2010 grade 4 - welcome to ksea | ksea · · 2018-03-05test grade : grade 4 ... test grade : grade...

KSEA National Mathematics Competition 2010

(Answer Sheet)Identification

Registration # English Name Korean Name

School Grade School Name KSEA Chapter

Answers

Level 1

1 a! b! c! d! e!

2 a! b! c! d! e!

3 a! b! c! d! e!

4 a! b! c! d! e!

5 a! b! c! d! e!

6 a! b! c! d! e!

7 a! b! c! d! e!

8 a! b! c! d! e!

9 a! b! c! d! e!

10 a! b! c! d! e!

Level 2

11 a! b! c! d! e!

12 a! b! c! d! e!

13 a! b! c! d! e!

14 a! b! c! d! e!

15 a! b! c! d! e!

16 a! b! c! d! e!

17 a! b! c! d! e!

18 a! b! c! d! e!

19 a! b! c! d! e!

20 a! b! c! d! e!

Level 3

21

22

23

24

25

26

27

28

29

30

· · · · · · · · · · · O!cial Use Only · · · · · · · · · · ·

N1 Level 1 = N2 Level 2 = N3 Level 3 =

G1 Total = G2 Level 3 = G3 Level 2 & 3 =


(Problem Sheets)

Test Grade : GRADE 4Identification



Instructions!

"

#

$

" Write your registration number, english name, school grade, school name, and the KSEA

chapter name in the “Identification” part of both the problem sheets and the answer

sheet. Korean name is optional.

" You will have 60 minutes to complete this exam. There are 30 questions, composed

of multiple choice problems and free-response problems. Every problem is worth 1

point. No partial credit will be given. However, ties will be broken using the following

procedure: (A) most correct answers on the test, (B) most correct Level 3 answers, (C)

most correct Level 2 answers.

" No calculator, dictionary, ruler, or notes are allowed. Use the provided space and

back of the problem sheets to solve the problems.

" You MUST write the answers into the “Answers” part of the answer sheet!

For the multiple-choice problems, write the letter (a, b, c, d, or e) of the correct answer.

For the free-response problems, write a simplified answer (fractions must be reduced

completely, etc).

" Do NOT write anything in the “O!cial Use” part of the answer sheet.

" When you are done with the test, return both the problem sheets and the answer sheet

to the instructor.

! ! ! ! !! Do Not Open Until Told ! " " " " ""

Grade 4 : Page 2 of 7

# # # Level 1 – Grade 4 # # #

1. What is the ten’s place digit for the following sum: 47 + 89?

(a) 1 (b) 2 (c) 3 (d) 4 (e) 5

2. Let a, b, and c be whole numbers satisfying: 1 < a < b < c < 7 and a $ c is odd.

What is b?

(a) 2 (b) 3 (c) 4 (d) 5 (e) 6

3. Round 2.13 and 4.55 to the nearest whole numbers and add them.

What is the sum of the rounded numbers?

(a) 3 (b) 4 (c) 5 (d) 6 (e) 7

4. Which of the following is not a divisor of 168?

(a) 4 (b) 5 (c) 6 (d) 7 (e) 8

5. What is the one’s place digit for 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9?

(a) 5 (b) 6 (c) 7 (d) 8 (e) 9


6. What is the average of the following scores: 10, 80, 40, 10, 70, and 90?

(a) 50 (b) 55 (c) 60 (d) 65 (e) 70

7. What is the median of the following numbers: 10, 80, 40, 10, 70, and 90?

(a) 50 (b) 55 (c) 60 (d) 65 (e) 70

8. What is 0.3 % of 10 dollars?

(a) 3 dollars (b) 0.3 dollars (c) 3 cents (d) 0.3 cents (e) 0.03cents

9. Which of the following is the best fit for x in the following pattern: 22, 29, 36, x, 50, · · ·?

(a) 43 (b) 44 (c) 45 (d) 46 (e) 47

10. Which of the following angles could be the smallest in an isosceles right triangle?

(a) 15o (b) 30o (c) 45o (d) 60o (e) 75o


# # # Level 2 – Grade 4 # # #

11. Let a, b, and c be whole numbers satisfying: 0 < a < b < c < 8; a $ b $ c is odd;

c % a is three times b % a. What is a?

(a) 1 (b) 2 (c) 3 (d) 4 (e) 5

12. What is the one’s place digit for 7$ 7 $ 7 $ 7 $ 7 $ 7 $ 7 $ 7$ 7 $ 7 $ 7 $ 7?

(a) 1 (b) 3 (c) 5 (d) 7 (e) 9

13. Let a be a whole number greater than 2. If b is two times a, what is the greatest common

divisor of a and b?

(a) 2 (b) a (c) b (d) 2a (e) b

a

14. Esther has 4 coins. The sum of all these coins is 50 cents. How many nickles does Esther

have?

(a) 1 (b) 2 (c) 3 (d) 4 (e) None

15. A positive whole number greater than 1 that is divisible by only 1 and itself is called prime.

How many prime numbers are there between 10 and 20?

(a) 1 (b) 2 (c) 3 (d) 4 (e) 5

16. Uncle Tom served in the military for 45 months, and then he attended college for 4 years.

How long did Uncle Tom spend in military and college?

(a) 49 months (b) 83 months (c) 7.25 years (d) 7.5 years (e) 7.75 years


17. Hana has 17 hours before her math exam. She plans to sleep for 8 hours, eat for 30 minutes

three times, exercise for one hour and 30 minutes twice, practice piano for an hour, and

play with her brother for 2 hours. How long can she study math?

(a) 5 hours and 30 minutes

(b) 4 hours and 30 minutes

(c) 3 hours and 30 minutes

(d) 2 hours and 30 minutes

(e) 1 hours and 30 minutes

18. At Hana’s birthday party, two pizzas were served: one cheese and one pepperoni. Both

were the same size, but the cheese pizza was cut into 8 equal pieces and the pepperoni

pizza was cut into 6 equal pieces. Hana ate 3 slices of the cheese pizza and 1 slice of the

pepperoni pizza. How much pizza did Hana eat?

(a) 3

8of a pizza (b) 1

6of a pizza (c) 1

3of a pizza (d) 13

24of a pizza (e) 1

12of a pizza

19. Which of the following statements is incorrect about triangles?

(a) Each triangle has three sides.

(b) Equilateral triangles can have a 90 degree angle.

(c) Equilateral triangles are always isosceles triangles.

(d) The sum of the angles in two triangles is 360 degrees.

(e) The greatest angle in any right triangle is 90 degrees.

20. A typical tennis ball can is in the shape of a cylinder, containing three tennis balls. If

each cylindrical can is exactly three balls high and one ball wide, which of the following

statements is correct about the height and the circumference of the can? We ignore the

thickness of the can.

(a) The height of the can is six times of the diameter of tennis balls.

(b) The circumference of the can is twice of the diameter of tennis balls.

(c) The height of the can is longer than the circumference.

(d) The circumference of the can is longer than the height.

(e) The height and the circumference of the can are the same.


# # # Level 3 – Grade 4 # # #

21. Simplify 1

2+ 1

4+ 1

8+ 1

16+ 1

32.

22. If 16 is the least common multiple of positive integers a and b and a is a prime number,

what is b?

23. Let a, b, and c be whole numbers satisfying: 1 < a < b < c < 10; a is prime; c is the least

common multiple of a and b. What is c?

24. A store charges 50 cents for an apple, and 5 dollars for a bag of a dozen apples. What is

the least cost to buy 100 apples?

25. Hana, Duna, and Sena collected rocks during Summer break. Hana collected 9 more rocks

than Duna, and Sena collected twice as many as Duna. If the total number of rocks collected

is 21, how many rocks did Hana collect?


26. Hana can make a pizza with the following toppings: pepperoni, onions, or mushrooms.

How many di"erent combinations of toppings can she make, if each pizza should have at

least one topping?

