2002 AMC 12B Problems and Solution.doc

2002 AMC 12B Problems and SolutionProblem 1 The arithmetic mean of the nine numbers in the set is a -digit number , all of whose digits are distinct. The number does not contain the digit Solution We wish to find , or . This does not have the digit 0, so the answer is Problem 2 What is the value of when ? Solution By the distributive property, Problem 3 For how many positive integers is a prime number? Solution Factoring, we get . Exactly of and must be and the other a prime number. If , then , and , which is not prime. On the other hand, if , then , and , which is a prime number. The answer is . Problem 4 Let be a positive integer such that is an integer. Which of the following statements is not true: Solution Since , From which it follows that and . Thus the answer is . Problem 5 Let and be the degree measures of the five angles of a pentagon. Suppose that and and form an arithmetic sequence. Find the value of . Solution The sum of the degree measures of the angles of a pentagon (as a pentagon can be split into triangles) is . If we let , it follows that Note that since is the middle term of an arithmetic sequence with an odd number of terms, it is simply the average of the sequence. Problem 6 Suppose that and are nonzero real numbers, and that the equation has solutions and . Then the pair is Solution Solution 1 Since , it follows by comparing coefficients that and that . Since is nonzero, , and . Thus . Solution 2Another method is to use Vietas formulas. The sum of the solutions to this polynomial is equal to the opposite of the coefficient, since the leading coefficient is 1; in other words, and the product of the solutions is equal to the constant term (i.e, ). Since is nonzero, it follows that and therefore (from the first equation), . Hence, Problem 7 The product of three consecutive positive integers is times their sum. What is the sum of their squares? Solution Let the three consecutive positive integers be , , and . So, . Rearranging and factoring, , so . Hence, the sum of the squares is . Problem 8 Suppose July of year has five Mondays. Which of the following must occur five times in August of year ? (Note: Both months have 31 days.) SolutionIf there are five Mondays, there are only three possibilities for their dates: , , and . In the first case August starts on a Thursday, and there are five Thursdays, Fridays, and Saturdays in August. In the second case August starts on a Wednesday, and there are five Wednesdays, Thursdays, and Fridays in August. In the third case August starts on a Tuesday, and there are five Tuesdays, Wednesdays, and Thursdays in August. The only day of the week that is guaranteed to appear five times is therefore . Problem 9 If are positive real numbers such that form an increasing arithmetic sequence and form a geometric sequence, then is SolutionSolution 1 We can let a=1, b=2, c=3, and d=4. Solution 2 As is a geometric sequence, let and for some . Now, is an arithmetic sequence. Its difference is . Thus . Comparing the two expressions for we get . The positive solution is , and . Solution 3 Letting be the common difference of the arithmetic progression, we have , , . We are given that = , or Cross-multiplying, we get So . Problem 10 How many different integers can be expressed as the sum of three distinct members of the set ? SolutionEach number in the set is congruent to 1 modulo 3. Therefore, the sum of any three numbers is a multiple of 3. We can make all multiples of three between 1+4+7=12 (the minimum sum) and 13+16+19=48 (the maximum sum), inclusive. There are integers we can form. Problem 11 The positive integers and are all prime numbers. The sum of these four primes is Solution Since and must have the same parity, and since there is only one even prime number, it follows that and are both odd. Thus one of is odd and the other even. Since , it follows that (as a prime greater than ) is odd. Thus , and are consecutive odd primes. At least one of is divisible by , from which it follows that and . The sum of these numbers is thus , a prime, so the answer is . Problem 12 For how many integers is the square of an integer? Solution Solution 1 Let , with (note that the solutions do not give any additional solutions for ). Then rewriting, . Since , it follows that divides . Listing the factors of , we find that are the only solutions (respectively yielding ). Solution 2 For and the fraction is negative, for it is not defined, and for it is between 0 and 1. Thus we only need to examine and . For and we obviously get the squares and respectively. For prime the fraction will not be an integer, as the denominator will not contain the prime in the numerator. This leaves , and a quick substitution shows that out of these only and yield a square. Problem 13 The sum of consecutive positive integers is a perfect square. The smallest possible value of this sum is Solution Let be the consecutive positive integers. Their sum, , is a perfect square. Since is a perfect square, it follows that is a perfect square. The smallest possible such perfect square is when , and the sum is . Problem 14 Four distinct circles are drawn in a plane. What is the maximum number of points where at least two of the circles intersect? Solution For any given pair of circles, they can intersect at most times. Since there are pairs of circles, the maximum number of possible intersections is . We can construct such a situation as below, so the answer is . Problem 15 How many four-digit numbers have the property that the three-digit number obtained by removing the leftmost digit is one ninth of ? Solution Let , such that . Then . Since , from we have three-digit solutions, and the answer is . Problem 16 Juan rolls a fair regular octahedral die marked with the numbers through . Then Amal rolls a fair six-sided die. What is the probability that the product of the two rolls is a multiple of 3? Solution Solution 1 On both dice, only the faces with the numbers are divisible by . Let be the probability that Juan rolls a or a , and that Amal does. By the Principle of Inclusion-Exclusion, Alternatively, the probability that Juan rolls a multiple of is , and the probability that Juan does not roll a multiple of but Amal does is . Thus the total probability is . Solution 2 The probability that neither Juan nor Amal rolls a multiple of is ; using complementary counting, the probability that at least one does is . Problem 17 Andys lawn has twice as much area as Beths lawn and three times as much area as Carlos lawn. Carlos lawn mower cuts half as fast as Beths mower and one third as fast as Andys mower. If they all start to mow their lawns at the same time, who will finish first? Solution We say Andys lawn has an area of . Beths lawn thus has an area of , and Carloss lawn has an area of . We say Andys lawn mower cuts at a speed of . Carloss cuts at a speed of , and Beths cuts at a speed . Each persons lawn is cut at a speed of , so Andys is cut in time, as is Carloss. Beths is cut in , so the first one to finish is . Problem 18 A point is randomly selected from the rectangular region with vertices . What is the probability that is closer to the origin than it is to the point ? Solution The region containing the points closer to than to is bounded by the perpendicular bisector of the segment with endpoints . The perpendicular bisector passes through midpoint of , which is , the center of the unit square with coordinates . Thus, it cuts the unit square into two equal halves of area . The total area of the rectangle is , so the area closer to the origin than to and in the rectangle is . The probability is . Problem 19 If and are positive real numbers such that and , then is Solution Adding up the three equations gives . Subtracting each of the above equations from this yields, respectively, . Taking their product, . Problem 20 Let be a right-angled triangle with . Let and be the midpoints of legs and , respectively. Given that and , find . Solution Let , . By the Pythagorean Theorem on respectively, Summing these gives . By the Pythagorean Theorem again, we have Problem 21 For all positive integers less than , let Calculate . Solution Since , it follows that Thus . Problem 22 For all integers greater than , define . Let and . Then equals Solution By the change of base formula, . Thus Problem 23 In , we have and . Side and the median from to have the same length. What is ? Solution Let be the foot of the median from to , and we let . Then by the Law of Cosines on , we have Since , we can add these two equati