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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 2) Express 5ê3 3ê2 J 86 – 1 –+ 29 – 2 N 1 2 and – 1 5 ? –1 as a rational number in lowest terms. 3) Define the binary operation Ñ by a Ñ b = a 2 + ab – b 2 . Find all real numbers x such that 7 Ñ x = 59 . 4) Find the positive real number x which satisfies the equation 1 1 1 = – . x 2 x+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 A E B 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. C D O 8) There are two candy jars, each of which contains the same number of pieces of candy. Bert takes 25 pieces of candy from the first jar and gives the rest to Karla, while Karla takes 17 pieces of candy from the second jar and gives the rest to Bert. When they are done, Karla discovers that Bert has more pieces of candy than she does. How many more pieces of candy does Bert have ? 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 . 127 2 – 123 2 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 ? B 13) In the figure, line segment BD bisects — ABC, AB = 7, BC = 12 and AC = 16. Find AD. A D C 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 p x number of accidents on city roads. Determine the value of y . Express your answer in the form q where 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 = 121 ? a-2 + b2 17) Find the coordinates of the point on the circle Hx + 1L 2 + Hy – 5L 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 a 0 = 2 , a 1 = 8 and a n = an–1 for n ¥ 2. Find a 2003 . an–2 20) In the isosceles trapezoid ABCD, BC and AD are parallel. If AD = 20, BC = 10 and AC = 25, find CD. 10 B C 25 A 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 ? D 20 B C A B 22) Find the degree measure of the acute angle b which satisfies the equation cos(81°) + cos(39°) = cos(b) . 23) Suppose that C 1 and C 2 are concentric circles, with C 1 being the larger circle. The length of a chord of C 1 which is tangent to C 2 is 28 cm. What is the area of the region which lies between C 1 and C 2 ? 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 ? 25) Suppose i = 26) If – 1 . Define a sequence of complex numbers by z 1 = 0 and z n + 1 = Hz n L 2 + i for n ¥ 1. Find z 2003 . x log 3 2 = 81, find x Ilog 3 2M 2 . x 8 27) Find the sum of the real solutions to the equation 4 + x = 6 . 4 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. y x 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. 31) Find the sum of all of the real zeros of the function f (x) = 2 cos(2x) + 1 in the interval 0 § x § 100 p . C 32) In D ABC, line segments DE and FG are parallel to AB and the three regions CED, DEGF and FGBA CD have equal areas. Find the value of . FA D E F G A B 33) The sum of all of the zeros of a cubic polynomial p (x) = x 3 + a x 2 + b x + c with c ∫ 0 is equal to twice the product of all of its zeros. The sum of the squares of all of the zeros of p(x) is equal to three times the product of all of its zeros. If p (1) = 1, find c. 34) Find the minimum value of xy if x + y2 2 2 5 §x§ 1 2 and 35) A circle is inscribed in D ABC where AB = 4, BC = 5 and AC = 3. Let t be the length of the line segment joining the points where the circle is tangent 1 3 3 § y § 8. B to sides BC and AC. Find t 2 . t A 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 ? 38) An obtuse isosceles triangle ABC with equal angles at B and C is inscribed in a circle. Tangents are drawn through B and C which intersect at a point D. Suppose that 3 — ABC = — BDC. If x is the radian measure of — BAC, determine the value of x. D A B C 39) If a and b are decimal digits (not necessarily distinct), let n(a,b) = aba + bab. For example, n(5,8) = 585 + 858 = 1443. Find ⁄ n(a,b), where the sum is computed over all ordered pairs of decimal digits a and b . 40) If x is a real number, let [ x ] be the largest integer less than or equal to x. Define f (x) = [ x ] x . What is the largest possible two digit integer value of f (x)? 41) A circle C 1 is inscribed in an equilateral triangle with side length 1 unit. Construct a circle C 2 that is tangent to C 1 and two sides of the triangle. Then construct a circle C 3 that is tangent to C 2 and two sides of the triangle. Continue constructing such circles indefinitely. Find the sum of the areas of this infinite sequence of circles. C1 C2