Math Olympiad Strategies

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Presentation transcript:

Math Olympiad Strategies Solving a Simpler Problem

Lucky Sevens The houses on Main Street are numbered consecutively from 1 to 150. How many house numbers contain at least one digit 7? 147 148 149

Lucky Seven solution You could examine each house number from 1 to 150, but that requires more work than is necessary. Let’s separate this problem into two simpler problems. 1. How many house numbers contain 7 in the ones place? This occurs once in every set of 10 consecutive numbers. So for 1 to 150 there are 15 sets of 10 consecutive numbers, so 15 house numbers contain the digit 7 in the ones place. 2. How many houses contain the digit 7 in the tens place? 70- 79 or 10 house numbers. Since we have already counted 77 as a 7 in the ones place, we will only add 9 to our total. So, 15 + 9 = 24 house numbers contain at least one digit 7.

Operation # Britney’s calculator has a special # key that obeys two rules: Rule 1: If the display show a one-digit number, pressing the # key replaces the display with twice its value. Rule 2: If the display show a two-digit number, pressing the # key replaces the display with the sum of the two digits. Suppose Britney enters the value 1 on the calculator and then presses the # key repeatedly. What does the display show after she presses the # key 50 times?

Operation # Solution To solve this type of problem, you should generate enough numbers to see if a pattern exists. First 1 is in the display, then the # button is pressed: 1,2,4,8,16,7,14,5,10;1,2,4,8,16,7,14,5,10… you can see that the pattern repeats after pressing the # button 9 times. So take the 50 and divide by 9 which equals 5 remainder 5. Remember the 1 was already displayed so after 9 presses the display shows a 1. It will also show a 1 on 18, 27, 39 and 45. So the remainder of 5 will occur on 7. The display shows 7 after she presses the # 50 times.

Solving a Simpler Problem Homework NAME:______________________________ Due:____________________________ Suppose that the houses on Main Street are numbered consecutively from 1 to 150. how many house numbers contain at least one digit 9?, 4?, 1? Suppose Britney enters the value 3. How many times must she press the # key in order to next show the value 3 again? Suppose Britney enters the value 7 and then presses the # key 100 times. How many times will the display show 5? Britney’s calculator also has a special @ key that increases each display by 3. Suppose she enters a value of 2 and then presses the @ key repeatedly. What value is displayed if she presses the @ key A) 3 times ? B) 5 times? C) 10 times? D) 100 times?