FORTY-SECOND ANNUAL OLYMPIAD HIGH SCHOOL PRIZE COMPETITION IN MATHEMATICS Conducted by

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1 FORTY-SECOND ANNUAL OLYMPIAD HIGH SCHOOL PRIZE COMPETITION IN MATHEMATICS Conducted by The Massachusetts Association of Mathematics Leagues (MAML) Sponsored by The Actuaries' Club of Boston FIRST LEVEL EXAMINATION Tuesday, October 5, 005

2 Olympiad Level 1 Examination ) If 5 gallons of a Y% salt solution are mixed with gallons of a.5(y)% salt solution, what percent of the final mixture is salt? 1 y 1 s l B)-Y, Q-Y 11 D )T7 F 1 E) 7.) Given a= and 6=5,fmd c expressed in terms of a. b c A)-if- B)-^_ C)-^L D)-?- E) 5a- 5a + a-5 5a- 5a +.) The graph of {x: 1 - x <5 and 9x j > 9 } on the number line consists of two line segments. Find the length of the longer of these two segments.., A), B >7^ 5 C >7^ N 7 D )T^- 8 E).) In a particular 65 day calendar year, October n fell on a Tuesday. In the next calendar <>nd year (also 65 days), November will fall on which day of the week? A) Sunday B) Tuesday C) Wednesday D) Friday E) Saturday nd. 5.) A car travels a certain distance at 50 mph. It travels this same distance in 10 minutes less time than at 0 mph. What is the number of miles in this distance? A)1# B)1% Q1 / D)15 E) 6.) A right circular cone has an altitude of 1 inches and a diameter of 10 inches. The cone is cut by a plane parallel to its base and 8 inches from its vertex, forming a smaller cone. Determine the number of square inches in the lateral surface area of the smaller cone. AX 10;r A) ^ 60;r B) ^ 8V6br C) 60;r D) ^ 60/r E) 9 9 7

3 7.) Find the area bounded by the graphs of y = -\x +, y = 0, and y = -. A) B)9 C)1 D) E)6 8.) If X ly = 8, find the value of - y X Ax+5y y A) 9 8 B)-«9 Q D) 8 E) ) A right prism is inscribed in a sphere of radius 10 inches. The bases of the prism are equilateral triangles and its height is 16 inches. Find the number of cubic inches in the volume of the prism. A) 16V B) V C) 86V D) 86 E) ) Sean, Gautham, Andrew, and Nick go to the movies, and whenever anyone purchases an item, the cost is split evenly among the four at a later time. While they are there, Sean buys the tickets for $0, Gautham buys the candy for $10, Andrew buys the popcorn for $15, and Nick buys the soda for $15. Amidst the confusion of who is owed what amount of money, Nick and Gautham both give Sean $10. How much money does Andrew owe, and to whom? A) $ 10 to Sean B) $ 10 to Nick C) $5 to Gautham D) $5 to Nick E) $ 10 to Gautham 11.) The Olympiad Level 1 Exam has 0 questions. A correct answer is worth 5 points, an answer left blank is worth 1 point, and an incorrect answer is worth 0 points. How many different scores are possible? A) 11 B)1 C) 1 D)1 E)15 1.) In circle O of radius 5 (as shown), PQ _L EF and PH - 8. Find the length of EW. m M A)6- B) 6- C)6- D)8- E)8- o Q

4 1.) Given two circles with one in the interior of the other (as shown). FI contains the center of the smaller circle, the measure of arc FG is 6, mzfhg = 70, and the measure of arc AB is 6. Find the number of degrees in the measure of arc BC. p 1 A) 5 B)0 C)5 D)0 E)5 1.) Let S n be the set of three- digit numbers in base n. For how many values of n is there an element in S which has the same value as the base 10 number 005? A) B) C) D)5 E)6 15.) Let a n be the number of distinct positive factors of 005" (including 1 and 005"). 1 Find a k. k=0 A) 105 B)8 C)819 D)101 E) ) Abby, Brenda, and Carla are students in a school. Abby has 0 friends, Brenda has 0 friends, and Carla has 5 friends. There are less than students who are friends with all of the girls. Friendship is mutual. Find the minimum possible number of students in the school. A) 5 B)6 C)7 D)8 E)9 17.) In AABC (as shown), AC =, D is the midpoint of AC and mzcbd = 0. Find the maximum length of AB. A) >/6 B) ^ C) ^+1 D)>/ + E)->/+ D.) Given AABC(as shown) with AC = 1 and BC = 6. Arcs ABC, ADB, and BDC are all semicircles. Find the area of the region common to all three semicircles. A)6;r-9V 15;r B) 9V C) 6;r 9V D) 15^-^ E) 15;r

5 ) Find the remainder when /»! is divided by 00. A) B)78 C)1007 D)1078 E)81 0.) How many positive integers less than 005 satisfy none of the following conditions: (i) Divisibility by. (ii) Divisibility by 7. (iii) Being a perfect square. A) 815 B)80 C)8 D)98 E)985 1.) Bill chooses two cards at random from a standard deck of 5 cards. They are both spades. He then chooses 5 more cards at random. The probability that at least 5 of his 7 cards N A are spades is. 50^5 Determine the value of N. A)165 B)1990 C)15597 D)111 E) 150.) Given a and b are positive numbers unequal to 1, log b (a l Sb ) = log a (6 log "), and log a (p - (b - 9) ) =. Find the minimum value for p. B) 7 Q 9 *7 E).) 8 partial circles of radius one are drawn (as shown), each centered on one of the vertices of a regular octagon with side length one. The area of the shaded region can be written in the form a + by/ - cv - n, where a, b, c, and d are whole numbers. Find the value of a+b+c+d A) B)1 C) D)7 E)0... _ x. n, cosx + sinv.) Given 0<x<, 0<y<, and = cotx, theny = an-bx. cosj-sinx Find the value of a + b. A) B) C) D)5 E)6 5

6 5.) Given / (z) = the real part of a complex number z. For example, / ( - /) =. If a is a positive integer, find > log /1 (1 + *\/) I. 6a A)a +16a B)6a-19 C) 5a-7 D)a -a E) 6a -19a 6.) Find the sum of all roots to the equation 0 = sin n log f X w A) B) D) 0 9 \x) where 0 < x < K. 7.) The real roots of the equation x - 19x + 57x + k = 0 form an increasing geometric sequence x p x, x. Find the value of x. A) C) 15 D) E) 1 x y 8.) A line is drawn through the point (5,0) so that it is tangent to the ellipse + - = 1 at a point below the x-axis. Find the value of the slope of this line. A >7 B) C) 1 E) 1 9.) Boys and/or girls sit along a row of chairs. A boy must always sit next to at least one other boy and a girl must always sit next to at least one other girl. The row begins with a boy. If there are 15 chairs, how many possible gender arrangements are there? For example: BBBGGGGBBGGGGBB is a possible arrangement. A) 10 B)9 C)77 D)8 E) 1! 0.)U«at =(lj + (ij, where k is a positive integer. These numbers are plotted on the complex plane at the corresponding points A x, ^, A, etc. Let A k A k+x represent the distance between the points A k and A k+l. The value of the infinite sum A i A + A A + A A +... can be written as e, where a, b, c, d, and e are integers and the radicals are in simplest form. Determine the sum : a + b + c + d + e. A) 66 B)68 C)70 D) 7 E) 7 6