THE KENNESAW STATE UNIVERSITY HIGH SCHOOL MATHEMATICS COMPETITION PART I MULTIPLE CHOICE NO CALCULATORS 90 MINUTES

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1 THE KENNESW STTE UNIVERSITY HIGH SHOOL MTHEMTIS OMPETITION PRT I MULTIPLE HOIE For each of the following questions, carefull blacken the appropriate bo on the answer sheet with a # pencil. o not fold, bend, or write stra marks on either side of the answer sheet. Each correct answer is worth 6 points. Two points are given if no bo, or more than one bo, is marked. Zero points are given for an incorrect answer. Note that wild guessing is apt to lower our score. When the eam is over, give our answer sheet to our proctor. You ma keep our cop of the questions. NO LULTORS 90 MINUTES. bos and girls club found it could achieve a membership ratio of girls to one bo either b inducting girls or b epelling bos. What is? () 6 () () 8 () (E) 6. In the figure, the measure of is three times the measure of, and the measure of E is twice the measure of F. What is the degree measure of? F () 6 () () 0 () 7 (E) 7 E. For, if 7 7, compute the product. 7 () () 7 () () (E) 7. In the sequence 007, a, b, 008, each term starting with the third term, b, equals the sum of the two previous terms. ompute the ratio of b to a. 008 () 007 () 009 () 0 0 () 008 (E) 0. prime-prime is a prime number that ields a prime when its units digit is omitted. (For eample, 7 is a three-digit prime-prime because 7 is prime and is prime). How man two-digit prime-primes are there? (Recall that is not a prime number.) () 7 () 8 () 9 () 0 (E)

2 6. lan earns $ more in hours than Jo earns in hours. Jo earns $0.0 more in hours than lan earns in hours. How much does lan earn per hour? () $.00 () $6. () $7.0 () $8.7 (E) $ Find the value of which satisfies log ( ) log. () () () 7 () 8 (E) 9 8. driver wishes to arrive at her destination at eactl :00 a.m. If she drives at 0 mph, she would get there at 0:00 a.m. If she drives at 0 mph, she would arrive at noon. How fast should she drive to arrive at :00 a.m. () mph (). mph () mph (). mph (E) 6 mph 9. The si basic trigonometric functions are sine, cosine, tangent, cosecant, secant, and cotangent. ompute the probabilit that a randoml chosen basic trigonometric function, when divided b a different randoml chosen basic trigonometric function, will result in a quotient that is itself one of the si basic trigonometric functions. () () () 7 () (E) None of these 0. On the graph shown, a line passes through the point (6, ) and intersects the positive - and -aes as shown. The shaded triangle has an area of 0. What is the -intercept of the line? () 8 () 8. () 9 () 9. (E) 0 (6, ). cartons, each containing the same number of marbles, have their contents emptied and repacked into smaller boes, each of which receives an equal number of marbles. What is the smallest possible (non-zero) total number of marbles involved? () 8 () () 86 () 7 (E) 967. For all real numbers, the function f() satisfies f(+) + f( ) =. What is the value of f()? () () () () (E) 6

3 . right triangle has sides of length, 6, and 0. point chosen on the shortest side of the triangle is equidistant from the other two sides of the triangle. ompute this distance. () () () (). Suppose N is a positive integer such that for which N is a non-reducible fraction. N (E) 6 <. ompute the number of values of N () () () () 6 (E) 7. One of the three integer roots of the equation a b a 0 (a 0) is the negative of a second root. ompute the value of b. () () () 0 () (E) 6. onsider all sets of nickels, dimes, and quarters having at least one coin of each tpe and which add up to one dollar. What is the probabilit that a set of this kind will consist of a prime number of coins? () () () () (E) 7 7. ompute the area of the region bounded b the graphs of = and =. () 6 () 6 () () (E) 8. The sum of the first ten terms of an arithmetic sequence is four times the sum of the first five terms. What is the ratio of the second term to the first. (Note: all terms of the sequence are non-zero.) () : () : () : () : (E) 6: 9. rectangular piece of paper with dimensions 6 b is folded along one diagonal, as shown. What is the sine of angle? () () () () (E) 6 6

