Brain Teaser Practice. Math Olympiad 2012

Transcription

1 Brain Teaser Practice Math Olympiad Ms. Arroyo asked the class to see if they could find the sum of the first 50 odd numbers. As everyone settled down to their addition, Terry ran to her and said, "The sum is 2,500." Ms. Arroyo thought, "Lucky guess," and gave him the task of finding the sum of the first 75 odd numbers. Within 20 seconds, Terry was back with the correct answer of 5,625. How does Terry find the sum so quickly? 2 I have a drawer full of socks. There are 40 socks of each colour, brown and grey, evenly mixed. In the morning, when it is very dark, I need to pull out a pair of socks of the same colour. How many must I pull out of the drawer to ensure that I have at least one pair of matching socks? 3 You have a piece of paper, 10cm by 10cm. Area = 100cm^2. For some reason, you need a square piece of paper with an area of 50cm^2. Using the paper you have, what's an easy way of getting the new square? 4 What is so unusual about these fractions? ,, There is only one pair of numbers that satisfies the equation x y y x where x does not equal y. Can you figure out which two numbers? 6 Can you place + and - signs between the following numerals to make the equation true? You can only use a total of 3 symbols and you cannot rearrange the numerals = 100

2 7 Checkers, the smartest dog in the world, was doodling in his math diary and come up with three equations: A B C D F E G H I He was about to write a fourth equation using multiplication when an interesting thought occurred to him. Instead of using a new set of numbers (J, K, L) for his fourth equation, he would use the answers of the three existing equations: C F I Now the only question in his mind was, "Can I replace the letters A through I with the digits 1 through 9 in such a way that all four of my equations are correct?" Can he? How? 8 If you reverse the digits of my age, you have the age of my son. A year ago, I was twice his age. How old are we both now? 9 I am the owner of a pet store. If I put in one canary per cage, I have one bird too many. If I put in two canaries per cage, I have one cage too many. How many cages and canaries do I have? 10 What's the largest natural number that satisfies this condition: Each digit from the third on equals the sum of its two preceding digits. 11 Sarah, Daniel and Megan had a contest to see who could bike the most miles. Together Sarah and Daniel biked 21 miles. The combined total for Sarah and Megan was 19. Together Megan and Daniel rode 22 miles. Who was the winner? 12 Waldo ate a total of 100 peanuts in 5 days. Each day, he ate 6 more peanuts than he did the day before. So, how many peanuts did Waldo start with on the first day? 13 Here is a sequence of words: gun, shoe, door, hive, kicks, heaven, gate, line, den Between which two words does the word free belong?

3 14 A Triple is a Series with 6 sets, each with 3 numbers apiece. The relationship between the first two numbers is the same throughout all of the sets, and the relationship between the last two numbers is also the same, however, it is different than the first relationship. Try and find the missing numbers: Follow these steps and see if you can figure this out. 1) Get a brown, cardboard box. 2) Get purple, orange, and turquoise paints. 3) Paint the box orange. 4) Paint on purple spots. 5) Paint on turquoise stripes. 7) Turn it upside down. 8) Lie on your side. What is missing from this sequence? 16 A family photo contained: One grandfather, one grandmother, two fathers, two mothers, six children, four grandchildren, two brothers, two sisters, three sons, three daughters, one father-in-law, one mother-in-law, one daughter-in-law. 29 people you may think, but no! What is the fewest number of people that could have been in the photo? 17 Below you will see six glasses. Three of these glasses contain orange. Moving only ONE glass, can you arrange the glasses such that those containing the orange are together? 18 Question 7 on the local BrainBashers pub quiz was: Name an ancient invention, which is still used in some parts of the world today, that allows people to see through walls. What can this be?

4 19 I went on a trip last week. The traffic was moderate and the journey took two and a half hours. On the return journey, the traffic was similar, but I made it back in 150 minutes. How come? 20 Can you move just one match to leave a perfect square? Draw your perfect square next to the original drawing. 21 Which of the four shapes comes next in this sequence?

5 22 Q1. Which is the first question where c) is the correct answer a) Q3 b) Q4 c) Q1 d) Q2 Q2. Which is the first question where a) is the correct answer a) Q4 b) Q2 c) Q3 d) Q1 Q3. Which is the first question where d) is the correct answer a) Q1 b) Q2 c) Q4 d) Q3 Q4. Which is the first question where b) is the correct answer a) Q2 b) Q4 c) Q3 d) Q1 23 Can you find a seven digit number which describes itself. The first digit is the number of zeros in the number. The second digit is the number of ones in the number, etc. For example, in the number 21200, there are 2 zeros, 1 one, 2 twos, 0 threes and 0 fours. 24 Can you find a seven digit number which describes itself. The first digit is the number of zeros in the number. The second digit is the number of ones in the number, etc. For example, in the number 21200, there are 2 zeros, 1 one, 2 twos, 0 threes and 0 fours.

6 25 Each of these sets of arrows represents a word - what are the words?