27. In a box, there are 5 pennies, 2 nickels, 5 dimes, and 3 quarters. Hana takes out coins one

by one, until one of the quarters is out, i.e. she cannot remove any more coins after one

quarter is out. If she took out 73 cents total, how many dimes remain in the box?

28. Extend the following triangle of numbers so that there are 10 total rows. What is the

rightmost number in the bottom (in the 10th row) in the extended triangle?

1

2 3

4 5 6

7 8 9 10

11 12 13 14 15

29. Hana is making a model of a staircase. She is using cubes for steps and each step is one

cube wide. How many cubes does she need to make 15 steps?

30. Dad needs to cover a rectangular wall with square tiles. The area of the wall is 3600 square

inches. Each square tile has perimeter of 12 inches. What is the smallest number of tiles

required to cover the wall with no gaps between tiles?





Answers

Level 1

1 a! b! c! d! e!

2 a! b! c! d! e!

3 a! b! c! d! e!

4 a! b! c! d! e!

5 a! b! c! d! e!

6 a! b! c! d! e!

7 a! b! c! d! e!

8 a! b! c! d! e!

9 a! b! c! d! e!

10 a! b! c! d! e!

Level 2

11 a! b! c! d! e!

12 a! b! c! d! e!

13 a! b! c! d! e!

14 a! b! c! d! e!

15 a! b! c! d! e!

16 a! b! c! d! e!

17 a! b! c! d! e!

18 a! b! c! d! e!

19 a! b! c! d! e!

20 a! b! c! d! e!

Level 3

21

22

23

24

25

26

27

28

29

30

· · · · · · · · · · · O!cial Use Only · · · · · · · · · · ·




(Problem Sheets)




Instructions!

"

#

$














completely, etc).



to the instructor.



# # # Level 1 – Grade 5 # # #

1. If the mean value of 4, 7, 10, 13, 16, and x is 10, what is the value of x?

(a) 7 (b) 9 (c) 10 (d) 11 (e) 12

2. What is

15

3+ 1

3

5=?

(a) 8$ 83!5

(b) 8 $ 3!58

(c) 163!5

(d) 83!5

(e) 1 $ 83!5

3. What is the next number in the following pattern?

5, 10, 16, 23, 31, ?

(a) 36 (b) 37 (c) 38 (d) 39 (e) 40

4. Esther has 4 coins. The sum of all these coins is 50 cents. How many nickels does Esther

have?

(a) None (b) 1 (c) 2 (d) 3 (e) 4

5. One side of a square is r cm long. If each side shrinks to half of the original length, what

is the ratio of the area of the resulting square to the area of the original square?

(a) 4r (b) r

2(c) 2r (d) 1

4(e) 2


6. One gallon is four quarts (qt), one qt is two pints (pt) and one pt is 2 cups. Three and

one-half gallons are equal to how many cups?

(a) 48 (b) 16 (c) 56 (d) 46 (e) 36

7. What is 0.25% of 0.25?

(a) 0.25$ 0.25 (b) 0.25100

(c) 0.25$ 0.25100

(d) 0.25$ 25100

(e) 0.025

8. Multiples of 3 are

3, 6, 9, 12, · · · ,

and multiples of 7 are

7, 14, 21, · · · .

What is the second least common multiple of 3 and 7?

(a) 21 (b) 24 (c) 32 (d) 42 (e) 63

9. If a and b are whole numbers satisfying 2 % a + b % 7 and 0 % b & a % 1, what is the

maximum possible value of a?

(a) 1 (b) 2 (c) 3 (d) 4 (e) 5

10. Which of the following is the same as

1a!b

1a

+ 1b

?

(a) 1a+b

(b) a + b (c) a+b

a!b(d) a!b

a+b(e) None of these


# # # Level 2 – Grade 5 # # #

11. The symbol ' means the following

f ' g = (f $ g)& (f + g).

For example, 3 ' 1 = (3$ 1)& (3 + 1) = &1. Which of the following is always true?

(a) f ' 1 = f (b) f ' (g ' h) = (f ' g) ' h (c) f ' g = g ' f

(d) f ' g & g ' f = 2(f ' g) (e) None of these

12. See the diagram below. Jinsoo wants to travel between points A and B. How many shortest

paths can Jinsoo choose to travel if he must travel on the grid lines?

(a) 7 (b) 8 (c) 9 (d) 10 (e) None of these

13. Imagine there is a triangle with two interior angles of 45". If the shortest side of the triangle

is 1cm long, what is the area of the triangle?

(a) 0.5 cm2 (b) 0.8 cm2 (c) 0.9 cm2 (d) 1.0 cm2 (e) 1.5 cm2

14. A prime number is a counting number greater than 1 that is divisible by only 1 and itself.

Which of the following statements is always true? The letters a and b in the following

statements are prime numbers.

(a) The product of two prime numbers is a prime number.

(b) The division of a prime number by another prime number is a prime number.

(c) The addition of two prime numbers is a prime number.

(d) If a + b is a prime number and b > a, then b & a is also a prime.

(e) None of these.


15. Jason, Susan and Greg earned money by car-washing on a Saturday. They earned $42 in

total. Jason worked twice the time Susan worked and Greg worked three times as much as

Susan. If they split their earning based on how much time they worked, how much should

Jason get?

(a) $14 (b) $12 (c) $10 (d) $8 (e) $6

16. The following chart shows the average high/low temperature of each month of New York

City in a certain year. The temperature is given in Celsius. Read the chart and determine

which of the following statements is not correct.

Month High/Low (Celsius)

January 4/& 3

February 5/& 3

March 8/1

April 16/6

May 21/11

June 27/17

July 29/18

August 28/19

September 25/16

October 19/10

November 12/5

December 5/& 1

(a) The month of July or August could be the hottest month of the year.

(b) The month of January or February could be the coldest month of the year.

(c) 25% of the months have minus temperature as their low temperatures.

(d) Most rapid temperature change between two consecutive months is about 8"C.

(e) 50% of the months have the high temperature above 20"C.

17. There is a certain number x and the square of x is &1, that is, x2 + 1 = 0. What is the

following

1 +1

x2+

1

x4+

1

x6=?

(a) &2 (b) &1 (c) 0 (d) 1 (e) 2


18. In a basket, there are 5 marbles, one red and four green marbles. If each of five people picks

one marble from the basket blindly without replacement, which of the following statements

is true?

(a) Everyone has the same chance to pick the red marble.

(b) The first person has the most chance to pick the red marble.

(c) The chance that the last person to pick the red marble is 10%.

(d) The chance the second person to pick the green marble is 90%.

(e) None of these.

19. The students of Mrs. Lee’s class are standing in a circle; they are evenly spaced and are

numbered in order starting with 1. The student with number 7 is directly across from the

student with number 18. How many students are there in the class?

(a) 22 (b) 20 (c) 19 (d) 18 (e) 17

20. In the following equation, two signs are hidden in # and '. These signs are among $,&, +,

and ÷.

1 # 1 ' 1 = 1.

For example, a possible answer for 1 # 1 ' 1 is 1 when # = $ and ' = ÷. Which of

the following is not a possible result of 1 # 2 ' 3 ?

(a) 0 (b) 2 (c) 23

(d) 4 (e) 6


# # # Level 3 – Grade 5 # # #

21. Angle ABC is 60". The line AD bisects angle BAC and the line CD bisects angle BCA.

Find the measure of angle ADC.

22. In a basket, there are 2 white balls and 1 red ball. Jason and Sunhee each pull a ball from

the basket without looking. What is the probability that the remaining ball is white?

23. Eight marbles look alike, but one is slightly heavier than the others. What is the minimum

number of weighings to determine the heavier one by using a balance scale?

24. Sarah’s team entered a mathematics contest where teams of students compete by answering

questions that are worth either 3 points or 5 points. No partial credit was given. Sarahs

team scored 44 points on 12 questions. How many 5 points questions did the team answer

correctly?


25. Susan needs to paint her house with a mixture of 30 liters of paints composed of 25% red

tint, 30% yellow tint and 45% water. After some liters of yellow tint are added to the

original mixture, yellow tint is now 40% of the new mixture. How many liters of yellow

tint were added to the original mixture?

