Yusuf Sila: February 2012

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Problem of the Week from CEMC : Prime Picking

8:48 PM

Muhammad Yusuf , Posted in Math Competitions , 1 Comment

Problem

A natural number greater than 1 is said to be prime if its only factors are 1 and itself. For example, the number 7 is prime since its only factors are 1 and 7. A perfect square is an integer created by multiplying an integer by itself. The number 25 is a perfect square since it is 5 × 5 or 5². Determine the smallest perfect square that has three different prime numbers as factors.

Solution

The problem itself is not very dificult once you determine what it is asking. So before looking at the solution, let us examine some perfect squares.

The numbers 4 and 9 are both perfect squares that have only one prime number as a factor, 4 = 2² and 9 = 3².

The number 36 is a perfect square since 36 = 62. However, the number 6 = 2×3 so 36 = (2×3)² = 2×3×2×3 = 2² ×3². So 36 is the product of the square of each of two different prime numbers. In fact, since 2 and 3 are the smallest prime numbers, 36 is the smallest perfect square that has two different prime factors. In order to create a perfect square we must find the product of the squares of prime numbers.

Using this idea, we can create the smallest perfect square with three different prime factors by squaring each of the three smallest primes, 2, 3, and 5, and then multiplying these squares together. So the smallest perfect square that uses three different prime factors would be 2² × 3² × 52 = 4 × 9 × 25 = 900. It

should be noted that 900 = 30² = (2 × 3 × 5)².

⟹ the smallest perfect square with three different prime factors is 900.

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CEMC Problem : Intermediete - Feb 11

8:36 PM

Muhammad Yusuf , Posted in Math Competitions , 0 Comments

1. If 24 is added to a number, the number is tripled. What is the original number?

2. A bowl contains 40g of white rice and 60g of brown rice. If 100g of white rice is added to the mixture, then the percentage of the new mixture that is white rice is

3. If 1/R is the average of 1/4 and 1/6 , then R equals

4. If px = 20; 6x - 3q = 30 and x = 4. then the value of p - q is

5. Positive numbers a, b, c, d, and e have the following property : ab = 2, bc = 3, cd = 4, de = 5. What is the value of e/a?

6. With how many zeros does the product of the first consecutive 10 prime numbers end?

7. An integer is chosen randomly from the integers 1 to 101 inclusive. What is the probability that at least one of the digits of the integer chosen is 7?

8. T and U are two digits of the number 9T68U. This five-digit number is divisible by 15. The number of different possible value of (T + U) is

9. In triangle PQR, PQ = 7, PR = 9, and median PM = 7. the length of QR is

10. The sum of the series

is

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Math Problems : 2011 UNB - Grade 9 (Part B)

12:05 AM

Muhammad Yusuf , Posted in Math Competitions , 0 Comments

University Of New Brunswick

Junior High School Mathematics Competition

2011 – Grade 9

PART B

11. Jane has 5 chocolate bars, Karen has 3, and Amel doesn’t have any. The three friends share them evenly. All the chocolate bars cost the same. Amel pays his friends a total of $4.00 for his share. How much of the $4.00 should go to Jane ?

(A) $2.50 (B) $2.75 (C) $3 (D) $3.25 (E) $3.50

12. A rectangle is 245 cm long and 175 cm wide. The rectangle is to be cut into squares that are all the same size. The entire rectangle is to be used. What is the largest possible area, in cm2, of each of the squares ?

(A) 5² (B) 7² (C) 15² (D) 25² (E) 35²

13. In a fenced yard there are sheep, goats, and cows. If all these animals but four are sheep, all these animals but six are goats and all these animals but eight are cows, how many cows are there in this yard ?

(A) 1 (B) 3 (C) 5 (D) 9 (E) None of these

14. How many triangles are there in the following diagram ?

(A) 20 (B) 25 (C) 30 (D) 35 (E) None of these

15. Sports cars are driven by men and each has two women as passengers. Sedan cars are driven by women and each has three men as passengers. If there are a total of 12 cars carrying a total of 43 persons, including the drivers, how many sports cars are there ?

