THE UNIVERSITY OF VERMONT
5 as a rational number in lowest terms.
2) Simplify the expression I8 27M
4ê3
I6481M– 3ê2. Express your answer as a rational number in lowest terms.
3) Express log2H7L
4) Larry and Jack are standing next to each other on a walking path. Jack begins walking north at a speed of 4 feet per second, while Larry begins walking south at a speed of 3 feet per second. After 2 minutes, Jack turns around and begins walking south at a speed of 5 feet per second. How far from the starting point will Larry have walked by the time Jack catches up to him?
5L Find the area of the polygon with verticesH1, 1L,H3, 2L,H2, 3L,H4, 4L andH5, 0L. See the sketch. Express your answer as a rational number in lowest terms.
6) To pass Mincle, a bridge troll, you must know his age and the number of his toes. He tells you that the product of these numbers is 4378. You know that he is at least 150 years old and that bridge trolls do not live past 300. How many toes does he have?
7) Express 2013 as an integer in base 7.
8) Your two robots, Gort and Klaatu, can vacuum your house in 12 minutes if they work together. Gort can vacuum the house in 28 minutes if it works by itself. How long does it take Klaatu to vacuum the house by itself?
9) In Ms. Direction’s special topics class, 20% of the students are juniors and 80% are seniors. On a recent test, the average score for the entire class was 85 and the average score for the seniors was 88.
14L The area of a circle with centerC is 72psquare units.
If the square ABCDhas one vertex atCand the opposite vertexAon the circle, what is the area of squareABCD?
C
A B
D
15LAn equilateral triangle with verticesA,BandCis inscribed in a circle. If the perimeter of the triangle is 18 cm, find the length of the arc of the circle between two adjacent vertices of the triangle. A
B C
16L ChordAB in circleChas length 12 cm. IfMis the midpoint of chordABandPis a point on the circle such that PMis
perpendicular toABandPM = 3 cm, find the diameter of
the circle.
A B
P
6 6
3
M
17) Find the smallest real number c such that the equation | x – 2 | + |3x + 4 | =c has at least one solution.
18) If a, b, c and d are positive real numbers such that logaHbL=8
9, logb(c) = –
3
4 and logcHdL=2 , find the value of logdHabcL.
19) The sequence 8an< is defined by a0=2, a1=4 and an=
6aHn-1L aHn-2L for n
¥ 2. Determine the value of a2013.
20) A bag contains 5 red marbles and 3 yellow marbles. Marbles are removed from the bag one at a time without replacement until either all of the red marbles have been removed or all of the yellow marbles have been removed. What is the probability that the last marble drawn from the bag is yellow ? Express your answer as a rational number in lowest terms.
21) Suppose that x and y are positive real numbers such that logIx y3M=2 and log x
2 y =3.
Determine the value of logHx yL.
22) Find the number of positive integers k with 10 000 k 99 999 such that the middle digit is the average
of the first and fifth digits.
23) The sum of 54 consecutive positive integers is a perfect cube. What is the smallest possible value of the sum ?
25) Find the value of a such that the three solutions of x3– 8x2 + ax– 12= 0 are positive integers.
26) If H1+iL 2013
H1 –iL2007 is expressed in the form a + bi where a and b are real numbers and i
2= – 1, find the value of a + b.
27) Six black checkers are placed on squares of a 6 by 6 checkerboard in the positions shown in Figure 1 and are left in place. A white checker begins on the square at the lower left corner of the board (marked A in Figure 1) and follows a path from square to square across the board, ending in the upper right corner of the board (marked B). How many different paths are there from A to B if at each step the white checker can move one square to the right, one square up or one square diagonally upward to the right and may not pass though any square occupied by a black checker? One such path is shown in Figure 2.
A
B
Figure 1 Figure 2
28LLetABCD be a square of edge length 6. UsingA Bas a diameter, draw a semicircle internal to the square. Using Das the center andD A as a radius, draw a quartercircle
internal to the square. The semicircle and the quartercircle intersect atE. What is the distance fromEtoA B?
A B
C D
E
29L LetABCDbe a square of side length 16. A circle of radiusris drawn through pointsCandDand is tangent to sideAB. Findr.
A B
C D
30) If
⁄
k=0 2013
Iik +i–kM is expressed in the form a + bi where a and b are real numbers and i2= – 1, find the value of a + b.
31) Find all ordered pairs (x,y) that satisfy the system of equations x
2+4x y-8x+4y2-16y+16=0
x y2-3x y+2x-2y2+6y-4=0.
32) If f (1) = 5 and f (n) =f (n–1) + 2n – 1 for all n¥ 2, find f (100).
33) In a random arrangement of the letters of FREEZEDRIED , what is the probability that all of the vowels will be together in a single consecutive grouping? Express your answer as a rational number in lowest terms.
34) Suppose that x and y are real numbers such that x+ y + x - y =10. What is the greatest possible
35LLetABCbe a nondegenerate right triangle. IfAB=30, AD=21, angleCAD= aand angleCDB=2a,
findCB.
A B
C
D
α 2α
36L LetABCbe the triangle with verticesH0, 0L, H4, 0L andH2, 3L. Find the coordinates of the pointPthat is equidistant fromA, BandC. Express your answer as an ordered pairHx,yL.
AH0,0L
CH2,3L
BH4,0L
PHx,yL
37) Suppose that a, b, cand d are positive integers such that a b + 2a + 2b =217
b c + 2b + 2c =81
c d + 2c +2d = 51
d a + 2d + 2a =139.
Determine the value of a +b + c+ d.
38) Let T be the triangular region whose vertices are H0, 0L, H4, 0L and H0, 5L . What is the probability that a randomly chosen point from T is closer to H4, 2L than to H0, 0L ? Express your answer as a rational number in lowest terms.
39LLetS1be a square of side length 3. TriangleT1is formed by joining the midpoint of the upper side ofS1to the endpoints of the lower side ofS1. LetA1be the area inside squareS1and outside triangleT1. SquareS2is inscribed inT1with one side on the lower side ofS1and triangleT2 is formed by joining the midpoint of the upper side ofS2to the
endpoints of the lower side ofS2. LetA2be the area inside squareS2and outside triangleT2. This process is successively repeated. The first four
iterations are shown in the figure. Compute
⁄
k=1 ¶
Ak.
40) Let t be the tens digit and u the units digit of 3333. Find t and u.
41L Three poles with circular cross sections are to be bound together with a wire. The radii of the circular cross sections are 1, 3 and 1 inches. The centers of the circles are on the same straight line as indicated in the sketch. If the length of
the wire is written in the form a 3 +bpwherea andb
