# MATHEMATICS CONTEST 2006 SOLUTIONS

PART II: 30 Minutes; CALCULATORS NEEDED

Section A. Each correct answer is worth 1 point.

 1. Evaluate: 45 − 54 Solution : This is simple on a calculator, but even without one, we can compute 45 = 42 · 42 · 4 = 162 · 4 = 256 · 4 = 1024, and 54 = 252 = 625. Therefore, 45 − 54 = 399. 2. The area of a triangle is 2006 sq. cm. Its base is 2006 cm. What is its height?Solution: Area equals 1/2bh = 1/2(2006)(h) = 2006, so h = 2 cm. 3. If 2/5 of a number is 2.5, what is the number?Solution: 4. Find the sum of x and y if 3x = 90, and 3y = x. Solution: Solve 3x = 90 to find x = 30, then solve 3y = x to find y = 10. Therefore, x + y = 30 + 10 = 40. 5. Find the measure (to the nearest degree) of the acute angle whose tangent is 1.732050808.Solution 1: Put your calculator in degree mode, and compute tan-1(1.732050808), which should give 60°(or 60.0000000062, but the problem asks for the answer to the nearest degree). Solution 2: If you recognize that you can draw a right triangle whose legs have length 1 and .so that the tangent of the larger acute angle would be . The hypotenuse length is therefore 2; these are the proportions for a 30–60–90 right triangle, meaning that the larger acute angle is a 60° angle. 6. The average of 11 numbers is 121. When one number is dropped, the average of the remaining set of numbers is 120. What number was dropped? Solution: If the average of 11 numbers is 121, their sum must be (11)(121) = 1331. If 10 numbers have average 120, their sum must be (10)(120) = 1200. Therefore, the missing number is 131. 7. Give an example of two numbers such that their product is positive, and their sum is negative.Solution: This is two for any two negative numbers, for example: (−1)(−2) = 2 is positive, but (−1) + (−2) = −3 is negative.

Section B. Each correct answer is worth 2 points.

 8. Take a three- digit number ending in 1, say ab1. The sum of the digits, a +b+1, is a two-digit number, cd. The product of those digits, c · d, equals 8. List all possible values for the original number.Solution: If c · d = 8, then the two-digit number cd must be one of these: 18, 81, 24, 42. But a ≤ 9 and b ≤ 9, so a + b + 1 cannot add up to more than 19. Therefore, a + b + 1 = 18, which means a and b are the digits 8 and 9 (in either order ); the original number was therefore 891 or 981. 9. A single die ( number cube ) is tossed three times. What is the probability that all three numbers are the same? Express as a ratio in simplest form . Solution 1: Here is one line of reasoning: There are 6·6·6 = 63 = 1296 possible outcomes from rolling the die three times. Of those 1296 possibilities, 6 have all numbers the same—either 3 Therefore, the probability that all faces are the same is Solution 2: Roll the die once. Regardless of how it turns up, there is a 1/6 chance that the second roll is the same as the first roll, and a 1/6 chance that the third roll is the same. Therefore, the probability that both rolls are the same as the first is 10. Doc Math chose a two-digit number. He subtracted it from 300 and doubled the result. Then he added the original two-digit number to this result. What is the largest number that Doc Math could get?Solution: If x is the two-digit number, Doc Math has computed (300 − x) · 2 + x = 600 − x. Because x ≥ 10, his final result is 600 − x ≥ 590. 11. While driving, Dale Jr. noticed that his car’s odometer reading, 47974 miles, was a palindrome (reads the same forward as backward). Dale Jr. continued driving, and two hours later the odometer showed the next possible palindrome. What was the average speed of his car during those 2 hours (in mph)?Solution: The next palindrome after 47974 is 48084, 110 miles later. Therefore, Dale Jr. traveled 110 miles in 2 hours, for an average speed of mph. 12. Find the largest integer n for which is a factor of Solution: We have That gives 12 “obvious” factors of 10, and 13 additional factors of 10 from combining paired factors of 2 and 5, for a total of n = 25 factors of 10.

Section C. Each correct answer is worth 3 points.

 13. In a set of ten whole numbers, the minimum number is 62, the range is 25, and the median is 82. Find the smallest and largest possible values for the mean of that set of numbers (given to the nearest tenth).Solution: The answer is 74.5 to 82. The smallest number is 62, the largest is 62 + 25 = 87, and the middle two numbers average to 82. That is, if listed in order, the 10 numbers are where The smallest (or largest) possible means happen when the blanks have the smallest (or largest) possible numbers in them: 62, 62, 62, 62, 82, 82, 82, 82, 82, 87, for which the mean is 74.5, and 62, 82, 82, 82, 82, 82, 87, 87, 87, 87, for which the mean is 82. 14. Given a triangle with sides 6, 7, and 9 units, write the length of the altitude to the shortest side. The answer may be exact, or to the nearest hundredth. Solution: Given the three side lengths a, b, c of a triangle, we can find its area using Heron’s formula : compute the semiperimeter s = (a+b+c)/2, then find For this triangle, s = 11, so the area is Using the more common area formula, with h as the altitude to the shortest side, we have Therefore, 15. Write the value of x. Solution: In general, if then Therefore, means that which in turn means that
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