(A) 3 (B) 5 (C) 7 (D) 9 (E) None of these

16. A cyclist is traveling along a path consisting of three sections of the same length. In the first section, pedaling against the wind, he goes at 10 km/h. In the second section, going up a hill, he goes at 5 km/h. In the third section, he bikes downhill at 30 km/h. What is the average speed of this cyclist on the whole path ?

(A) 5 km/h (B) 9 km/h (C) 10 km/h (D) 15 km/h (E) 20 km/h

17. Daryl leaves Fredericton by car and drives at constant speed. At noon, his distance traveled, in kilometers, is a two digit number. At 1 PM, the distance traveled is the same two digits, reversed. At 2 PM, the distance traveled is the same two digits as at noon, but separated by a zero. At what speed is Daryl driving ?

(A) 45 km/h (B) 50 km/h (C) 55 km/h (D) 61 km/h (E) 106 km/h

18. Mark and Tom are playing with two six sided coloured dice. On each die the faces are painted blue or red. They throw both dice at once and Mark wins if the upper faces of the dice are of the same colour, while Tom wins if they are different colours. Each player has exactly the same chance to win. If the first die has one blue face and five red faces, how many red faces are there on the second die ?

(A) 1 (B) 2 (C) 3 (D) 4 (E) 5

19. The government decides to use new coins so that only 3 cent and 7 cent coins are used. Some amounts, like 5 cents, cannot be made exactly using the new coins. Which is the largest amount that cannot be made exactly with the new coins ?

(A) 8 cents (B) 9 cents (C) 10 cents (D) 11 cents (E) 12 cents

20. Albert, Bob and Carl are digging identical holes in a field. When Albert works with Bob, they dig a hole in four hours. When Albert works with Carl, they dig a hole in three hours. When Bob works with Carl, they dig a hole in two hours. How many hours does it take Albert to dig a hole when he works alone ?

(A) 9 hours (B) 12 hours (C) 24 hours (D) 36 hours (E) 40 hours

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Math Problems : 2011 UNB - Grade 9 (Part A)

11:56 PM

Muhammad Yusuf , Posted in Math Competitions , 0 Comments

University Of New Brunswick

Junior High School Mathematics Competition

2011 – Grade 9

1. If x is equal to ½ , what is the value of

(A) 2/5 (B) 9/10 (C) 10/9 (D) 5/2 (E) None of these

2. One third of the birds in a cage are blue. Forty of the 60 females are blue, while 25% of the males are blue. How many birds are in the cage ?

(A) 60 (B) 120 (C) 180 (D) 240 (E) 300

3. Define a new arithmetic operation a ✴ b = b² - 2a. Then (3 ✴ 4) ✴ 5 is equal to:

(A) 5 (B) 15 (C) 23 (D) 60 (E) 90

4. The largest of four consecutive integers is twice as large as the smallest. The sum of these four integers is

(A) 10 (B) 14 (C) 18 (D) 24 (E) None of these

5. Three planets revolving in the same direction around the same star are in a straight line with the star. The first planet completes one revolution in 4 years, the second one in 6 years and the third one in 9 years. In how many years will the three planets return to their current position ?

(A) 18 (B) 24 (C) 30 (D) 36 (E) 42

6. This year, my father’s age is twice my age. Ten years ago my age was one-third the age of my father. The sum of our ages is:

(A) 30 (B) 40 (C) 50 (D) 60 (E) 70

7. In a test consisting of 15 multiple choice questions, four points are awarded for each correct answer, and two points are deducted for each wrong answer. Alex answered all the questions and scored 30. How many questions did he answer correctly ?

(A) 8 (B) 10 (C) 12 (D) 14 (E) 16

8. Of the numbers below, which is the largest that could be the perimeter of some triangle of which two sides have lengths 4 and 5 ?