26. A toll gate on a highway has 5 toll booths. Each toll booth has a lamp on its top. If the

lamp is lit, the booth is represented by “1” (the toll booth is open) and “0” (the toll booth

is closed) otherwise. On the first day of April, the lamp pattern of toll booths was 00001

(as shown below), on the second day of April it was 00010, on the third day of April it was

00011, and on the 10th day of April it was 01010. How many toll booths were open on the

17th day of April?

April 1 : 0 0 0 0 1

April 2 : 0 0 0 1 0

April 3 : 0 0 0 1 1

27. 10 children are riding on bicycles and tricycles. If there are 25 wheels in total, how many

children are riding on tricycles?

28. How many ordered pairs (x, y) of counting numbers (x = 1, 2, 3, · · · , and y = 1, 2, 3, · · ·)

satisfy the equation x + 2y = 100?


29. Three right isosceles triangles are constructed on the sides of a 3 & 4 & 5 right triangle, as

shown below. A capital letter represents the area of each triangle. What is Z & (X + Y )?

30. It is true that

1 + 3 + 1 = 12 + 22,

1 + 3 + 5 + 3 + 1 = 22 + 32,

1 + 3 + 5 + 7 + 5 + 3 + 1 = 32 + 42.

What is the following sum

1 + 3 + · · ·+ 19 + 21 + 19 + · · ·+ 3 + 1 =?





Answers

Level 1

1 a! b! c! d! e!

2 a! b! c! d! e!

3 a! b! c! d! e!

4 a! b! c! d! e!

5 a! b! c! d! e!

6 a! b! c! d! e!

7 a! b! c! d! e!

8 a! b! c! d! e!

9 a! b! c! d! e!

10 a! b! c! d! e!

Level 2

11 a! b! c! d! e!

12 a! b! c! d! e!

13 a! b! c! d! e!

14 a! b! c! d! e!

15 a! b! c! d! e!

16 a! b! c! d! e!

17 a! b! c! d! e!

18 a! b! c! d! e!

19 a! b! c! d! e!

20 a! b! c! d! e!

Level 3

21

22

23

24

25

26

27

28

29

30

· · · · · · · · · · · O!cial Use Only · · · · · · · · · · ·




(Problem Sheets)




Instructions!

"

#

$














completely, etc).



to the instructor.



# # # Level 1 – Grade 6 # # #

1. Which of the following fractions has the largest value?

(a)19

8(b)

7

3(c) 2

2

5(d) 2

3

7(e) None of these

2. Jennifer bought 3 gallons of milk for $2.80 per gallon. The sale’s tax rate is 5% of the

purchase. How much change would Jennifer get back if she gave the cashier a $10 bill?

(a) $1.60 (b) $1.46 (c) $1.18 (d) $2.00 (e) None of these

3. Simplify: 33 $!

24 ÷ |$ 4|"

% 5 + 20

(a) 13.75 (b) 8 (c) 7 (d) 18 (e) None of these

4. 51 is what percent of 60?

(a) 80% (b) 85% (c) 90% (d) 95% (e) None of these

5. On a map, 1/4 of an inch represents 250 miles. If two cities are 875 miles apart, how far

apart are the two cities on the map?

(a)3

4of an inch (b)

4

5of an inch (c) 1

1

4of an inch

(d)7

8of an inch (e) None of these


6. A square table individually holds four people and only one person can be seated on each

side. How many people can be seated at 12 square tables lined up end to end?


7. If the dimensions of Room 1 in a hotel are 12ft%12ft%8ft, and the dimensions of Room 2

are 18ft%18ft%24ft, what is the ratio of the volume of Room 2 to the volume of Room1?

(a) 4 to 27 (b) 27 to 4 (c) 27 to 8 (d) 8 to 27 (e) None of these

8. A straight line passes through the three points (3,$4), (5, 1) and (1, y). What is the value

of y?

(a) 6 (b) $1 (c) $5 (d) $9 (e) None of these

9. A person 120 centimeters tall casts a shadow of 32 centimeters at the same time that a flag

pole casts a shadow of 120 centimeters. How tall is the flag pole in centimeters?


10. The students of Mrs. Lee’s class are standing in a circle; they are evenly spaced and are

numbered in order starting with 1. The student with number 7 is directly across from the

student with number 18. How many students are there in the class?



# # # Level 2 – Grade 6 # # #

11. A square of the backyard contains a circular pool with a diameter of 20 feet. The area of

the yard not covered by the pool is 2,186 square feet. Find the width of the yard in feet.

Use 3.14 for !.

(a) 46.75 feet (b) 58.67 feet (c) 31.84 feet (d) 50 feet (e) None of these

12. Let x be a positive number and y be its reciprocal. Compute1

x + 1+

1

y + 1.

(a) 1 (b)2

(x + 1)(y + 1)(c)

2

x + 1(d)

x + y + 2

x2 + 1(e) None of these

13. A clock is malfunctioning such that the minute hand moves 51 minutes every hour. If you

set the clock at the correct time of 10:00am, what time will the clock read when the actual

time is 7:20pm of the same day?

(a) 6:15pm (b) 6:00pm (c) 5:56pm (d) 5:30pm (e) None of these

14. 38 students were asked about their use of the gymnasium at their school:

• 20 students use the weight room

• 15 students use the pool

• 12 students use the climbing wall

• 5 students use the pool and the weight room

• 7 students use the pool and the climbing wall

• 5 students use the climbing wall and the weight room, but don’t use the pool

• 4 students use the climbing wall, the weight room, and the pool

How many students don’t use any of these facilities?



15. The original price of an item was $50. The store deducted 20%, and then deducted an

additional 20% o" the reduced price. How much more does a buyer save if the store had

simply reduced the original price by 40%?

(a) None (b) $2 (c) $8 (d) $18 (e) None of these

16. A red cube and a white cube, each numbered with the integers 4 through 9, are tossed.

What is the probability that the sum of two numbers rolled is greater than 16?

(a) 0 (b) 1/9 (c) 1/12 (d) 1/4 (e) None of these

17. As x increases from 0.01 to 1, which of the following also increases?

(a) 1$1

x2(b) 1

x(c) 1$ x (d) $x2 (e) None of these

18. John divided his souvenir hat pins into two piles. The two piles had an equal number of

pins. He gave his brother one-half of one-third of one pile. John had 66 pins left. How

many pins did John give to his brother?


19. Suppose that a & b = ab + b. What is the value of 1 & (1 & (1 & (1 & 1)))?


20. Sarah’s team entered a mathematics contest where teams of students compete by answering

questions that are worth either 3 points or 5 points. No partial credit was given. Sarah’s

team scored 44 points on 12 questions. How many 5 points questions did the team answer

correctly?



# # # Level 3 – Grade 6 # # #

21. Given:5

13=

n

39=

n + m

156=

p $ m

104. What is the value of p?

22. A toll gate on a highway has 5 toll booths. Each toll booth has a lamp on its top. If the

lamp is lit, the booth is represented by ”1” (the toll booth is open) and ”0” (the toll booth

is closed) otherwise. On the first day of April, the lamp pattern of toll booths was 00001

(as shown below), on the second day of April it was 00010, on the third day of April it was

00011, and on the 10th day of April it was 01010. How many toll booths were open on the

17th day of April?

April 1 : 0 0 0 0 1

April 2 : 0 0 0 1 0

April 3 : 0 0 0 1 1

23. When its digits are reversed, a particular positive two-digit integer is increased by 20%.

What is the original number?

24. Jack starts at the summit of Mt. Mckinley (elevation, 20,320 feet) and travels down the

mountain at a rate of 4 feet in elevation per second. Jill starts at the bottom at the same

time and travels up the mountain at a rate of 1 foot per second. At what elevation, in feet,

will they meet?

25. A five-digit number is represented by ABCDE, where each letter represents a digit. If we

add the digit 1 in front of ABCDE, we get 1ABCDE. The product of 1ABCDE and 3 is

the six-digit number ABCDE1. Find the value of the original five-digit number ABCDE.


