Online Math LeagueA Division of AcademicLeagues.com

Algebra Sample Contest

Student Name DateRules: You have 30 minutes to complete the test. You must work independently.Calculators and reference tools are not permitted. Each problem has exactlyone correct answer.

1. Zoe picks a number, multiplies it by 3, adds 3, then multiplies by 3 again.When she’s done, she has 333. What number did Zoe originally pick?

A. 33 B. 36 C. 48 D. 66 E. 104

2. The next number in the sequence 153, 155, 159, 165, 173, . . . is

A. 179 B. 181 C. 183 D. 185 E. 189

3. One of the following numbers is prime. Which one is it?

A. 121 B. 87 C. 1895 D. 143 E. 491

4. For which of the following x values does 3x − 3 = 5x + 1?

A. -2 B. -1 C. 0 D. 1 E. 2

5. The side length of square B is twice the side length of square A. The sidelength of square C is twice the side length of square B. The area of squareC divided by the area of square A is:

A. 2 B. 4 C. 8 D. 16 E. 32

6. Suppose you play a game where you roll three standard dice (fair dicenumbered 1 through 6). You win the game if all three come up with oddnumbers. You also win the game if all three come up with even numbers.Otherwise, you lose. What is the probability that you win this game?

A. 18 B. 1

6 C. 14 D. 1

3 E. 12

7. Beth buys math and physics books at the bookstore. Each math bookcosts $13.50 and each physics book costs $12. If Beth’s 5 books cost $63,she has bought:

A. 0 math and 5 physics books B. 1 math and 4 physics booksC. 2 math and 3 physics books D. 3 math and 2 physics booksE. 4 math and 1 physics books

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8. One of the side lengths of a rectangle is 3 cm. The perimeter of therectangle (in cm) equals the area of the rectangle (in cm2). The sides ofthe rectangle that are not 3 cm long have a length of

A. 4 cm B. 6 cm C. 9 cm D. 12 cm E. 18 cm

9. A square is drawn on a coordinate grid. The endpoints of one side of thesquare are (1, 2) and (4, 6). What is the area of the square?

A. 5 B. 7 C. 16 D. 25 E. 49

10. Which of the following expressions is equal to 210 + 210?

A. 210 B. 211 C. 220 D. 2100 E. None of these.

11. Suppose two perpendicular lines `1 and `2 intersect at the point (0,4). Ifthe equation of `1 is y = 3x + 4, what is the equation of `2?

A. y = 3x − 4 B. y = −3x − 4 C. y = −3x + 4D. y = 1

3x + 4 E. y = − 13x + 4

12. Suppose I write the numbers 1 through 4 on separate pieces of paper andput the pieces of paper into a hat. If I draw one piece of paper out of thehat, throw it away, then draw a second piece of paper out of the hat, whatis the probability that the number on the second piece of paper is greaterthan the number on the first piece of paper?

A. 14 B. 1

3 C. 12 D. 2

3 E. 34

13. The two solutions to the equation 2x2 − 2x − 2 = 2 are:

A. 2 and 1 B. −1 and 1 C. −2 and 1 D. 2 and −1 E. −2 and −1

14. A line passes through the point (10,−18) and has an equation in the formy = mx + b. Which of the following points cannot lie on the line?

A. (0, 0) B. (12,−8) C. (10, 4) D. (12, 0) E. (0,−18)

15. Which of the following must be true about the sum of three consecutiveintegers?

A. The sum is odd.

B. The sum is even.

C. The sum is divisible by 3.

D. The sum is divisible by 4.

E. None of the above must be true.

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Online Math LeagueAlgebra Sample Contest

Answers

1. B 4. A 7. C 10. B 13. D2. C 5. D 8. B 11. E 14. C3. E 6. C 9. D 12. C 15. C

Explanations for some of the more difficult questions:

6. Half the numbers on a die are odd, so the probability that a single die willcome up odd is 1

2 . The probability that all three dice will come up odd is12 ·

12 ·

12 , or 1

8 . Similarly, the probability that one die will come up even is 12 , so

the probability that all three dice will come up even is 18 . Combining these, the

probability that you will win the game is 18 + 1

8 = 14 .

8. Suppose the unknown side length of the rectangle is x. The area of therectangle is 3x cm2, and the perimeter of the rectangle is 6 + 2x cm. Therefore3x = 2x + 6, and x = 6.

9. Using the Distance Formula or the Pythagorean Theorem, we can determinethat the distance from (1, 2) to (4, 6) is 5. Thus the side length of the square is5, and its area is 25.

10. 210 + 210 = 2 · 210 = 21 · 210 = 211.

11. The slopes of perpendicular lines are opposite reciprocals. The slope of `1is 3, so the slope of `2 must be − 1

3 . `2 also passes through the point (0, 4), soits y-intercept is 4. Thus the equation of `2 is y = − 1

3x + 4.

12. The two pieces of paper I draw have different numbers on them. One num-ber must be greater than the other, but the papers were drawn at random. Sothe first one is just as likely to be greater than the second as the second is likelyto be greater than the first. The probability of each outcome is 1

2 .

14. A line with an equation in the form y = mx + b cannot be a verticalline. (There is no real value for m that would make the line vertical.) If theline passed through (10,−18) and another point whose x-coordinate were 10, itwould be vertical. So the line cannot pass through (10, 4).

15. Suppose the first of the consecutive integers is x. Then the next integer isx + 1, and the integer after that is x + 2. The sum of these three integers isx + (x + 1) + (x + 2), or 3x + 3. Since x is an integer, 3x + 3 must be a multipleof 3.

Note that one way to solve this problem is to try summing up various setsof three consecutive integers, and see what you notice about the sums.

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