MATH DAY 2010 at FAU Competition A Individual

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1 NOTE: MATH DAY 2010 at FAU Competition A Individual 1. Enter your name on the answer sheet. Detach the answer sheet from the rest of the test before handing it in. You may keep the test as such. 2. Starred Problems Twenty of the problems are multiple choice. For the other five problems (identified with a star beside their number) the answer is in every case a positive integer which you enter directly beside the problem number on the answer sheet. Make sure you write clearly. 3. In the multiple choice questions, the option NA stands for None of the previous answers is correct. 4. In all questions,i stands for the imaginary unit;i 2 = log b a denotes the logarithm in base b ofa; log b a = c if and only ifb c = a. 6. If n is a non-negative integer, then n! stands for the product of all positive integers in the range1 n ifn 1, with0! defined to be 1. That is: 0! = 1,1! = 1,2! = 2,3! = 2 3 = 6,4! = = 24,5! = = 120,etc. 7. Do NOT assume that pictures are drawn to scale. They are merely intended as a guide. THE QUESTIONS 1. The least number of students that must be in a classroom to ensure that there are at least 10 boys or at least 10 girls is (A)10 (B)11 (C)19 (D)20 (E) NA 2. If greeting cards cost $2.50 for a box of 12, $1.25 for a packet of three, or 50 cents each, determine the greatest number of cards that can be purchased for $ A store cuts the price of an article by 25%. To restore its price to its original value, the store must increase the price by: (A)24 2 3% (B)25% (C)30% (D)33 1 3% (E) NA 4. Adam ran up a hill at a speed of 3 miles per hour, and then ran down at a speed of 6 miles per hour. His average speed for the round trip was (in miles per hour) (A)4 (B)4.5 (C)5 (D)5.5 (E) NA

2 5. The picture below shows grid lines in the plane, all grid lines intersect at points with integer coordinates. The lower left corner of the picture is the origin, the point at (0,0), the point in the upper right corner has coordinates (20,10). We will call a path from the origin to the point(20,10) proper if it stays on the grid lines and has a total length of 30. This means that a proper path can only go up or to the right, never down or to the left. One such path is shown with thick lines. How many proper paths are there from(0, 0) to(20, 10)? (20, 10) (0,0) (A) 30! 10!20! (B) 30! 10! (C) 30! 20! (D) 10!30! 20! (E) 30! (F) NA 6. The number10! = 3,628,800 ends in two zeros. In how many zeros does500! end. 7. For a certain integer n, the numbers5n+16 and8n+29 have a common factor larger than one. That common factor is: (A)11 (B)13 (C)17 (D)19 (E) NA 8. In how many ways can one buy 44-cent and 90-cent stamps with exactly 50 dollars? (A)1 (B)2 (C)3 (D)4 (E) NA 9. Determine the two last digits of (A)03 (B)21 (C)27 (D)49 (E)81 (F) NA 10. Iflog a x = 28 andlog b x = 14, thenlog b a equals (A) 1 2 (B) 2 (C) 42 (D) 24 (E) Can t be determined (F) NA

3 11. Let f(x) be a function such that f(x)+2f( x) = sinx for every real number x. What is the value off ( π 2)? (A) 1 (B) 1 2 (C) 1 2 (D)1 (E) NA 12. Determine the coefficient ofx 5 if the expression ( (1+x 2 ) 3 2x ) 4 is expanded and written in standard polynomial form. (A)0 (B) 187 (C) 300 (D) 384 (E) NA 13. Suppose it is known that one root of the equation3x 2 +ax+b = 0, where a,b are real numbers, is 2 + 3i. The value of b is (A) Undetermined (B) 3 (C) 13 (D) 26 (E) 39 (F) NA 14. The remainder of dividingx 33 4x x 7 byx 2 is (A)0 (B)2 31 (C) 7 (D)57 (E)64 (F) NA 15. Tom has13 numbers. He adds up all possible products of the numbers and finds a sum S. Now, he adds a 14-th number to the list, adds up all possible products of the numbers, and finds a sumt. What is the14-th number? (A) T S (B) 1+T 1+S (C) T S 1+S (D) T +S S 1 (E) NA Note. Among all possible products we have to include products of a single factor; that is, the 13 numbers themselves so that if the numbers are a 1,...a 13, then S = a 1 + +a 13 +a 1 a 2 + +a 12 a 13 +a 1 a 2 a 3 + +a 1 a 2 a Determine the largest number n such that 1 n 1000 and n has an odd number of positive divisors. (A) 789 (B) 841 (C) 900 (D) 999 (E) NA

4 17. How many positive integers divide at least one of the following two numbers: , ? (Just to make sure, the first number is three to the power ten, times thirty five to the power twenty; the second number is six to the fifteen, times five to the ten.) 18. Determine the smallest positive number with exactly 56 divisors. 19. The equation x 5 +ax 4 +bx 3 +cx 2 +dx+e = 0 has the roots1,2,3,4, and5. Determinec. (A)0 (B) 121 (C) 200 (D) 220 (E) 225 (F) NA 20. Suppose it is known that an equation of the form x n +a n 1 x n 1 + a 1 x+10 = 0, where all coefficientsa n 1,a n 2,...,a 1 are integers, has four distinct positive integer rootsm 1,m 2,m 3 andm 4. Determinem 1 +m 2 +m 3 +m 4. (A)10 (B)18 (C)20 (D)30 (E)36 (F) NA 21. Let x,y,z be real numbers such thatx+y +z = 1 and 1 x + 1 y + 1 z = 1. Then (A) They are all different, and none of them is 1. (B) All three of them must coincide. (C) One of them must be 1. (D) Two of them must equal 1. (E) NA

5 22. Two congruent rectangles of dimensions a b share a common diagonal as shown in the diagram below. Find the area of their overlap. b a (A) a(a2 +b 2 ) b (B) b(a2 +b 2 ) a (C) a(a2 +b 2 ) 2b (D) b(a2 +b 2 ) 2a (E) NA 23. Two poles, one 30 feet tall, the other one 45 feet tall, are set up near each other. From the top of each pole a wire is stretched to the base of the other pole. How high above the ground do the two wires meet (intersect)? (Determine x in the picture below.) 30 ft x 45 ft 24. A circle is drawn through vertices A and D of a square ABCD in such a way that the circle is tangent to sidebc. D C A B If the length of a side of the square is l, the radius of the circle equals (A) l 2 (B) l 2 2 (C) 5l 8 (D) 5l 2 8 (E) 3l 8 (F) NA

6 25. Between two congruent circles that are tangent to each other at A and to a linel at B,C, a third circle is constructed tangent to both circles and also to l. What is the ratio of the area of the small circle to the area of the curvilinear triangle ABC? A l B C (A) 1 2 (B) π 8(4 π) (C) π 8(4+π) (D) 1 3 (E) NA