26. A rhombus is formed by two chords and two radii of a circle with radius 5 meters. What

is the number of square meters in the area of the rhombus?

27. Eight marbles look alike, but one is slightly heavier than the others. What is the minimum

number of weighings to determine the heavier one by using a balance scale?

28. A runner in a three-mile race averaged 6 mph for the first mile, 5 mph for the second mile,

and 3.75 mph for the third mile. How long did it take the runner to complete the race?

29. Mary and Pat decided that they needed more boys in the Glee Club. Currently only 1/4 of

the members are boys. They would like to increase enrollment until at least 1/3 of members

are male. If they currently have 400 members, at least how many boys would be needed if

35 additional girls also joined during the membership drive?

30. A fox is forty leaps [that is, fox leaps] ahead of a hound. The fox makes three leaps in the

time it takes the hound to make two leaps. Two of the hound’s leaps cover as much ground

as four of the fox’s. If both are hopping as described above, how many leaps will the hound

take to catch the fox?





Answers

Level 1

1 a! b! c! d! e!

2 a! b! c! d! e!

3 a! b! c! d! e!

4 a! b! c! d! e!

5 a! b! c! d! e!

6 a! b! c! d! e!

7 a! b! c! d! e!

8 a! b! c! d! e!

9 a! b! c! d! e!

10 a! b! c! d! e!

Level 2

11 a! b! c! d! e!

12 a! b! c! d! e!

13 a! b! c! d! e!

14 a! b! c! d! e!

15 a! b! c! d! e!

16 a! b! c! d! e!

17 a! b! c! d! e!

18 a! b! c! d! e!

19 a! b! c! d! e!

20 a! b! c! d! e!

Level 3

21

22

23

24

25

26

27

28

29

30

· · · · · · · · · · · O!cial Use Only · · · · · · · · · · ·




(Problem Sheets)




Instructions!

"

#

$














completely, etc).



to the instructor.



# # # Level 1 – Grade 7 # # #

1. Find the value of 214 $

!

313 $ 1

12

"

.

(a) $ 512 (b) $1 (c) $5

4 (d) $213 (e) $11

4

2. If Brittany gets a 97 on her next math test, her average will be 89. However, if she gets

72, her average will be 84. How many tests has Brittany already taken?

(a) 3 (b) 4 (c) 5 (d) 7 (e) 8

3. Martha’s parents presented her a saving account with the balance of $81, 000 on her birth-

day this year. When she was born, her parents deposited $3, 000, which has been tripled

every four years. How old is Martha?

(a) 4 years old (b) 8 years old (c) 12 years old

(d) 16 years old (e) 27 years old

4. The graph of 2x $ ay = 3 passes through (a, 1) and ($b, 3). Find a + b.

(a) $6 (b) $3 (c) 3 (d) 6 (e) 9

5. We partition the rectangle ABCD into seven congruent rectangles like the figure below. If

the area of the #ABCD is 84 cm2, what is the perimeter of a small rectangle in cm?

(a) 9 (b) 7 (c) 14 (d) 19 (e) 38


6. Let A and B be natural numbers such that A

2 = B

3 . Let M be the least common multiple

of A and B and let D be the greatest common divisor of A and B. If the product of M

and D is 216, then find B $ A.

(a) 6 (b) 12 (c) 18 (d) $6 (e) $12

7. Let the measure of $AOC be 78!, denoted as m$AOC = 78!. For the line AB, m$COD =

3m$DOE, m$FOB = 2m$EOF and m$COD =m$FOB. Find the measure of $DOF .

(a) 12! (b) 18! (c) 30! (d) 36! (e) 48!

8. Last year, 1150 students attended Hill Middle School. This year, the number of boys

decreased 3% and the number of girls increased 2%. The total number of students becomes

1143. Find the number of girls this year.

(a) 600 (b) 550 (c) 582 (d) 560 (e) 561

9. A town has a basketball team and a football team, and both teams play once a week. The

basketball team wins 1 out 3 games and the football team loses 2 out of 5 games. What is

the probability that both teams win on the same week?

(a) 1115 (b) 14

15 (c) 38 (d) 2

15 (e) 15

10. Tom plants 5 white rosebushes and 2 red rosebushes in a row. How many di"erent ways

can he plant at least four white bushes consecutively?

(a) 21 (b) 4 (c) 6 (d) 9 (e) 12


# # # Level 2 – Grade 7 # # #

11. A lattice point is an ordered pair of integers (x, y). How many lattice points(including the

end points) are there on the line segment joining the points (4, 0) and (18, 175)?

(a) 7 (b) 8 (c) 9 (d) 12 (e) 13

12. Let a#

b = ab + a $ 2. If 3#

(x#

($2)) = 4, then find x.

(a) $5 (b) $2.5 (c) $3 (d) $6 (e) $1

13. Complete the proof.

Consider a circle (center is O) that inscribes the triangle #RST and has the diameter ST .

We want to show that

$RST =1

2$ROT

Proof: We begin by constructing radius OR.

Since $ROT is an exterior angle of % ROS

$ROT = (1)

Since the radius of the circle is OR = OS

% ROS is (2)

Since % ROS is (2)

$ORS = (3)

Then $RST = 12$ROT .

(1) (2) (3)

(a) $ORS isosceles $OSR

(b) $ORS + $OSR isosceles $OSR

(c) $ORS + $OSR equilangular $OSR

(d) $ORS isosceles $OSR

(e) $ORS + $OSR isosceles $ROS


14. If Kevin passes in Algebra class, then he gets at least 80 points out of 100 in this exam.

Which of the following is equivalent to this statement?

(a) If Kevin gets at least 80 points out of 100 in this exam, then he passes in Algebra class.

(b) If Kevin gets less than 80 points out of 100 in this exam, then he fails in Algebra class.

(c) Kevin passes in Algebra class only if he got at least 80 points in this exam.

(d) If Kevin fails in Algebra class, then he gets less than 80 points out of 100 in this exam.

(e) It Kevin passes Algebra class, then he will get at least 80 points in this exam.

15. Brian and Emma share the labor to build a hut. It would take 10 days to complete the hut

if Brian works the first 6 days and then Emma works the rest 4 days after Brian. However,

it would take 11 days if Brain works the first 9 days and then Emma works the rest 2 days

after Brian. If Brain works alone, how many days will it take to complete the hut?

(a) 10 days (b) 11 days (c) 12 days (d) 13 days (e) 14 days

16. The probability that Michael can win a game against Jack is 0.3. What is the smallest

number n for which the following statement is true? If Jack and Michael play the game n

times, Michael has more than 70% chance to win at least one game.

(a) 4 (b) 3 (c) 5 (d) 2 (e) 6

17. Each of two jugs has a capacity of x gallons. One is filled with wine and the other with

water. Three gallons are taken from each jug and transferred to the other. After the

contents are thoroughly mixed, two gallons are taken from each jug and transferred to the

other. Given that the amount of wine in each jug is the same, find the possible value of x.

(a) 3 (b) 5 (c) 6 (d) 8 (e) 10


18. Suppose that f(x) = ax + b where a and b are real numbers. Given that f(f(f(x))) =

27x + 39, what is the value of a + b ?

(a) 5 (b) 6 (c) 8 (d) 16 (e) 21

19. Evaluate

12 $ 22 + 32 $ 42 + · · ·$ 9982 + 9992.

(a) 1, 500, 500 (b) 499, 999 (c) 500, 999 (d) 500, 500 (e) 499, 500

20. A box contains 4 red balls and 6 white balls. Three balls are drawn from the box without

replacement. What is the probability of obtaining 1 red ball and 2 white balls, given that

at least 2 of the balls out of three are white?

(a) 12 (b) 2

3 (c) 34 (d) 4

5 (e) 56


# # # Level 3 – Grade 7 # # #

21. Zac drove a car at 36 km per hour and then walked at 4 km per hour to go to work. From

his home to his o!ce, the total distance was 26 km and he took no more than 1 hour and

10 minutes. What is the minimum distance in kilometers that he drove his car?

