UNIVERSITY OF VERMONT
DEPARTMENT OF MATHEMATICS AND STATISTICS FIFTY-FOURTH ANNUAL HIGH SCHOOL PRIZE EXAMINATION
MARCH 9, 2011 1) Evaluate 1 1
10 1 1 11 1
1 12 1
1
13 . Express your answer as a rational number in lowest terms.
2) Simplify the expression
2011 2012 2011 2010
2011 . Express your answer as a rational number in lowest terms.
3) The difference of two positive numbers is 4 and the product of the two numbers is 19. Find the sum of the two numbers.
4) Find the value of a + b2 if ab – 14 and a2 b2 30 .
5 Consider the circle with diameterAD. If ABC 130 °, CDA 50 ° and BCA 20 °, find BAD. Express your answer in degrees.
A
B
C
D
6) If the length of a rectangle is increased by 40% and the width is decreased by 15%, what is the percentage change in the area of the rectangle?
7) One day last month, Ray's Reasonably Reliable Repair Service offered the following Saturday special: "Buy 3 shock absorbers at the regular price and receive an 80% discount on the fourth."
Jerome bought 4 shock absorbers on that day and paid a total of $176. What was the regular price of one shock absorber?
8) The two roots of the quadratic equation x2 85x c 0 are prime numbers. What is the value of c?
9) For real numbers x, yand z, define F x, y,z x y y z z x. For which real numbers a is F 2,a,a 1 F 5,a,a 1 ?
10) Michelle has a collection of marbles, all of which are either blue or green. She is creating pairs of 1 blue marble and 1 green marble.
After a while, she notices that 2
3 of all the blue marbles are paired with 3
5 of all the green marbles. What fraction of Michelle’s
marble collection has been paired up? Express your answer as a rational number in lowest terms.
11) Find the integer value of the expression log7 1
8 log825 log25 log549 .
12) If sin(x) + cos(x) = 1
2, find the value of sin
13 Find the area of the region bounded by the lines x 2y 2, – 4x y 1, x 2y 11 andx–y 2.
14 Find the area of the circle that contains the pointQ 9, 8 and that is tangent to the line x– 2y 2 at the pointP 6, 2 .
P Q
15) If logx y2 3, determine the value of log
y x
2 . Express your answer as a rational number in lowest terms.
16) When a complex number z is expressed the form z a b where a and b are real numbers, the modulus (or absolute value)
of z is defined by z a2 b2 . Suppose that z z 3 9 . Determine the value of z 2.
17) Twenty balls numbered 1 to 20 are placed in a jar. Larry reaches into the jar and randomly removes two of the balls. What is the probability that the sum of the numbers on the two removed balls is a multiple of 3? Express your answer as a rational number in lowest terms.
18) The three vertices of a triangle are points on the graph of the parabola y x2. If the x-coordinates of the vertices are the roots of the cubic equation x3 60x2 153x 1026 0, find the sum of the slopes of the three sides of the triangle.
19) For how many real numbers x will the mean of the set 6, 3, 10, 9,x be equal to the median?
20) Each side of square ABCD has length 3. Let M and N be points on sides BCand CD respectively such that BM ND 1 and let MAN. Find sin .
21) Find the value of the real number x such that 5 + x, 11 + x and 20 + x form a geometric progression in the given order.
22 Find the number of paths from the lower left corner to the upper right corner of the given grid, if the only allowable moves are along grid lines upward or to the right.
23) If x and z are real number such that 2 x – 3 z – 9 and x z 23 , find x + z .
24) If sin( ) = 1
4, find sin(3 ). Express your answer as a rational number in lowest terms.
25) Let f(x,y) x–y
x y. Define the sequence an by a1 f(3,1) and an 1 f an, 1 forn 1. Find a2011.
26) Find the sum of all of the positive real solutions of x2– 4 – 4 1 .
27 TriangleABCis a 3 4 5 right triangle withAB 4.
Construct the perpendicularAD1and letAD1 x1. Construct the
perpendicular D1D2and letD1D2 x2. Construct the perpendicular
D2D3and letD2D3 x3. If this process is continued forever,
30) How many liters of a brine solution with a concentration of 30% salt should be added to 300 liters of brine with a concentration of 23% salt so that the resulting solution has a concentration of 26% salt?
31) Four horses compete in a race. In how many different orders can the horses cross the finish line, assuming that all four horses finish the race and that ties are possible?
32 As shown in the sketch, on each side of a square with side length 4, an interior semicircle is drawn using that side as a diameter. Find the area of the shaded region.
33) Two large pumps and one small pump can fill a swimming pool in 4 hours. One large pump and three small pumps can fill the same swimming pool in 4 hours. How many minutes will it take four large pumps and four small pumps, working together, to fill the swimming pool? (Assume that all large pumps pump at the rate R and all small pumps pump at the rate r.)
34) There are 40 students in the Travel Club. They discovered that 17 members have visited Mexico, 28 have visited Canada, 10 have been to England, 12 have visited both Mexico and Canada, 3 have been only to England and 4 have been only to Mexico. Some club members have not been to any of the three foreign countries and an equal number have been to all three countries. How many students have been
35) A bag contains 11 candy bars: three cost 50 cents each, four cost 1 dollar each and four cost 2 dollars each. Three candy bars are randomally chosen from the bag, without replacement. What is the probability that the total cost of the three candy bars is 4 dollars or more? Express your answer as a rational number in lowest terms.
36) If x4 + x3 + x2 + x + 1 0 and x + 1
x > 0, determine the value of x + 1 x.
37) Suppose that a and r are real numbers such that the geometric series whose first term is a and whose ratio is r has a sum of 1 and the geometric series whose first term is a3 and whose ratio is r3 has a sum of 3. Find a.
38 PointsA 0, 0 ,B 18, 24 andC 11, 0 are the vertices of triangleABC. PointPis chosen in the interior of this triangle so that the area of trianglesABP,APCandPBCare all equal. Find the coordinates ofP.
Express your answer as an ordered pair x, y.
A 0,0
B 18,24
C 11,0 P
x
y
39 Find the distance between the centers of the inscribed and circumscribed circles of a right triangle with sides of length 3, 4 and 5.
A B
C
40) Let S be the set of all 11–digit binary sequences consisting of exactly two ones and nine zeros. For example, 00100000100 and 10000100000 are two of the elements of S. If each element of S is converted to a decimal integer and all of these decimal integers are summed, what is the value of the sum? Express your answer as an integer in base 10.
41 As shown in the sketch, circular arcsA C and B Chave respective centers at
BandA. Suppose thatSis a circle that is tangent to each of these arcs and also to the line segment joiningAandB. Find the radius ofSif AB 24.
A B
