Pizza and Problems
Pizza and Problems
Pizza and Problems
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<strong>Pizza</strong> <strong>and</strong> <strong>Problems</strong>Fall 2009Assigned on: December 1, 2009 Due on: December 12, 2009Problem 1 What is the remainder when 3 0 + 3 1 + 3 2 + · + 3 2009 is divided by 8?Problem 2 Triangle △ABC has a right angle at B, AB = 1, <strong>and</strong> BC = 2. The bisector of ∠BACmeets BC at D. What is the length BD?A1BDProblem 3 Boris has an incredible coin changing machine. When he puts in a quarter, it returnsfive nickels; when he puts in a nickel, it returns five pennies; <strong>and</strong> when he puts in a penny,it returns five quarters. Boris starts with just one penny. Which of the following amountscould Boris have after using the machine repeatedly?(a) $3.63 (b) $5.13 (c) $6.30 (d) $7.45 (e) $9.07Problem 4 Five unit squares are arranged in the coordinate plane as shown, with the lowerleft corner at the origin. The slanted line, extending from (a, 0) to (3, 3), divides the entireregion into two regions of equal area. What is the value of a?y(3, 3)C(a, 0)xProblem 5 The first term of a sequence is 2005. Each succeeding term is the sum of the cubesof the digits of the previous term. What is the 2005 th term of this sequence?Problem 6 For k > 0, let I k = 10 . . . 064, where there are k zeros between the 1 <strong>and</strong> the 6. LetN(k) be the number of factors of 2 in the prime factorization of I k . What is the maximumvalue of N(k)?Assigned: December 1, 2009 1 Due: December 12, 2009
<strong>Pizza</strong> <strong>and</strong> <strong>Problems</strong>Problem 7 Many Gothic cathedrals have windows with portions containing a ring of congruentcircles that are circumscribed by a larger circle. In the figure shown, the number of smallercircles is four. What is the ratio of the sum of the areas of the four smaller circles to the areaof the larger circle?Problem 8 The diagram shows 28 lattice points, each one unit from its nearest neighbors.Segment AB meets segment CD at E. Find the length of segment AE.AECDProblem 9 Let x <strong>and</strong> y be two-digit integers such that y is obtained by reversing the digits ofx. The integers x <strong>and</strong> y satisfy x 2 −y 2 = m 2 for some positive integer m. What is x+y+m?Problem 10 Find all positive integers n such that 20 n − 13 n − 7 n is divisible by 309.BAssigned: December 1, 2009 2 Due: December 12, 2009
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