22. Suppose that a box contains seven cards, labeled 0, 1, 2, 3, 4, 5, and 7. Three of the seven

cards are selected at random without replacement to make a three digit number. How

many of the possible three digit numbers are divisible by 3?

23. Let f(x) = 1(2x"1)(2x+1). Find f(1) + f(2) + · · ·+ f(100).

24. Let abc be a three digit number while a, b and c are distinct digits. If acb + bca + bac +

cab + cba = 3194, then what is the number abc?

25. Let P (x) be the number of divisors of x ; for example, P (9) = 3. Find the sum of the first

and second smallest value of x which satisfies P (180)P (x) = 72.

26. On a dark and lonely night, a man is standing 5 meters away from a street light. The man

is 2 meters tall and the light is 6 meters high. How long is the man’s shadow in meter?


27. If x is a number such that x2 + x + 1 = 0, then find the numerical value of$

x +1

x

%2

+

$

x2 +1

x2

%2

+

$

x3 +1

x3

%2

+ · · ·+

$

x30 +1

x30

%2

.

28. Consider a sequence of numbers, A1, A2, A3 · · · . The entries of the sequence satisfy the

following properties. Find A12.

A1 = 1, and A2 = 2

An+2 = 3An+1 $ 2An for all n > 0

29. Two parallel chords in a circle have lengths 24 and 10. The distance between them is 7.

Then find the diameter of the circle.

30. In a triangle ABC, D and E are the midpoints of AB and AC respectively. Also F and G

are the midpoints of DB and EC respectively. Let N be the intersection of FG and CD.

Let M be the intersection of FG and EB. When EB intersects with CD at O, find the

ratio of the area of %OMN to the area of %ABC .





Answers

Level 1

1 a! b! c! d! e!

2 a! b! c! d! e!

3 a! b! c! d! e!

4 a! b! c! d! e!

5 a! b! c! d! e!

6 a! b! c! d! e!

7 a! b! c! d! e!

8 a! b! c! d! e!

9 a! b! c! d! e!

10 a! b! c! d! e!

Level 2

11 a! b! c! d! e!

12 a! b! c! d! e!

13 a! b! c! d! e!

14 a! b! c! d! e!

15 a! b! c! d! e!

16 a! b! c! d! e!

17 a! b! c! d! e!

18 a! b! c! d! e!

19 a! b! c! d! e!

20 a! b! c! d! e!

Level 3

21

22

23

24

25

26

27

28

29

30

· · · · · · · · · · · O!cial Use Only · · · · · · · · · · ·




(Problem Sheets)




Instructions!

"

#

$














completely, etc).



to the instructor.



# # # Level 1 – Grade 8 # # #

1. Andrew has 23 coins worth $3.65 consisting of only quarters and dimes. How many of the

coins are dimes?

(a) 14 (b) 9 (c) 8 (d) 4 (e) 5

2. Katie, Sunny, and 3 other boys make a team for a quiz show. In how many di"erent ways

can 5 people be seated in a row if Katie and Sunny sit next to each other?

(a) 24 (b) 30 (c) 48 (d) 52 (e) 60

3. Which of the following is equivalent to the statement for the particular computer?

If the program is running, then there must be at least 250K of RAM in this computer.

(a) If there is at least 250K of RAM, then the program is running.

(b) If there is less than 250K of RAM, then the program is not running.

(c) The program will run only if there is at least 250K of RAM.

(d) If the program is not running, then there is less than 250K of RAM.

(e) Even if there is greater than 250K of RAM, then the program is not running.

4. Suppose $2 < b < a < 0. Find the second smallest number among A, B, C, D, and E.

A =!

($a)2 B = $%

a2 C = |b| D = $%

b2 E = $!

(4 $ b)2

(a) A (b) B (c) C (d) D (e) E

5. Let 25x $ 0.2 > 0.3(x$ 1) + 3

5and 2

5x > a $ x!a

2have the same solution set. What is a?

(a) 1 (b) 2 (c) 3 (d) 4 (e) 5


6. Jinyoung is going to install floor tiles in his rectangular room. He chose A inch square tiles

for his 96 in by 60 in room. What is the least number of tiles to cover the floor completely

without cutting tiles?

(a) 12 (b) 36 (c) 40 (d) 52 (e) 60

7. The quadratic equation 2x2 $ 6x + 3 = 0 has two solutions ! and ". Find 1!

+ 1".

(a) $6 (b) $3 (c) $1 (d) 3 (e) 2

8. A function f has the following properties. What is the value of f(243)?

f(3) = 1

f(xy) = f(x) + f(y) for any real numbers x, y

(a) 15 (b) 12 (c) 10 (d) 5 (e) 1

9. In a 100-yard race, Justin beat Bob by 10 yards and Bob beat Jon by 20 yards. Assuming

that each runner ran at a constant speed, by how much did Justin beat Jon in yards?

(a) 35 (b) 30 (c) 28 (d) 25 (e) 20

10. Let f be given by the graph in below. If g(x) = 2 + 3 · f(x + 1), find g($2).

(a) $10 (b) $4 (c) $1 (d) 5 (e) 8


# # # Level 2 – Grade 8 # # #

11. Each square in the partially filled grid shown contains a number from 1 to 25, each number

appearing exactly once. The numbers in each row and each column add up to 65. What is

the value of x + y ?

(a) 15 (b) 17 (c) 19 (d) 21 (e) 23

12. The area bounded by the lines y $ 2x + 6 = 0, y $ ax + 6 = 0, and the x-axis is 21. Find

all possible x-intercepts of the line y $ ax + 6 = 0.

(a) $1.5 (b) 0.6 (c) $1.5, 0.6 (d) $10, $4 (e) 10, $4

13. There are two di"erent density saltwater A and B. If we mix 200g of A and 300g of B then

it becomes 8% saltwater. If we mix 300g of A and 200g of B then it becomes 7% saltwater.

Find the density percentage of A.

(a) 5% (b) 7% (c) 8% (d) 9% (e) 10%

14. We will select five di"erent digits from {0, 2, 3, 4, 5, 7, 9} to make 5-digit numbers. How

many of the possible 5-digit numbers are divisible by 4?

(a) 2160 (b) 420 (c) 320 (d) 312 (e) 240

15. An equilateral triangle and a regular hexagon have equal perimeters 18 cm. What is the

ratio of the area of the triangle to the area of the hexagon?

(a) 1 (b) 12

(c)"

32

(d) 13

(e) 23


16. A weighted coin is twice as likely to turn up tails as heads. If the coin is tossed five times,

what is the probability that the third head occurs on the fifth toss?

(a) 881

(b) 40243

(c) 1681

(d) 80243

(e) 35

17. For how many di"erent positive integers n, does%

n di"er from%

100 by less than 1?

(a) 9 (b) 10 (c) 19 (d) 20 (e) 39

18. Let f(x) = 1"

x+1+"

x. Find f(0) + f(1) + · · ·+ f(99).

(a) $10 (b) $9 (c) 9 (d) 10 (e)%

99 $ 1

19. The probability that a randomly chosen male has a headache problem is 0.25. Men who

have a headache problem are twice as likely to be smokers as those do not have a headache

problem. What is the probability that a male has a headache problem, given that he is a

smoker?

(a) 14

(b) 13

(c) 25

(d) 12

(e) 23

20. In sequence 1, 2, 5, 10, 17, 26, 37, 50, · · · , find the 50th number of the sequence.

(a) 2400 (b) 2401 (c) 2402 (d) 2501 (e) 2502


# # # Level 3 – Grade 8 # # #

21. Let P and Q be points in line segment AB such that AB = 3PB and 4AQ = 3QB. Let M

be the midpoint of AP and N the midpoint of QB. If MN = 163

, find the length of AB.

22. Suppose that 10 teams participated in a soccer tournament where each team played exactly

one game against each of the other teams. The winner of each game received 3 points, while

the loser received 0 points. In case of a tie, both teams received 1 point. At the end of the

tournament, the 10 teams received a total of 126 points. How many games ended in a tie?

