# Math Olympiad Fall 2017

#### For last 2 weeks, we reviewed ways of writing numbers in different bases. We started by noting that in base-ten, a number can be written as the sum of powers of 10, with each power of 10 multiplied by a number from 0 to 9. For instance,

#### 395 = 3*10^2 + 9*10^1 + 5*10^0.

#### Problem 1) Write each of the following numbers as the sum of powers of 10, with each power of 10 multiplied by a number from 0 to 9:

#### a) 779

#### b) 1255

#### c) 12425

#### Armed with this knowledge, we reviewed how numbers in base-ten can be converted to binary numbers noting that each of the digits represents a power of 2 times 0 or 1. We started by nothing that the rightmost digit is 2^0 times 1 or 0, the next digit 2^1 times 1 or 0, and so on. As an example, we will rewrite the following binary number as the sum of powers of 2, with each power of 2 multiplied by 1 or 0:

#### 10101 = 1*2^4 + 0*2^3 + 1*2^2 + 0*2^1 + 1*2^0

#### This sum is 21 in base-ten.

#### Problem 2)

#### a) Convert the following base-ten number into binary: 225

#### b) What is the following binary number in base-ten? 1001011

#### Lastly, we showed how a base-ten number can be converted into a number of a different base, from 2 to 9, by using the same principles. So we noted that for a different base, the rightmost digit would be that base to the 0 power. The next would be that base to the power of 1, and so on. We also noted that for whatever base, the highest number that can be in a single position is the base – 1. For instance, the highest number that can be in a single position for a binary number is 1. For base-ten, the highest digit is 9. For base-seven, the highest digit is 6, and so on.

#### Problem 3)

#### a) Can the following number be in base-seven? 965

#### b) Convert 569 into base-six

#### c) Convert the following base-five number into base-ten: 432

# Math Genius

**HL 1 Revise problems we went over in class:**

1) At Springfield Elementary School, a shelf contains either 4 math books or 9 spelling books. On 20 shelves at Springfield Elementary, there are 117 spelling books and some number of math books. All 20 shelves are full. How many math books are on the shelves?

Solution:

Dividing 117 by 9, we get 13. So 13 shelves have 9 spelling books on them. That means there are 7 shelves left with math books. Since there are 4 math books per shelf for those shelves containing math books, there must be 28 math books.

2) Ari, Barry, and Carrie agree on a 4-digit number. Ari said the number is a multiple of 3, 5, and 11. Barry said the number is a multiple of 2 and 7. Carrie said that exactly one of the 4 digits is 9, but none of the digits is 6. What is the number they chose?

Solution:

Start by multiplying all of the numbers 2, 3, 5, 7, and 11 to get 2310. This is the smallest number that’s a multiple of all of the numbers. Now, if we multiply this number by any number greater than 4, we have a number that has more than 4 digits. We then try the four possibilities:

1*2310

2*2310

3*2310

4*2310

Only the last one fits the criteria, so the number is 9240.

3) What is the greatest multiple of 37 that has exactly four digits?

#### Solution:

The largest possible four-digit number is 9999, so this puts a boundary on the number. I started with 3 * 37, which is 111. If you multiply this number by 10, you get 1110. If you multiply this number by 9, you get 9990. If you add one more 37, you go over the boundary number, so this number must be it.

**HL 2 Video: Watch Silly video about 7*13**

**HL 3 – Game:**

Powers of 2. In class, we went over powers. In the game at the link below, the goal is to reach powers of 2. You do this by using the arrow keys. If a power of 2 is on top of the same power of 2, if you press the down or up arrow key, the two powers of 2 will be summed to get a larger power of 2. If a power of 2 is next to the same power of 2, if you press the left or right arrow key, the two numbers will be summed.

https://poweroftwo.nemoidstudio.com/1024

**HL 4- Solve the following 2 questions**

#### 1) Amy has a nickel, a dime, a quarter, a half-dollar, and a silver dollar. After she lost one coin, she had exactly seven times as much money as her brother had. Which coin did she lose?

2) In a four-digit number, the sum of the thousands and hundreds digits is 3. The tens digits is 4 times the hundreds digit. The ones digit is seven more than the thousands digit. No two digits are equal. What is the four-digit number?

**HL 5 – Review the following test questions discussed in class **

#### 1) What is the value of: 80 x 3 – 70 x 4 + 60 x 5 – 50 x 6 + 40 x 7 – 30 x 8 + 20 x 9?

*Solution:*

*Solution:*

#### Start by grouping:

#### (80 x 3) – (70 x 4) + (60 x 5) – (50 x 6) + (40 x 7) – (30 x 8) + (20 x 9)

#### Now rearrange so that it’s clear which terms cancel each other out:

#### (80 x 3) – (30 x 8) + (40 x 7) – (70 x 4) + (60 x 5) – (50 x 6) + (20 x 9)

#### Starting from the left, every two numbers cancel each other out except for the last number, (20 x 9). So the answer is 180.

#### 2) The cost of 3 pencils and 2 markers is $1.30.

#### The cost of 2 pencils and 3 markers is $1.20.

#### What is the total cost of 7 markers and 7 pencils?

**Solution:**

**Solution:**

#### We can solve this using a system of equations. We will review this problem in class.

#### Let p represent a pencil and m a marker. We can rewrite the above in the following way:

#### 3p + 2m = 1.30

#### 2p + 3m = 1.20

#### We rewrite the second equation in the following way:

#### 3m = 1.20 – 2p

#### m = .40 – (2/3)p

#### Now we plug the value of m into the first equation:

#### 3p + 2(.40 – (2/3)p) = 1.30

#### 3p + .80 – (4/3)p = 1.30

#### 3p – (4/3)p = .50

#### (5/3)p = .50

#### p = .30

#### Since we have the value of p, we can plug this into the first equation:

#### 3(.30) + 2m = 1.30

#### 2m = .40

#### m = .20

#### We plug both into the second equation to make sure they work:

#### 2(.30) + 3(.20) = 1.20

#### So a pencil is 30 cents, and a marker is 20 cents.

#### The solution is 7(.30) + 7(.20) = $3.50.

**HL 6 – Homework problems:**

##### 1) Bryan can buy candy canes at 4 for 50 cents and can sell them at 3 for 50 cents. How many canes must Bryan buy in order to make a profit of 5 dollars?