4 0. The pages of a book are numbered through n. When the page numbers of the book were added, one of the page numbers was mistakenl added twice, resulting in the incorrect sum of 008. What was the number of the page that was added twice? () () 6 () 7 () 7 (E) 8. In triangle, sin:sin:sin = ::6, while cos:cos:cos = ::. Find the ordered pair (, ). () (, ) () (, ) () (9, 6) () (0, ) (E) (, 9). Quadrilateral is inscribed in a circle. The degree measures of angles,, and, in order, are integers that form an increasing geometric sequence. ompute the sum of all possible values for the measure of angle. () 6 () 80 () () 7 (E) 60. In a list of 00 numbers, ever one (ecept the end ones) is equal to the sum of the two adjacent numbers in the list. The sum of all 00 numbers is equal to the sum of the first 00 of them. Find that sum if the thirt-sith number in the list is 008. () 60 () 06 () 0 () 06 (E) 60. If a, b and c are three distinct numbers such that c ab 7, then compute a b c. a bc 7, b ac 7, and () () 7 () () 8 (E). straight canal is eactl 0 miles long and mile wide. Point is located miles inland from one end of the canal and point is located miles inland from the other end of the canal on the opposite bank. competitor starts from point, runs to the canal, swims directl across the canal and then runs to point. (His path is shown as the dotted segments.) ompute the least number of miles such a competitor ma run. () 0 () 0. () (). (E) ½ 0 mile

5 PRT I - Solutions: THE KENNESW STTE UNIVERSITY HIGH SHOOL MTHEMTIS OMPETITION. If = the number of bos and G = the number of girls, then G + = and G = ( ). Thus, G + = G +, so =. F. Represent the measures of the angles as shown. Then = + and = 80. From these two equations, = 6 and = 7 and m = 80 (6 + 7) = 7. E ( ) ( 7( ) ) 7. E a = b and 008 = a + b. Solving gives a and 0 b b so that 0. a. For a two-digit number to be a prime-prime, its ten digit must be a prime. Thus onl numbers in the 0s, 0s, 0s, and 70s are eligible. The primes in each of these groups are:, 9,, 7,, 9, 7, 7, 79, for a total of Let J = amount Jo earns in hour and let = amount lan earns in hour. Then = J + and J = +.. Solving these two equations together gives = $ log ( + ) = + log log ( + ) = log 8 + log log ( + ) = log (8). Therefore, + = 8 and = Let t represent the time it takes when she drives 0 mph. Then 0t = 0(t+) and t =, so the distance is (0)() = 0 miles. Then in order to arrive at :00, she must travel 0 for hours at = mph.

6 9. There are (6)() = 0 possible quotients. Using sine, cosine, and tangent in the numerator, the quotients that result in one of the basic functions are: sin sin cos cos tan tan tan, cos, tan, sin, csc, sin. cos tan sin cot sin csc Each of these si equations above can be replaced b a corresponding equation in which all of the trig functions used are replaced b their reciprocal functions. Thus, there are a total of, and the required probabilit is. 0 b b 0. E rea of the triangle = ab = 0, so ab = 0. lso,. 6 a Therefore, -ab a = -6b ab = 6b a or 0 = 6b a. 0 0 Since a =, we have 0 = 6b ( ) and 6b 0b 00 = 0. b b ividing b and factoring, (b + 0)(b 0) = 0, and b = 0. (0,b) (a,0) (6, ). E = ( )()() and = ( )(). The smallest possible total number of marbles is the LM of the two numbers, ( )()()() = Using = in the given equation, f(+) + f( ) = () and f() + f(0) = Using =, f( ) + f(+) = ( ) and f(0) + f() =. Solving the two equations for f() gives f() =.. onstruct a line segment from the point to the opposite verte. The two triangles formed are congruent (HL). Thus the hpotenuse is divided into segments of 6 and. Using the Pthagorean Theorem, + = (-) from which = There are positive integers N <. Since = ( )(), we must eliminate all those that are multiples of or. There are multiples of (including and ). There are two odd multiples of that are less than. Therefore, the total number of non-reducible fractions is ( + ) =.. In an equation of the form + a + b + c = 0 with roots p, q, and r, (i) p + q + r = -a, (ii) pq + pr + qr = b, and (iii) pqr = c Represent the roots of the given equation a + b + a = 0 b r, r, and p. Using (i) r + ( r) + p = a and p = a. Using (iii) (r)( r)(p) = (r)( r)(a) = -a. Therefore, r = and r =. Thus, two of the roots of the given equation are and -. Using (ii) ()( ) + ()(p) + ( )(p) = b and b =.