23. There are 16 points in a plane where the distance of nearest two points are 1 cm and the

rows and columns are parallel respectively. How many squares can we make by connecting

any 4 points?

• • • •• • • •• • • •• • • •

24. Consider the triangle bounded by 3 lines x + y = 5, x + 3y = 3, and 2x $ y = 1. Suppose

(x, y) = (a, b) is on or inside the triangle. Find the maximum value of a + 2b.

25. Let a, b, c, d and e be digits such that M = ab, a two digit number, and N = cde, a three

digit number. If 9(M & N ) = abcde, find M + N .


26. Urn I contains ten balls; four red and six black balls, and Urn II contains sixteen red and

an unknown number of black balls. A single ball is drawn from each urn. If the probability

that both balls are the same color is 0.44, find the number of black balls in Urn II.

27. The area of the parallelogram ABCD is 84 cm2. The point P is on CD such that CP :

PD = 1 : 3, and AP and BD intersects at Q. Find the area of # AOQ.

28. Solve the equation for x if

x =6

1 + 6

1+6

....

29. A right circular cone is inscribed in a sphere whose radius is 6 in. If the slant height of the

cone has a length equal to the diameter of the base, find the volume of this right circular

cone.

30. Given a sequence of numbers, A1, A2, A3 · · · which satisfies the following properties, find

A1 + A2 + · · ·+ A10.

A1 = 1, and A2 = 3

An+2 = 3An+1 $ 2An for all n > 0





Answers

Level 1

1 a! b! c! d! e!

2 a! b! c! d! e!

3 a! b! c! d! e!

4 a! b! c! d! e!

5 a! b! c! d! e!

6 a! b! c! d! e!

7 a! b! c! d! e!

8 a! b! c! d! e!

9 a! b! c! d! e!

10 a! b! c! d! e!

Level 2

11 a! b! c! d! e!

12 a! b! c! d! e!

13 a! b! c! d! e!

14 a! b! c! d! e!

15 a! b! c! d! e!

16 a! b! c! d! e!

17 a! b! c! d! e!

18 a! b! c! d! e!

19 a! b! c! d! e!

20 a! b! c! d! e!

Level 3

21

22

23

24

25

26

27

28

29

30

· · · · · · · · · · · O!cial Use Only · · · · · · · · · · ·




(Problem Sheets)




Instructions!

"

#

$














completely, etc).



to the instructor.



# # # Level 1 – Grade 9 # # #

1. If$

48 + a$

3%$

12 = 6$

3, then which of the following is$

a ?


2. Paul drove from Des Moines to Detroit via Chicago at an average speed of 50 miles per

hour. It took 5 hours to drive from Des Moines to Chicago at an average speed of 70 miles

per hour. If the whole trip took 13 hours, how far is it from Chicago to Detroit?

Note: The answer may be di"erent from the actual distance.

(a) 250 miles (b) 300 miles (c) 350 miles (d) 400 miles (e) None of these

3. If x =1

2+ 2

$6 and y =

1

2% 2

$6, which of the following is x2 % y2 ?

(a) 2$

3 (b) 4$

3 (c) 2$

6 (d) 4$

6 (e) None of these

4. If the average of a, b, c, d, and e is x, which of the following is the average of a + 2b% 3, b+

2c % 1, c + 2d, d + 2e + 1 and e + 2a + 3 ?

(a) x (b) 3x (c) x + 3 (d) 3x + 3 (e) Cannot be determined

5. If 3 % 2$

2 solves x2 % 6x + a = 0, then which of the following is a?

(a) 1 (b)$

2 (c)$

3 (d) 2 (e) None of these


6. Find the sum of the angles #a, #b, #c, #d,#e and #f .

(a) 360! (b) 450! (c) 540! (d) 630! (e) Cannot be determined

7. If 4 Americans, 3 Koreans, and 3 Frenchmen are to be seated in a row, how many seating

arrangements are possible when people of the same nationality must sit next to each other?

Note: n! = n · (n % 1) · · ·2 · 1 for a positive integer n. For example, 4! = 4 · 3 · 2 · 1.

(a) 4 · 3 · 3 (b) 4!3!3! (c)10!

4!3!3!(d) 10! (e) None of these

8. If a < b < 0, then which of the following is/are true?

I. %$

2a > %$

2b II. ab > b2

III.a + 2b

b> 3 IV.

!

(a % b)2 = a % b

(a) I only (b) II only (c) I,II and III only

(d) I, II and IV only (e) I, II,III and IV

9. How many natural numbers n are there satisfying 4 %$

7 < n < 5 +$

11?


10. Determine the number of digits of 28 & 3 & 510.



# # # Level 2 – Grade 9 # # #

11. At the beginning of this year a stock was worth $25. On January 31, its price went up

20%. On February 28, its price dropped 20% and on March 31 its price went back up 20%.

What was the single percent increase in the price of the stock over the three-month period?

(a) 0% (b) 13.4% (c) 15.2% (d) 18.6% (e) 20.0%

12. A mixture containing 6% salt is to be mixed with 2 ounces of a salt solution containing

15% salt in order to obtain a 12% salt solution. How much of the first solution must be

used?

(a) 1 ounce (b) 2 ounces (c) 3 ounces (d) 4 ounces (e) None of these

13. Which one is the same as

1

1 +$

2+

1$

2 +$

3+

1$

3 +$

4+ · · ·+

1$

24 +$

25?


14. The numbers 1, 2, 3, 4, and 5 are written on balls in a jar. If you are to draw two balls from

the jar at the same time, what is the probability that the product of two numbers you will

draw is a prime number?

(a) 1/10 (b) 1/5 (c) 3/10 (d) 2/5 (e) None of these


15. If AC and BD are perpendicular, AD = 6, and BC = 4, which of the following is AB2

+

CD2?

(a) 12 (b) 24 (c) 52

(d) 144 (e) Cannot be determined

16. Let a point Q be on the line segment connecting A(0, 12) and B(4, 0). Find the maximum

area of the rectangle $ OPQR.


17. For a positive integer n, let a = (n% 1)2 + 4n + (n + 1)2. Which of the following for$

a is

always true?

(a) Even number (b) Odd number (c) Rational number which is not an integer

(d) Irrational number (e) None of these

18. For f(x) =x % 3

x + 1, define f1(x) = f(x), f2(x) = f(f1(x)), f3(x) = f(f2(x)), · · · , and so on.

Evaluate f2009(f2010(3)).

(a) %3 (b) 0 (c) 3 (d) 2010 (e) 4019


19. If a and b are positive integers such that1

a+

1

2a+

1

3a=

1

b2 % 2b, then which of the following

is the smallest possible value of a + b?


20. Leslie, Elgin, Fritz, Howard, and Gary each choose to take 3 lessons from a choice of tennis,

violin, art, Spanish, and piano. Only one of them does not take tennis lessons and three

of them take lessons for violin, art, and Spanish. Leslie takes Spanish and piano lessons,

Elgin takes art and Spanish lessons, Fritz takes violin and art lessons, Howard takes tennis,

violin and art lessons, and Gary takes tennis, violin, and piano lessons. Which of following

is NOT true?

(a) Leslie does not take tennis lessons.

(b) Elgin does not take violin lessons.

(c) Fritz takes neither tennis nor piano lessons.

(d) Howard takes neither Spanish nor piano lessons.

(e) Only two of them take piano lessons.


# # # Level 3 – Grade 9 # # #

21. Roger and Maria are running around a track having a perimeter of 300 meters. If they start

running from the start line simultaneously in the same direction, Roger will pass Maria in

1 minute. If they run in di"erent directions, they will meet in 20 seconds. Find the speed

of Roger in meters per second.

22. There are 4 light bulbs in a row with switches. If they represent integers by 3 and 2, for

example

on o" o" on = 3, 223 o" o" on o" = 2, 232

what is the sum of all numbers that the light bulbs can represent?

23. Let x be the decimal part of (2 %$

3)2010, y = (2 %$

3)2010 % x, and z = (2 +$

3)2010.

