sponsored by Wake County Public School System College of Physical and Mathematical Sciences at North Carolina State University

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1 1997 NC STATE UNIVERSITY MATHEMATICS COMPETITION (Previously the Frank MKee Exellene in Mathematis Competition) November 8, 1997 Department of Mathematis North Carolina State University sponsored by Wake County Publi Shool System College of Physial and Mathematial Sienes at North Carolina State University DIRECTIONS: Please do not turn this page until told to do so. The test is in two parts: Part I onsists of 16 multiple hoie problems. Eah problem is worth 2 points. You are asked to irle the orret answer. Part II onsists of 12 problems for whih you are asked to show your method of solution learly. Eah of these problems is worth 4 points. Please work eah problem from Part II on a separate sheet of white paper provided to you. The yellow paper is for srathwork and will not be sored. You may alloate the 90 minutes allotted for this test in any manner you wish, but a reasonable strategy would be to spend 30 minutes on Part I and 60 minutes on Part II.

2 PART I: MULTIPLE CHOICE ( Cirle the orret answer.) ²b³ ²³ 1. b is equal to ²³ ²b³ (a) (b) () (d) 0 (e) 2. Solve for % if log² log² log %³³ ~. (a) (b) () (d) (e) 8 3. If % is used to approximate the value of ÁO%O, the ratio of the error made b% to the orret value is (a) % (b) % () (d) % (e) % 4. If % %b and 7 ~ % b %b, then b% b% b% (a) 7an take any real value (b) 7 () 7 (d) 7 (e) 7 5. Find the minimum value of l% b & if % b & ~ À (a) (b) () (d) (e) 6. Let ²%³ ~ log 4b% 5 %b% and ²%³ ~. Then ²²%³³ ~ % b% (a) ²%³ (b) ²%³ () ²%³ (d) ²%³! ²%³ (e) ²%³! 7. If and are roots of % b%b~á Á Á then the sum of the roots is (a) (b) () (d) (e) 8. In the expansion of ² %³ the oeffiient of % is (a) (b) () (d) (e) 2

3 9. If os ~ where, then the value of os is (a) (b) () (d) (e) 10. When the base of a triangle is inreased by 10% and the altitude is dereased by 10%, the hange in the area is (a) 1% inrease (b) % inrease () 0% (d) % derease (e) 1% derease 11. If represents a two-digit number in the base, and if is twie, then is: (a) 7 (b) 8 () 9 (d) 11 (e) The area of the ring between two onentri irles is square inhes. The length of a hord of the larger irle tangent to the smaller irle, in inhes, is (a) (b) () l l (d) (e) l 13. A triangle has angles of 30 and 45. If the side opposite the 45 angle has length, then the length of the side opposite the 30 angle is (a) (b) l () l (d) l (e) Let - ~ ÀÄ, with digits and repeating. Then when - is written as a fration in lowest terms, the denominator exeeds the numerator by (a) (b) () (d) (e) 15. The expression o% n% m% l%, written as a power of %, is (a) % (b) % () % (d) % (e) % 16. If %~b and &~b then & in terms of % is %b % % (a) (b) () (d) (e) % % %b % % 3

4 PART II: BEGIN EACH PROBLEM ON A NEW SHEET OF WHITE PAPER To reeive full redit, you must arefully organize your work on your answer sheets. 1. Hugh, Al, and Milt, working together, an omplete a job in % hours. Working alone, Hugh ould omplete the job in %bhours. Similarly, Al ould do the job alone in %bhours, and Milt ould do the job alone in % hours. Solve for %À 2. The number ~ is exatly divisible by two numbers between and. Determine these numbers. 3. A piee of string is ut in two at a point seleted at random. Find the probability that the longer piee is at least % times as long as the shorter pieeá where %. (The probability that a random point is in an interval is proportional to its length.) 4. Lines are drawn through the point ²Á ³ and the trisetion points of the segment joining the points ²Á ³ and ² Á ³. Find the equations of these two lines. 5. Let 6()* be a unit square in the %&-plane with verties 6²Á ³Á (²Á ³Á )²Á ³Á and *²Á ³. Let " ~ % & and # ~ %& be a transformation of the %&-plane into the "#-plane. Find the image of the square in the "#-plane and sketh it. 6. If % & ' ~ and % b & ' ~, ', then determine the value of % b%& & b'. 7. An equilateral triangle ()* with side length inhes is plaed inside a square with side length inhes as shown in the figure. The triangle is rotated lokwise about ), then about *, and so on along the sides of the square until * first omes to the original position of (. Find the length of the path in inhes traversed by vertex *À C A B 4

5 8. Find all values % suh that tan ²% b ³ b tan ²% ³ ~ tan 8 9À % & 9. The ellipse b ~ has fous - at ²Á³. For a variable point 7 on the ellipse, let 4 be the midpoint of 7-. Show that as 7 varies, the lous of 4 is an ellipse, and find its enter. 10. Find all real solutions of the system of equations % % ' ~ & ' ~ %! % b & b ' ~ À 11. Prove that if O%O and O&O, then O% b &O O b %&O. 12. To number the pages of a bulky volume, the printer used digits. How many pages has the volume? 5