UNIVERSITY OF VERMONT
DEPARTMENT OF MATHEMATICS AND STATISTICS FORTY-SIXTH ANNUAL HIGH SCHOOL PRIZE EXAMINATION
MARCH 12, 2003 1) What rational number is equidistant from 1
2 and – 1 5 ?
2) Express 8
5 3– 93 2
6– 1+2– 2 – 1
as a rational number in lowest terms.
3) Define the binary operation Ñ by a Ñ b = a2 + ab – b2. Find all real numbers x such that 7 Ñ x = 59 .
4) Find the positive real number x which satisfies the equation 1
x+2 = 1
x –
1 2.
5) 2000 different episodes of a talk show originally aired on television network A. Cable channel B then rebroadcast 200 episodes of the show. Cable channel C later rebroadcast 460 episodes of the show, including all but 20 of the episodes previously shown on cable channel B. What percentage of the show's episodes have been rebroadcast by at least one of the cable channels ?
6) On a balance scale, 3 green balls balance 6 blue balls, 2 yellow balls balance 5 blue balls and 6 blue balls balance 4 white balls. How many blue balls are needed to balance a set of 4 green balls, 2 yellow balls and 2 white balls ?
7) The semicircle has radius 3 . Chord AB has length
3 and is parallel to CD. If O is the center of the
semicircle and OE is perpendicular to AB, find the
length of OE.
A
B
C
D
O
E
9) Last week, Earl took three quizzes in his algebra class. He earned 13 points out of 20 on the first quiz and 41 points out of 50 on the second quiz. There were 30 points available on the third quiz. When each quiz score was converted to a percent, the average percent score for the three quizzes was 79%. How many points did Earl earn on the third quiz ?
10) Find the value of 500 2
1272– 1232 .
11) A square in the plane has a pair of opposite vertices at the points (2 , 4) and (– 2 , 2). If the points (a , b) and (c , d) are the other two vertices, determine the value of a + b + c + d.
12) If you first multiply a number x by 4 and then subtract 12, you get twice as much as you get when you first subtract 12 from x and then multiply by 4. What is x ?
13) In the figure, line segment BD bisects—ABC, AB= 7, BC= 12 and AC =16. Find AD.
A
B
C
D
14) Consider the arrangement AAABBBCC. In how many ways can the eight letters be rearranged so that each position in the rearrangement is occupied by a type of letter which is different from the type of letter which occupied that position in the given arrangement?
15) Statistics for road use in a certain county show that in the past year, there were 32 accidents per 100,000 miles driven on rural roads and 18 accidents per 100,000 miles driven on city roads. Combined statistics for both rural and city roads show that there were 24 accidents per 100,000 miles driven. Let x be the total number of accidents on rural roads and y be the total
number of accidents on city roads. Determine the value of x
y. Express your answer in the form p
qwhere p and q are positive
integers having no common divisor other than 1.
16) How many pairs of positive integers ( a , b ) with a + b § 1000 satisfy a2+b 2
a 2+b2 = 121 ?
17) Find the coordinates of the point on the circle x +1 2 + y – 5 2 = 10 that is closest to the line y = 3x + 20 .
18) If 4 balls are randomly drawn without replacement from a box containing 5 red balls, 4 blue balls, 4 green balls and 2 yellow balls, what is the probability that for each color, the selection contains no more than 2 balls of that color ?
19) Define a0 = 2 , a1 = 8 and an = an– 1 an– 2 for
20) In the isosceles trapezoid ABCD, BC and AD are
21) Seven congruent rectangles form a larger rectangle ABCD. If the area of rectangle ABCD is 756 square units, what is the perimeter of ABCD ?
A
B
C
B
22) Find the degree measure of the acute angle b which satisfies the equation cos(81°) + cos(39°) = cos(b) .
23) Suppose that C1 and C2 are concentric circles, with C1 being the larger circle. The length of a chord of C1 which is
tangent to C2 is 28 cm. What is the area of the region which lies between C1 and C2 ?
24) Suppose that S is a set of 5 distinct positive integers such that when any 4 of the integers are added together, the possible sums are 169, 153, 182, 193 and 127. What is the largest integer in S ?
28) In the rectangle with vertices (0,0), (24,0), (24,32) and (0,32), form a triangle by connecting the midpoints of the sides containing (24,32). Then inscribe a circle in the triangle as indicated in the sketch. Find the coordinates of the center of this circle.
x
y
29) Recall that for a positive integer n, n! = n · (n – 1) · (n – 2) · · · 3 · 2 · 1. Find the number of zeros at the end of 2003! .
30) On a test, the average score of those who passed was 75, the average score of those who failed was 35 and the average score of the entire class was 60. What fraction of the class passed ? Express your answer as a rational number in lowest terms.
32) In D ABC, line segments DE and FG are parallel to
AB and the three regions CED, DEGF and FGBA
have equal areas. Find the value of CD FA. segment joining the points where the circle is tangent
to sides BC and AC. Find t2.
t
A
B
C
36) Suppose D ABC is inscribed in a circle whose radius is 10 cm. If the perimeter of the triangle is 31 cm, determine the value of sin(A) + sin(B) + sin(C) .
37) In D ABC, the respective coordinates of A and B are (0 , 0) and (15 , 20) . It is known that C has integer coordinates. What is the minimum positive area of D ABC ?
Sup-41) A circle C1 is inscribed in an equilateral triangle with
side length 1 unit. Construct a circle C2 that is tangent to C1 and two sides of the triangle. Then construct a circle C3 that is tangent to C2 and two sides of the triangle. Continue constructing such circles indefinitely. Find the sum of the areas of this
infinite sequence of circles.
