UNIVERSITY OF VERMONT
DEPARTMENT OF MATHEMATICS AND STATISTICS
FORTY-SEVENTH ANNUAL HIGH SCHOOL PRIZE EXAMINATION
MARCH 4, 2004

1)  Express     1/2 + 1/(1/3 + 1/(1/4 + 5))  as a rational number in lowest terms.

2)  Solve for n:   1/(1 + 1/n) = 3/4.   Express your answer as a rational number in lowest terms.

3)  Express   (2^( 2004) + 2^( 2002))/(2^( 2003) – 2^( 2001))   as a rational number in lowest terms.

4)  If  8^( x) = 125,  what is the value of  4^( – x) ?  Express your answer as a rational number in lowest terms  .

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  7)  A triangle has angles  α,  β and  γ.  Angle γ is 10° larger than α and the difference between α and β is 40°.  If β is the smallest
       angle, find the degree measure of angle β.

  8)  Tank A contains a mixture of lime juice and water, where 80% is water and the rest is lime juice.  Tank B contains pure
        lime juice.  If the contents of Tank A and Tank B are combined to fill a 200 liter tank with a mixture which is half lime
        juice and half water, how many liters were in Tank A ?

  9)  Find the distance from the point (– 4 , 1) to the center of the circle whose equation is   x^( 2) + y^( 2) + 2x + 6y = 15.  

10)  Find all values of x that satisfy  (5x)/(x + 3) + (x + 2)/x – 6/(x^( 2) + 3x) = 5 .

11)  The lengths of two opposite sides of a rectangle are each increased by 25% while the lengths of the other two sides are each
        decreased by 40% .  Find the percent decrease in the area of the rectangle.

12)  When the two wheels shown in the figure are spun,   
        two numbers are selected.  If the wheels are spun,
        what is the probability that the sum of the two
        numbers is even ?                                               
[Graphics:HTMLFiles/test_04_15.gif]

13)  Find the smallest positive integer divisible by each of  2, 6, 10, 15, 35 and 45.

14)  Find the largest integer n such that  1/2 + 1/n > 1/4 + 2/5.

15)  As shown in the sketch, a line is tangent to a circle
       centered at the origin.   The point of tangency is
       (4 , 3).  The line intersects the x-axis at x = a.  
       Find a.                                            
[Graphics:HTMLFiles/test_04_20.gif]

16)  Find all positive real numbers a such that the points  ( a , 12 )  and  ( 5 , a )  lie on a straight line of slope a.

17)  If a, b, c and d are real numbers such that  a/b = 2/3,  c/d = 4/5  and  d/b = 6/7, determine the value of a/c.  Express your
       answer as a  rational number in lowest terms.

18)  As shown in the sketch, a square is cut into  three
       congruent rectangles by two lines parallel to a side.  
       If each of the three rectangles has a perimeter of
       40 cm, determine the area of the square.
[Graphics:HTMLFiles/test_04_28.gif]

19)  Express   1/2004 + (2005 × 2003)/2004 + 2004  as a rational number in lowest terms.

20)  Express the value of s as a rational number in lowest terms where
       s = sin^( 2)(10°) + sin^( 2)(20°) + sin^( 2)(30°) + sin^( 2)(40°) + sin^( 2)(50°) + sin^( 2)(60°) + sin^( 2)(70°) + sin^( 2)(80°) + sin^( 2)(90°).

21)  As shown in the figure, six circles of radius 1 are
       arranged so that each circle is tangent to two
       others, while a seventh circle of radius 1 is tangent  
       to each of the other six.  What is the area of the
       shaded region ?
[Graphics:HTMLFiles/test_04_40.gif]

22)  Let θ be an acute angle such that tan(θ) = 3.  Find the value of cos(2θ).  Express your answer as a rational number in lowest terms.

23)  Let x and y be positive real numbers such that  log_ ( x) (y) = – 1/4.  Find the value of  log_ ( x) (xy^( 5)) – log_ ( y) (x ^2/y^(1/2)) .  Express your answer
       as a rational number in lowest terms.

