University of South Carolina
High School Math Contest
December 2, 1995

1. Suppose that * is an operation on the integers defined by a*b = a² + b. What is the value of 3*(2*1)?
   
   (a) 12        (b) 14        (c) 54        (d) 170        (e) 172
   
   

2. Suppose that f(x) is a function from the real numbers to the real numbers such that f(x+f(x)) = 4f(x) and f(1) = 1. What is the value of f(5)?
   
   (a) 16        (b) 18        (c) 20        (d) 22        (e) 24
   
   

3. What is the largest prime divisor of 2¹⁶ - 16?
   
   (a) 7        (b) 11        (c) 13        (d) 17        (e) 23
   
   

4. In the figure below, the lines AN, AM and BC are tangent to the circle, and the length of AN = 7. What is the perimiter of triangle ABC?
   
   (a) 12        (b) 13        (c) 14        (d) 15        (e) 16
   
          
   
   
   

5. The average of some set of n positive numbers is 60. After removing one of the numbers, the average of the remaining n-1 numbers is 70. What is the largest possible value of n?
   
   (a) 22        (b) 20        (c) 12        (d) 8        (e) 6
   
   

6. In an infinite sequence of squares, every square except the first is formed by joining the midpoints of the sides of the previous square. What is the sum of the perimiters of the squares given that the first square has perimiter 1?
   
   (a) 2 - 1        (b) 1 + 2        (c) 1 +         (d) 2 +         (e) 2(1 + )
   
   
   
7. An square with the entries is said to be a magic square if the sum of the entries of any row or column is always the same. Examples of a magic square and a magic square are shown below. Find the value of A in the (partially filled) magic square.

   4 3 8 9 5 1 2 7 6

   16 3 2 13 5 10 11 8 9 6 7 12 4 15 14 1

   - - - - - - - - - - - - - - - 20 13 6 4 A - - - - -

   (a) 10        (b) 18        (c) 22        (d) 17        (e) 15
   
   
   

8. If are the roots of , then
   (a) 6        (b) -6        (c) 3        (d) -3        (e) 2
   
   

9. If , then
   
   (a) .88        (b) -.88        (c) .44        (d) -.44        (e) 0.4
   
   

10. Two vertical poles of different heights stand on level ground. Straight chords form the top of each pole to the base of th eother pole cross at a point 24 meters above ground. If the shorter pole is 40 meters tall, what is the height of the taller pole, in meters?
    
    (a) 48        (b) 52        (c) 56        (d) 60        (e) 64
    
    

11. The value of is 0.301... (accurate to as many digits as shown) How many base 10 digits does the number 5⁸⁰ have?
    
    (a) 56         (b) 57        (c) 58        (d) 59        (e).60
    
    

12. Which of the following is closest in value to ?
    
    (a) 1.002        (b) 1.0003        (c) 1.00045        (d) 1.0008        (e) 1.003
    
    

13. What is the remainder when 17¹⁰⁰ is divided by 7?
    
    (a) 1        (b) 3        (c) 4        (d) 5        (e) 6
    
    

14. If A + B = 12, B + C = 10, C + D = 16,
    then A + D = ?
    
    (a) 12        (b) 14        (c) 16        (d) 18        (e) 20
    

15. Two jugs each have a capacity of x gallons. One is filled with wine and the other with water. Two gallons are taken from each jug and then transferred to the other, after which each jug has its contents thoroughly mixed. Next, two gallons are again taken form each jug and transferred to the other. Given that the amount of wine in each jug is now the same, find x.
    
    (a) 3        (b) 4        (c) 5        (d) 6        (e) 7
    
    

16. The perimiter of a the trapezoid below is 35. Side CD has length 5. Also, and . What is the length of side AD?
           
    
    (a) 15        (b) 16        (c) 18        (d) 20        (e) 22
    
    

17. Atlanta is playing Cleveland in the World Series. Each team has an even chance to win any given game. The World Series is won by the first team to win 4 games, so at most seven games can be played. Atlanta is leading the series 3 to 2. What is the probability that Atlanta wins the series?
    
