 14
13. Let m be a onstant. The graphs of the lines y = x 2 and y = mx+3
interset atapointwhose x-oordinate andy-oordinate areboth positive
if and only if
(a) m = 1 (b) m < 1 () m > 3=2
(d) 3=2 < m < 0 (e) 3=2< m < 1
14. Let f(x) be a funtion suh that, for every real number x,
f(x)+2f( x) = sinx:
15. What is the largestnumber ofirles of radius1that antinside airle
of radius 2:4 with no two of the irles of radius 1 having any ommon
points?
(a) 2 (b) 3 () 4 (d) 5 (e) 6
16. Suppose 4ABC has area p
of \ACB (in degrees)?
N = 123456789101112424344
What is the remainder when N is divided by 45 ?
(a) 4 (b) 9 () 14 (d) 18 (e) 19
18. What is the value of the following produt?
tan5
point on the x-axis or the y-axis, then what is the least valuefor the sum
of the distanes PA and PB ?
(a) 9 (b) 10 () 11 (d) 12 (e) 13
20. Eah ar of a ve-ar train must be painted a solid olor. The only olor
hoies are red, blue, and yellow. If eah of these olors must be used for
at least one ar, then how many ways are there to paint this train?
(a) 100 (b) 125 () 150 (d) 175 (e) 200
21. If integers m, n, a, and b satisfy the equation 36ma 2
6nb 2
= 1008, then
what is the greatest integer that an divide both a and b ?
(a) 1 (b) 2 () 3 (d) 6 (e) 8
22. What is the number of distint real numbers x whih have the property
that the median of the ve numbers x;6;4;1;9 is equal to their mean?
23. Whih one of the following integers does not divide 2 1 ?
25. Suppose4ABC isan equilateraltriangle withA = ( 1;0)and withboth
points B and C onthe right half ofthe hyperboladened by the equation
x
26. A point (x;y) is alled integral if both x and y are integers. How many
points on the graph of
1
27. Four triangular gardening plots form a square as in the
gure shown. A gardenerwill plant owers so that eah
plot ontains one kind of ower, and owers in plots
whih share a ommon edge are dierent. How many
hoies does the gardener have if the owers an be
roses, arnations, daisies, lilies, or tulips?
south
north
east
west
solution whih is a permutation of the numbers 1, 2, 3, 4, and 5. Five
ontestants in a math tournament were asked to solve this system, but
they were only able to provide the following partial answers before the
end of the allotted time:
Amy: z = 1, u = 2 Bob: y = 3, v = 2 Cathy: z = 5, u = 3
Dave: x = 4, v = 2 Eva: x = 4, y = 1
In their partial answers above, eah of the ontestants had found the
orret value of exatly one unknown. What is the value of x+ z+v ?
(a) 6 (b) 7 () 8 (d) 9 (e) 10
29. In 4ABC, AB = 20, BC = 7, and CA = 15.
Side BC is extendedtopoint D sothat4DAB
is similar to 4DCA. What is DC ?
C
D
15
7
B
A
20
(a) 9 (b) 10 () 11 (d) 12 (e) 13
30. Let f(x) = x 4
+ax 3
+bx 2
+x+d, where a, b, , and d are real numbers.
Suppose the graph of y = f(x) intersets the graph of y = 2x 1 at
x = 1, 2, and 3. What is the value of f(0)+f(4) ?