(A) 13 (B) 15 (C) 17 (D) 19 (E) 21

9. There are 15 marbles in a box. They come in three colors: green, blue and red. There are seven times as many blue marbles as red marbles. How many green marbles are in the box ?

(A) 3 (B) 4 (C) 5 (D) 6 (E) 7

10. John spent almost all his money in four stores. In each of these stores, he spent half of the money that he had going in plus $1. At the end, he was left with $1. How much money did John have at the beginning?

(A) $4 (B) $10 (C) $22 (D) $46 (E) $94

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Math Problems : Junior School

9:35 PM

Muhammad Yusuf , Posted in Math Competitions , 0 Comments

1. After finding the average of a list of 35 numbers, a student places the average on the list. The student then calculates the average of the 36 numbers on the list. What is the ratio of the original average to the new average?

(A) 1:1 (B) 35:36 (C) 36:35 (D) 2:1 (E) None of these

2. What is the last digit of the product of the first 2005 prime numbers?

(A) 0 (B) 1 (C) 2 (D) 3 (E) 5

3. How many 5-digit numbers are there that are both a perfect square and perfect cube?

(A) 1 (B) 2 (C) 3 (D) 4 (E) 5

4. Let P equal the product of 3,659,893,456,789,325,678 and 342,973,489,379,256. The number of digits in P is

(A) 32 (B) 33 (C) 34 (D) 35 (E) 36

5. An 18 inch by 24 inch painting is to be placed into an open wooden frame with the longer dimension vertical. The painting and frame do not overlap. The wood at the top and bottom is twice as wide as the wood on the sides. If the frame area equals that of the painting itself, the ratio of the smaller to the larger dimension of the framed painting is

(A) 1:3 (B) 1:2 (C) 2:3 (D) 3:4 (E) 1:1

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Math Problem : Middle School Problems

9:32 PM

Muhammad Yusuf , Posted in Math Competitions , 0 Comments

________mL 1) You have a recipe that requires 0.25 L of milk. Your measuring cup is

marked only in milliliters. How many milliliters of milk do you need?

__________ 2) How many of the numbers 11, 21, 31, 41, 51, 61, 71, 81, 91 are prime?

__________ 3) What is the numerical difference between 5/6 and its reciprocal?

__________ 4) When I divide my age by 5, the remainder is 3. Your age is twice my

age. If I divide your age by 5, what will the remainder be?

__________ 5) Your bad hair day began 720 minutes before 7:20 p.m. At what time

did your bad hair day start? INCLUDE ALL NECESSARY UNITS IN YOUR ANSWER.

________ft 6) The area of a square room is 256 square feet. How many feet are in

the perimeter of the room?

__________ 7) A number n is subtracted from 18. The result is four less than n. What

is the value of n?

$_________ 8) The first 12 contestants won an average of $80. The next 20 won an

average of $70. What was the average amount won by all 32 contestants?

__________ 9) Solve: (301 + 302 + 303 + . . . + 325) – (1 + 2 + 3 + . . . 25)

__________ 10) Order from least to greatest: 0.2 ¼ 3.3% 2^-3

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Math Problems : Practices

9:27 PM

Muhammad Yusuf , Posted in Problem Of The Day , 0 Comments

1. Two players, A and B, are playing the following game. They take turns writing down the digits of a six-digit number from left to right; A writes down the first digit, which must be nonzero, and repetition of digits is not permitted. Player A wins the game if the resulting six-digit number is divisible by 2, 3 or 5, and B wins otherwise.

Prove that A has a winning strategy.

2. Prove that nn - n is divisible by 24 for all odd positive integers n.

3. Let a and b be real numbers. Prove that the inequality

holds.

When does equality hold?

4. Let ABCD be a quadrilateral. The circumcircle of the triangle ABC intersects the sides CD and DA in the points P and Q respectively, while the circumcircle of CDA intersects the sides AB and BC in the points R and S. The straight lines BP and BQ intersect the straight line RS in the points M and N respectively. Prove that the points M, N, P and Q lie on the same circle.