24)  Find all real numbers x that satisfy   x/2^( – 3x) – 16x^( 3) = 16x – x^( 3)(8^( x)) .

25)  If x is a number such that  3x + 1/(2x) = 4 ,  what is the numerical value of   27x^( 3) + 1/(8x^( 3))?  

26)  A function f satisfies  f (x + y) = 4 f (x) f (y) for all real numbers x and y.  If  f (3) = 32,  find f (1).  

27) Express the product  (5^(1/2) + 7^(1/2) + 11^(1/2)) (5^(1/2) + 7^(1/2) – 11^(1/2)) (5^(1/2) – 7^(1/2) + 11^(1/2)) (–5^(1/2) + 7^(1/2) + 11^(1/2))  as an integer .

28) Suppose that x and y are real numbers such that  x  y = 8  and  x^2y + (xy)^( 2) + x + y = 144.  
       Determine the value of  x^( 2) + y^( 2).

29)  Eve has a collection of 120 different sweaters.  Each sweater is made of either cotton or wool, comes in one of three
       styles (full length sleeve, three quarter length sleeve or sleeveless), displays one of four geometric patterns (squares,
       triangles, stars, hexagons)  and is one of five colors (red, green, blue, yellow, gray).  How many of these sweaters
       differ from the red, sleeveless wool sweater with the triangle pattern in exactly two ways ?

30)  Find positive integers x and y such that  x^( 2) – y^( 2) = 11^( 3) with x as small as possible.

31)  Let f (x) = x/(9 + 8 x^( 2)).  Find the largest value of n such that f (n) = f (2n).

32)  If it takes 2004 digits to number the pages of a book, how many pages does the book contain ?

33)  Express   (1/2 – 1/3)/(1/3 – 1/4) · (1/4 – 1/5)/(1/5 – 1/6) · (1/6 – 1/7)/(1/7 – 1/8) · · ·  (1/2004 – 1/2005)/(1/2005 – 1/2006)  as a rational number in lowest terms.

34)  In a random arrangement of the letters  AAAABBBCCD, what is the probability that no two A's are next to each other ?

35)  After a difficult mathematics competition between two teams of nine students each, the eighteen competitors are ranked
       1 through 18 (there are no ties).  The team score is the sum of the ranks of the team members and the team with the
        lower score is the winner of  the competition.  How many different winning scores are possible ?

36)  Let S_ ( 1) be a square with side length 1.  In one
       of the triangles formed by the diagonals of S_ ( 1)
       inscribe a square S_ ( 2).  In one of the triangles
       formed by the diagonals of S_ ( 2) inscribe a square
       S_ ( 3). This process is continued indefinitely.
       If  A_ ( k) is the area of square S_ ( k) , find  Overscript[Underscript[∑, k = 1], ∞] A_ ( k) .
[Graphics:HTMLFiles/test_04_73.gif]
37)  Let ABC be a right triangle with
       hypotenuse 4.  On each side of Δ ABC an
       equilateral triangle is constructed outward
       as shown in the figure.  Find the sum of
       the areas of the equilateral triangles.
[Graphics:HTMLFiles/test_04_74.gif]

38)  For how many integers n,  1 ≤ n ≤ 2004, is the rational number  (n^( 2) + 11)/(n + 6)   NOT in lowest terms ?

39)  From a point P which lies outside a circle of
       radius r units, two secants are drawn.  The first
       secant intersects the circle at points A and B
       and the second secant intersects the circle at
       points C and D. Given that AB = 14,  CD = 2,  
       PA = 6 and  ∠APC = 60°, find r^( 2).                        
[Graphics:HTMLFiles/test_04_77.gif]

40)  Find the area of the region in the plane bounded by the straight lines  y = x ,  y = 1 – x ,  y = 1/2x  and  y = 1 – 2x .

41)  Find all ordered pairs (x , y) of real numbers such that  {9 7 2 2 -- + -- = 2 x + 2 y 4x 4y 9 ... 2 --   –   -- = – x + y 4x 4y  .

Created by Mathematica  (March 15, 2004)