Find xy + yz + zx.

24. For x =

"

2 +

#

2 +$

2 + · · · and y =

"

3 %#

3 %$

3 · · ·, define a = x2%x and b = y2+y.

Evaluate a + b.

Note: It is known that x and y are real numbers.

25. Suppose a candy cane costs 15 cents and a jellybean costs 3 cents. If Cli"ord wants to

buy a total of 30 candy canes and jellybeans with $3.30, what is the maximum number of

candy canes he can buy?


26. If x2 + 3x + 1 = 0, evaluate x2 + x + 2 +1

x+

1

x2.

27. Scott woke up between 7 and 8 AM when two hands of a clock coincided. When he arrived

at school between 8 and 9 AM, he noticed that two hands coincided again. How long did

it take for Scott to arrive at school after waking up?

28. If OA and OB are tangential to a circle with radius 3 and #AOB = 60!, what is the area

of 'OAB?

29. For positive integers a and b, a new operation ( is defined as follows:

a ( b =a + b

ab + a + b + 2

For example, 2(3 =2 + 3

2 · 3 + 2 + 3 + 2=

5

13. For some values of x and y, the value of x(y

is equal tox + y

16. Evaluate x ( y.

30. Let a and b be integers satisfying (1+$

2)2010 = a+b$

2. Find the greatest common divisor

of a and b.





Answers

Level 1

1 a! b! c! d! e!

2 a! b! c! d! e!

3 a! b! c! d! e!

4 a! b! c! d! e!

5 a! b! c! d! e!

6 a! b! c! d! e!

7 a! b! c! d! e!

8 a! b! c! d! e!

9 a! b! c! d! e!

10 a! b! c! d! e!

Level 2

11 a! b! c! d! e!

12 a! b! c! d! e!

13 a! b! c! d! e!

14 a! b! c! d! e!

15 a! b! c! d! e!

16 a! b! c! d! e!

17 a! b! c! d! e!

18 a! b! c! d! e!

19 a! b! c! d! e!

20 a! b! c! d! e!

Level 3

21

22

23

24

25

26

27

28

29

30

· · · · · · · · · · · O!cial Use Only · · · · · · · · · · ·




(Problem Sheets)




Instructions!

"

#

$














completely, etc).



to the instructor.



# # # Level 1 – Grade 10 # # #

1. Find 2x + 3y given that x + y = 6 and x2 + 3xy + 2y2 = 60.

(a) 14 (b) 16 (c) 18 (d) 20 (e) 22

2. Determine

i2007 $ i2008 $ i2009 $ i2010,

where i2 = %1.

(a) 1 (b) i (c) %i (d) %1 (e) None of these

3. Determine (&

2%&

3)2010 given (&

2 +&

3)2010 = a.

(a) %a (b)1

a(c) %

1

a(d) a2 (e) 1 % a

4. For a graph of y = f(x) shown below, find the number of real numbers x satisfying

f(f(x)) = 1.

(a) 3 (b) 5 (c) 6 (d) 8 (e) 10

5. Find the remainder when

(x % 1) + 2(x % 1)2 + 3(x % 1)3 + · · ·+ 15(x% 1)15 + 16(x% 1)16

is divided by x % 2.

(a) 153 (b) 68 (c) %32 (d) 136 (e) %136


6. If x < 0 and %1 < y < 0, which of the following is true?

(a) x < xy2 < xy (b) x < xy < xy2 (c) xy2 < x < xy

(d) xy2 < xy < x (e) None of these

7. Consider the circle (x % 3)2 + (y % 4)2 = 8. Find an equation of the line tangent to the

circle at the point (1, 2).

(a) y = %x + 3 (b) y = %2x + 4 (c) y = %1

2x +

5

2(d) y =

1

2x +

3

2(e) None of these

8. To buy a membership at a recreation center, people must pay a one-time registration fee

plus a regular monthly fee. The table below shows the total amount a person pays, including

the registration fee, for di"erent numbers of months of membership.

Membership Fees

Number of Months Total Amount Paid

6 $140

12 $230

According to the information in the table, which of the following statements is true?

(a) The monthly fee is $10. (b) The monthly fee is $15.

(c) The registration fee is $20. (d) The registration fee is $30.

(e) None of these

9. Find the real number t for which

3t $ 3t+1 $ 3t+2 $ 3t+3 $ 3t+4 = 243.

(a) %1 (b) %1

2(c) %

2

9(d)

5

9(e) %

11

10

10. If a function f(x) satisfies 2f(x) + 3f(1% x) = x2 for all real numbers x, what is f(0)?

(a)5

3(b) %

5

3(c)

3

5(d) %

3

5(e) None of these


# # # Level 2 – Grade 10 # # #

11. Suppose logy x = logx y, where x and y are positive real numbers other than 1 with x '= y.

Which of the following is true?

(a) xy = 1 (b) xy = yx (c) x + y = 1 (d) x + y = 2 (e) None of these

12. Of the numbers 6222, 5333, 4444, 3555, and 2666 which is the greatest?

(a) 6222 (b) 5333 (c) 4444 (d) 3555 (e) 2666

13. Find the coe!cient of x2 in the expansion of

!

2x +3

x

"4

.


14. In the circle below, AB and CD are chords intersecting at E (Not drawn to scale). If

AE = 5, BE = 14, and CE = 7, what is the length of DE?

(a) 7 (b) 9 (c) 10 (d) 12 (e) 13


15. If positive integers x and y satisfy the following, find x % y.

3 +1

x +1

y +1

40

=776

245

(a) 1 (b) 2 (c) 3 (d) 4 (e) %1

16. Find the solution of the following equation

x %1 %

3x

24

%2 %

x

43

= 2.

(a) 1 (b) %1 (c) 2 (d) %2 (e) 3

17. From the figure below (Not shown to scale) AB = BC = AD, #ABC = ! and #CAD = ",

which of the following is true?

(a) ! + " = 180! (b) 2! + 2" = 180! (c) ! + " = 180!

(d) 3! + 2" = 180! (e) 2" = !

18. Find the constant term in a polynomial of lowest degree with rational coe!cients that has

1 %&

2 and 1 + 2i as two of its zeros and with leading coe!cient of 1.

(a) 8 (b) %4 (c) %8 (d) %5 (e) 2


19. The surface area of a large spherical balloon is doubled. By what factor is the volume of

the balloon increased?

(a) 8 (b) 4 (c) 2&

2 (d) 3&

4 (e) 2

20. John forgot to write down a very important phone number. All he remembers is that it

started with 866 and that the next set of 4 digits involved 1, 3 and 9 with one of these

numbers appearing twice. If he guesses a phone number, what is the probability that he

gets the number correct?

(a)1

36(b)

1

72(c)

1

24(d)

1

81(e)

1

27


# # # Level 3 – Grade 10 # # #

21. In the diagram below of (ABC (Not drawn to scale), three triangles (CEF , (BDF , and

(CBF have areas 30, 40, and 60 respectively. Find the area of (ABC.

22. Find the largest integer n for which40!

3nis an integer.

23. Find the maximum ofy

xsatisfying (x % 3)2 + (y % 3)2 = 6.

24. Two circles with radius of 1 and 2 are inscribed so that their centers lie along the diagonal

of the square shown (Not drawn to scale). Each circle is tangent to two sides of the square

and they are tangent to each other. Find the shaded area between the circles and the

square.


25. Kim starts at point A travels 5 km east, then turns and continues 3 km north, then turns

southwest and continues&

8 km and ends at point B as in the figure below (Not drawn to

scale). How far in km is point A from point B?

26. For a real number x find the sum of the maximum and minimum of

y =(x2 % 2x % 3)

(2x2 + 2x + 1).

27. If !n and "n are the solutions of x2 + (n + 1)x + n2 = 0 for a positive integer n, find the

following sum

1

(!2 + 1)("2 + 1)+

1

(!3 + 1)("3 + 1)+ · · ·+

1

(!20 + 1)("20 + 1).