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Math Problems : Line and Area

8:55 PM

Muhammad Yusuf , Posted in Math Competitions , 0 Comments

In the diagram, D and E are the midpoints of AB and BC respectively.

(a) Determine an equation of the line passing through the points C and D.

(b) Determine the coordinates of F, the point of intersection of AE and CD.

(c) Determine the area of ΔDBC.

(d) Determine the area of quadrilateral DBEF.

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Problem Of The Day - Filled each circle

8:48 PM

Muhammad Yusuf , Posted in Problem Of The Day , 0 Comments

Each of the integers 1 to 7 is to be written, one in each circle in the diagram. The sum of the three integers in any straight line is to be the same. In how many different ways can the centre circle be filled?

(A) 1 (B) 2 (C) 3

(D) 4 (E) 5

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Math Problems : Grade 7 (2)

10:43 PM

Muhammad Yusuf , Posted in Math Competitions , 0 Comments

1. A 4 x 4 x 4 cube consisting of smaller cubes is painted and then broken apart. How many of the smaller cubes will have exactly 2 painted sides?

(A) 8 (B) 16 (C) 20 (D) 24 (E) 32

2. How many three digit numbers can be constructed using the digits 1, 2, 3, 4 and 5 if the same digit cannot appear twice in a row in any of the numbers?

(A) 60 (B) 65 (C) 80 (D) 120 (E) None of these

3. A rectangular floor is completely covered with tiles whose size is 1 x 2. If the tiles are not cut and do not overlap, the size of the floor cannot be

(A) 4 x 9 (B) 8 x 8 (C) 11 x 7 (D) 16 x 5 (E) None of these

4. How many ways can the number 10 be written as the sum of exactly three positive and not necessarily different integers if the order in which the sum is written does not matter? For instance, 10 = 1 + 4 + 5 is one such sum. This sum is the same as 10 = 4 + 1 + 5.

(A) 5 (B) 6 (C) 7 (D) 8 (E) 10

5. Paul’s calculator can make only two operations: add 12 to the number displayed, or subtract 7 from it. Today, it shows the number 1998. What is the minimal number of steps needed to display the number 2000?

(A) 4 (B) 12 (C) 16 (D) 21 (E) 24

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Math Probems : Grade 7

10:41 PM

Muhammad Yusuf , Posted in Math Competitions , 0 Comments

1. Everyday, Lisa puts her spare change (nickels and dimes) in a piggy-bank. This weekend she decides to count her savings. She finds that she has 72 coins with a total value of $4.95. How many dimes does she have?

(A) 14 (B) 23 (C) 25 (D) 27 (E)

2. Ève has two more marbles than Solène. Solène has twice as many marbles as Steve. Steve has 7 less marbles than Ève. How many marbles do they have between them?

(A) 13 (B) 20 (C) 27 (D) 34 (E) None of these

3. One day in math class, Shelley asks the teacher: “Mr. Nelson, how old are you?” Mr. Nelson responds: “This year I am three times as old as my sister. However, six years ago, I was five times as old as she was.” How old is the mathematics teacher?

(A) 36 (B) 40 (C) 49 (D) 55 (E) None of these

4. Four tennis players enter a tournament. How many different ways can the pairings be made for the first round games?

(A) 3 (B) 6 (C) 8 (D) 12 (E) 24

5. A box contains some apples. Andrée takes ½ of them along with one extra apple. Beatrice takes 1/3 of the remaining apples along but put two apples back in the box and finally, Corrine takes 5/6 of the remaining apples along with one more apple. There are now seven apples left in the box. How many apples were in the box before Andrée took her share?

(A) 16 (B) 44 (C) 110 (D) 140 (E) None of these

6. The “floor” of a fraction is defined to be the largest integer which is not greater than that fraction. For instance, floor (10 / 3) = 3. Evaluate

floor ( floor ( 1000 / 7 ) / (floor ( 71 / 2 ) ).

(A) 4 (B) 5 (C) 7 (D) 10 (E) 500

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