    (a) .50        (b) .60        (c) .65        (d) .75        (e) .80
    
    

18. Consider the following subsets of the integers: S₁ = {1}, S₂ = {2, 3}, S₃ = {4, 5, 6}, S₄ = {7, 8, 9, 10}, S₅ = {11, 12, 13, 14, 15}... Where each S_n contains one more element that the precedding S_n-1 and begins with the smallest integer not in S_n-1. What is the largest element of S₁₀₀?
    
    (a) 4000        (b) 4050        (c) 5000        (d) 5050        (e) 6000
    
    

19. If S is a set containing three or more integers, then there must be two integers in S whose sum is divisible by 2. For example, if S = {2, 11, 17}, then 11 + 17 is divisible by 2. What is the smallest positive integer n such that if S is a set containing n or more integers, then there must be three integers in S whose sum is divisible by 3?
    
    (a) 3        (b) 4        (c) 5        (d) 6        (e) no such n
    
    

20. The number of even positive integers that are divisors of is
    
    (a) 15        (b) 16        (c) 24        (d) 25        (e) 29
    
    
    

21. There are 44 permutations of the numbers 1, 2, 3, 4, 5 in which no number appears in its proper position; i.e., 1 is not first, 2 is not second, 3 is not third, etc.
    How many permutations of 1, 2, 3, 4, 5, 6, 7 have exactly two numbers appearing in their proper positions? (For example, 2 1 3 5 7 6 4 and 1 3 7 4 2 5 6
    
    (a) 900        (b) 924        (c) 930        (d) 936        (e) 954
    
    

22. Suppose that x is a complex number such that x² - x + 1 = 0. What is the value of x³?
    
    (a).-1        (b) 0        (c) 1        (d) 1.5        (e) 2
    
    
23. Suppose that the array below is completed so that each row and each column of the final array is a permutation of the numbers 1, 2, 3, 4. Then a + b =
    

    1	2	3	4
    2	-	-	-
    3	-	a	-
    4	b	-	-

    (a) 2        (b) 3        (c) 4        (d) 5        (e) 6
    
    

    
    

24. The probability that Michael can win a game (any game) against Dave is 0.20. What is the smallest integer n for which the following statement is true?
    "If Dave and Michael play n games, then Michael has a better than 60% chance to win at least one."
    
    (a) 2        (b) 3        (c) 4        (d) 5        (e) 6
    
    

25. Five points P₁, P₂, P₃, P₄ and P₅ are positioned in such a way that no three are collinear. Some lines are drawn with each line drawn passing through two of the five points and no two lines passing through the same two points. Suppose that for , that is the number of lines drawn that pass through P_j, and suppose further that . What is the value of ?
    
    (a) 0        (b) 1        (c) 2        (d) 3        (e) 4
    
    

26. If , then
    
    (a) -1        (b) i        (c) -i        (d) 1        (e) none of these
    
    

27. The last digit in 57⁸⁹ is
    
    (a) 1        (b) 3        (c) 5        (d) 7        (e) 9
    
    

28. Suppose that . What is the value of ?
    
    (a) 2150        (b) 2154        (c) 2158        (d) 2162        (e) 2166
    

29. Which of the following divides evenly into ?
    
            I.
            II.
            III.
            IV.
    
    (a) I and II only        (b) III and IV only        (c) I and III onlt
    (d) II and IV only        (e) II III and IV only
    
    

30. Given that the equations below all hold for x, y z, t, and w, determine which statement (a), (b), (c) , (d), or (e) is correct.
    
           
    
    (a) x must be -2        (b) z must be either 0 or positive
    (c) y must be negative        (d) none of x, y, z, t, and w can be uniquely determined.
    (e) There are no x, y, z, t, and w satisfying the equations.