##### 2) Suppose Sandy writes every whole number from 1 to 100 properly and without skipping any numbers. How many times will Sandy write the digit 2?

**HL 7 – Math fun:**

##### Start with a large number. Count the number of even digits, the number of odd digits, and the sum of the two numbers (number of even digits and number of odd digits). Put the numbers next to each other with the number of even digits first, the number of odd digits next, and finally the sum. Now repeat the process for this number. Soon enough, you will arrive at a number that produces itself.

##### Example: 98127665321

##### Number of even digits: 5

##### Number of odd digits: 6

##### Sum: 11

##### New number: 5611

##### Number of even digits: 1

##### Number of odd digits: 3

##### Sum: 4

##### New number: 134

##### Number of even digits: 1

##### Number of odd digits: 2

##### Sum: 3

##### New number: 123

##### Number of even digits: 1

##### Number of odd digits: 2

##### Sum: 3

##### New number: 123

##### So with this process, we have arrived at a number that will produce itself.

**HL 8 Watch Video:**

## https://www.ted.com/talks/arthur_benjamin_does_mathemagic

**HL 9 Complete the following problems:**

1) Each letter in the ordered list A, B, C, D, E, F, G, H represents a number. The numbers are not necessarily different, which means that two letters can represent the same number. The sum of the values of any three adjacent letters is 20. When B = 6 and D = 9, what is the value of F?

2) The Funny Book has its pages numbered in the following way: 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5…That is, there is one 1, two 2s, three 3s, four 4s, five 5s, and so on. How many actual pages (including front and back) will the funny book have if it contains all possible pages that are numbered “1” through “20”, but not any that are numbered “21”?

**HL 10 Math fun: Palindromes with multiplication:**

1 x 1 = 1

11 x 11 = 121

111 x 111 = 12321

1111 x 1111 = 1234321

11111 x 11111 = 123454321

**HL 11 Watch Video** What’s special about 196?

**HL 12 Puzzle: 1996 Puzzle**

### Visit https://www.mathsisfun.com/puzzles/1996.html

**HL 13 – Riddle:**

### You want to boil a two-minute egg. If you only have a three-minute timer (hourglass), a four-minute timer and a five-minute timer can you boil the egg for only two minutes?

**HL 14- Watch Video: Math trick**

**HL 15 – Solve the following:**

### A normal duck has two legs. A lame duck has one leg. A sitting duck has no legs. Donald has 33 ducks. He has two more normal ducks than lame ducks and two more lame ducks than sitting ducks. How many legs in all do the 33 ducks have?

**HL 16 – Solve the following:**

### Tracy has A quarters and B dimes with a total value of $3.45. Tracy has more quarters than dimes. How many different values of A can Tracy have?

1) Solve the following:

- a) Staci looks at the first and fourth pages of a chapter in her book. The sum of their page numbers is 47. On what page does the chapter begin?
- b) Different letters represent different digits. AB is an even two-digit number. EEE is a 3-digit number. M has a single digit. Find AB, EEE, and M given than EEE = M * AB

2) You are given the following pattern: 2, 4, 6, 8…

Person A says, “If the pattern continues, the next number must be 10.” Is he right? If not, why not?

3) Summing the first x numbers is given by the formula (x*(x + 1))/2. For instance, the sum of the first ten numbers is given by (10 * 11)/2, which is 55. What about summing the first x numbers, where each of the numbers is cubed? That is, what about summing 1^3 + 2^3 + 3^3+…x^3? For instance, the sum of the first three numbers with each of them cubed is 1^3 + 2^3 + 3^3 = 1 + 8 + 27 = 36.

Hint: what happens if you raise the formula for summing the first x numbers to a power?

**HL 17 – Solve the following:**

### a) Staci looks at the first and fourth pages of a chapter in her book. The sum of their page numbers is 47. On what page does the chapter begin?

b) Different letters represent different digits. AB is an even two-digit number. EEE is a 3-digit number. M has a single digit. Find AB, EEE, and M given than EEE = M * AB

**HL 18 – Pattern recognition :**

### You are given the following pattern: 2, 4, 6, 8…

### Person A says, “If the pattern continues, the next number must be 10.” Is he right? If not, why not?

**HL 19 – Solve the following:**

### Summing the first x numbers is given by the formula (x*(x + 1))/2. For instance, the sum of the first ten numbers is given by (10 * 11)/2, which is 55. What about summing the first x numbers, where each of the numbers is cubed? That is, what about summing 1^3 + 2^3 + 3^3+…x^3? For instance, the sum of the first three numbers with each of them cubed is 1^3 + 2^3 + 3^3 = 1 + 8 + 27 = 36.

**Hint:** what happens if you raise the formula for summing the first x numbers to a power?

**HL 20 – Math Jokes**

### a) Why didn’t the quarter roll down the hill with the nickel? It had more cents.

### b) What happened to the plant in math class? It grew square roots.

### c) What do you call a number that can’t keep still? A roamin’ numeral.

### d) Why is 6 afraid of 7? Because 7 8 9.