7 6. elow is a list of all the combinations of nickels, dimes, and quarters that meet the requirements of the problem. nickels dimes quarters # of coins prime 7 9 prime prime 6 7 prime Therefore, the desired probabilit is. 7. The region is a square with sides of length. The desired area is ( ) =. = = 8. Represent the sequence b a, a + d, a + d,..., a + 9d. Then the sum of the first 0 terms is 0a + d and the sum of the first terms is a + 0d. Therefore, 0a + d = (a + 0d) d = a. Hence, the first two terms are a and a, with a ratio of :. 9. learl, = 6. Since E, represent the lengths as shown. Using the Pthagorean Theorem in right, = = 7. and =.. Therefore, 6. sin() = Let K be the number of the page that is added twice. Then 0 < K < n+ and n + K is between n and n + (n+). Using the well known formula for the sum of the first n (or n+) positive integers, this n(n ) (n )(n ) becomes 008 and n(n + ) < 06 < (n + )(n + ). Since (6)(6) 06 6, we find b inspection that n = 6. Therefore, K = E

8 . E the Law of Sines, the sides of triangle are also in the ratio ::6. Using the Law of osines, 6 ()(6) cos cos ()(6) cos cos and 6 6 ()() cos cos 8 Therefore, cos:cos:cos = :9: and (, ) = (, 9). 6. Since the opposite angles of an inscribed quadrilateral are supplementar, m< + m< = 80. Since the measures of angles,, and form a geometric sequence, represent the angles in order as a, ar, and ar. Then a + ar = a( + r ) = 80 =. Hence, ( + r ) is a factor of 80. The onl factors of 80 which are more than a perfect square are = +, = +, and 0 = +. Now, r since the progression increases. If r =, then a = 6 and m< = 6, m< = 7, m< =, and m< = 08. If r =, then a = 8 and m< = 8, m< =, m< = 6, and m< = 6. Therefore, the possible values for the measure of < are 08 and 6, and the desired sum is.. E Let n represent the n th number in the list. If a, b, then b a, a, b, 6 a b, 7 a, 8 b, etc. Therefore, the sequence has period 6 (repeats itself ever si terms). If S n represents the sum of the first n terms, then S 6 = 0, S 00 = b a, S 00 = a + b. Since S 00 = S 00, then a b = 0. We are given 6 = 008, so that a b = 008. Thus a = 008, b = 06, and S 00 = ( ) = 60.. Subtract the nd and rd equations to get 0 b c a( b c) ( b c a)( b c). Since b and c are distinct, b + c = a. dd all three equations and regroup: a b c ac ab bc a b c a( b c) bc a b c a bc a b c 7. Thus, a b c.. The two right triangles and EF are similar since EF and E. The ratio of their corresponding sides is. Therefore, (0) E F mile and EF (0) 6. Using the Pthagorean Theorem ½ on each triangle, =, F = 7, and the shortest distance the competitor can run is. miles. 0