28. Each of two boxes contains 10 marbles, and each marble is either black or white. The total

number of black marbles is di"erent from the total number of white marbles. One marble

is drawn at random from each box. If the probability that both marbles are white is 0.24,

what is the probability that both are black?


29. Let X = {a, b, c, d, e} and Y = {1, 2, 3}. Determine the number of all possible functions f

from X to Y such that f(a) + f(b) + f(c) + f(d) + f(e) = 7.

30. Let [x] denote the greatest integer that is less than or equal to x. For example, [3.1] = 3,

[%1.2] = %2, and [2] = 2. Suppose that a function f is defined by f(x) = x% [x] and for a

positive integer k

Sk = {x|f(kx) = 0, 0 ) x ) 1}.

Count the number of elements in the intersection of two sets, S3 * S5.





Answers

Level 1

1 a! b! c! d! e!

2 a! b! c! d! e!

3 a! b! c! d! e!

4 a! b! c! d! e!

5 a! b! c! d! e!

6 a! b! c! d! e!

7 a! b! c! d! e!

8 a! b! c! d! e!

9 a! b! c! d! e!

10 a! b! c! d! e!

Level 2

11 a! b! c! d! e!

12 a! b! c! d! e!

13 a! b! c! d! e!

14 a! b! c! d! e!

15 a! b! c! d! e!

16 a! b! c! d! e!

17 a! b! c! d! e!

18 a! b! c! d! e!

19 a! b! c! d! e!

20 a! b! c! d! e!

Level 3

21

22

23

24

25

26

27

28

29

30

· · · · · · · · · · · O!cial Use Only · · · · · · · · · · ·




(Problem Sheets)




Instructions!

"

#

$














completely, etc).



to the instructor.



# # # Level 1 – Grade 11 # # #

1. It is known that the equation x2 + x+ 1 = 0 has two complex number solutions, say ! and

" are the solutions. Find !3 + "3.

(a) 0 (b) 1 (c) 2 (d) $1 (e) $2

2. Find the solution set of the equation!

ex+1"x+2

= 1.

(a) {1, 2} (b) {$1,$2} (c) {32} (d) {$3

2} (e) { } (No solution)

3. A committee composed of Alice, Mark, Ben, Connie, and Francisco is about to select

three representatives randomly. What is the probability that Connie is excluded from the

selection?

(a) 45 (b) 3

5 (c) 25 (d) 1

5 (e) 0

4. A ball is thrown vertically upward and its height after t seconds is h = 24+64t$16t2 feet.

Find the maximum height reached by the ball.

(a) 24 feet (b) 64 feet (c) 80 feet (d) 88 feet (e) 104 feet

5. Evaluate the following:

log2

#

1

2

$

+ log2

#

2

3

$

+ log2

#

3

4

$

+ · · ·+ log2

#

1023

1024

$

.

(a) $10 (b) $5 (c) 0 (d) 5 (e) 10


6. How many one-to-one functions are there from the set X = {a, b, c} to the set Y =

{1, 2, 3, 4}.

(a) 9 (b) 16 (c) 24 (d) 27 (e) 64

7. Find the sum of all solutions of the equation log x + log(x + 21) = 2. (Note that log is the

common logarithmic function with base 10.)

(a) $21 (b) $4 (c) 0 (d) 4 (e) 21

8. Let z = 2 + bi, where b is a positive real number and i2 = $1. If the imaginary parts of z2

and z3 are equal, then what is b?

(a) 1 (b)%

2 (c) 2 (d) 2%

2 (e) 4

9. What is the di"erence between the sum of the numbers in the set {2, 6, 10, 14, · · · , 38} and

the average of the set of numbers?

(a) 220 (b) 200 (c) 180 (d) 160 (e) 140

10. Let f(x) = x2 + 1. If a function g satisfies the following two conditions:

(a) f(g(x)) = x4 + 2x2 + 2, and

(b) g(x) & 0 for any x.

Then what is g(2)?

(a) 0 (b) $1 (c) $3 (d) $5 (e) $7


# # # Level 2 – Grade 11 # # #

11. The rational function f(x) =ax + 2

x + bhas the inverse function f!1(x) =

ax + 2

x + b. Find a+b.

(a) $2 (b) $12 (c) 0 (d) 1

2 (e) 2

12. Find the remainder when x2010 + x4 $ x2 + 1 is divided by x2 + 1.

(a) 0 (b) 1 (c) 2 (d) 1005 (e) 2011

13. If 4x = 7, then what is 8x!1?

(a) 7%

7 $ 8 (b) 8%

7$ 7 (c) 7"

78 (d) 8

"

77 (e) 7

8

14. If sinx $ cosx = 23 , then what is the value of sin3 x $ cos3 x?

(a) $2327 (b) $13

27 (c) 127 (d) 13

27 (e) 2327

15. Find the di"erence of the maximum value and minimum value of # + sinx + 3 cosx.

(a) 2%

2 (b)%

10 (c) 2%

2 # (d) 2%

10 (e) 2%

10 #


16. If log2 5 = a, then what is log25 20 in terms of a?

(a) 12 + 1

a(b) 1

2 + 1a2 (c) 1

2 + 12a

(d) 12 + a (e) 1

2 + a2

17. An employee of a computer store is paid a base salary $1,500 a month plus a 2% commission

on all sales over $4,000 during the month. How much must the employee sell in a month

to earn a total of $2,000 for the month?

(a) $25, 000 (b) $26, 000 (c) $27, 000 (d) $28, 000 (e) $29, 000

18. A speedboat takes 1 hour longer to go 16 miles up a river than to return. If the boat cruises

at 12 miles per hour in still water, what is the rate of the current?

(a) 2 miles/h (b) 4 miles/h (c) 5 miles/h (d) 6 miles/h (e) 8 miles/h

19. The set of all real numbers x for which

log2010 (log2009 (log2008 (log2007 x)))

is defined is {x|x > c}. What is the value of c?

(a) 0 (b) 1 (c) 20072008 (d) 20092010 (e) 200720082009

20. Find the number of distinct real solutions for the equation 1 $ 2x log2 x = 0.

(a) No solution (b) 1 solution (c) 2 solutions (d) 4 solutions (e) Infinite


# # # Level 3 – Grade 11 # # #

21. Find the number of distinct real solutions for the equation 5x5 + 3x3 $ 2x2 $ x = 0.

22. Let x be 1 +1

2 +1

1 +1

2 +1.. .

. It is known that x is a real number. Evaluate (2x $ 1)2.

23. If sinx = 3 cosx, then what is the value of sinx cosx?

24. Let f be a function such that f(x) + f

#

4

2 $ x

$

=1

xfor all real numbers x except 0 and

2. What is the value of f(4)?

25. Let p be a prime number and k be an integer. If the equation x2 + kx + p = 0 has two

positive integer solutions, then what is the value of k + p?


26. How many pairs (x, y) of real numbers satisfy

x2008 + y2008 = x2009 + y2009 = x2010 + y2010.

27. Let (x $ 1)10 = x10 + a9x9 + · · ·+ a1x + a0. Find a1 + a3 + a5 + a7 + a9.

28. Let m and n be integers greater than 1. In base n system, m3 is written as 10(n). Then

what is the base m representation for n2?

(In base r representation system, we use exactly r symbols to express numbers. For exam-

ples, we use the symbols 0, 1, 2, 3 and 4 to express number in base 5 system, and we use

the symbols 0, 1, 2, 3, 4, 5 and 6 in base 7 system. Thus, the base 5 representation for the

integer 25 is 100(5), and the same integer 25 can be expressed as 34(7) in base 7 system.)

29. There are 24 four-digit numbers that can be formed by permuting 1, 2, 3 and 4. In fact,

they are

1234, 1243, 1324, 1342, · · · , 4321.

Find the sum of all these numbers.

30. A circle of radius r is entirely contained in the interior of a circle of radius R. Consider the

region lying outside the first circle but inside the second. If that region’s area is 2# and

the length of its boundary is 5#, what is the bigger radius R?

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