Transcript Math Olympiad Strategies

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? 7079 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
Due:____________________________
NAME:______________________________
1.
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?
2.
Suppose Britney enters the value 3. How many times must she press the # key in order to next show the value 3
again?
3.
Suppose Britney enters the value 7 and then presses the # key 100 times. How many times will the display show
5?
4.
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?