Math G9 Challenge practice (34) Name_____________

1. Mary thought of a positive two-digit number. She multiplied it by 3 and added 11. Then she switched the
digits of the result, obtaining a number between 71 and 75, inclusive. What was Mary's number?
(A) 11 (B) 12 (C) 13 (D) 14 (E) 15

2. Sofia ran 5 laps around the 400-meter track at her school. For each lap, she ran the first 100 meters at an
average speed of 4 meters per second and the remaining 300 meters at an average speed of 5 meters per
second. How much time did Sofia take running the 5 laps?
(A) 5 minutes and 35 seconds (B) 6 minutes and 40 seconds
(C) 7 minutes and 5 seconds (D) 7 minutes and 25 seconds
(E) 8 minutes and 10 seconds

3. Real numbers x, y, and z satisfy the inequalities0 < 𝑥 < 1, −1 < 𝑦 < 0, and 1 < 𝑧 < 2. Which of the
following numbers is necessarily positive?
(A) 𝑦 + 𝑥 2 (B) 𝑦 + 𝑥𝑧 (C) 𝑦 + 𝑦 2 (D) 𝑦 + 2𝑦 2 (E) 𝑦 + 𝑧

3𝑥+𝑦 𝑥+3𝑦
4. Suppose that x and y are nonzero real numbers such that 𝑥−3𝑦 = −2. What is the value of 3𝑥−𝑦?

(A) −3 (B) −1 (C) 1 (D) 2 (E) 3

5. Camilla had twice as many blueberry jelly beans as cherry jelly beans. After eating 10 pieces of each kind,
she now has three times as many blueberry jelly beans as cherry jelly beans. How many blueberry jelly beans
did she originally have?
(A) 10 (B) 20 (C) 30 (D) 40 (E) 50

6. What is the largest number of solid 2” × 2” × 1” blocks that can fit in a 3” × 2” × 3” box?
(A) 3 (B) 4 (C) 5 (D) 6 (E) 7

7. 𝑆𝑎𝑚𝑖𝑎 set off on her bicycle to visit her friend, traveling at an average speed of 17 kilometers per hour.
When she had gone half the distance to her friend's house, a tire went flat, and she walked the rest of the
way at 5 kilometers per hour. In all it took her 44 minutes to reach her friend's house. In kilometers rounded to
the nearest tenth, how far did 𝑆𝑎𝑚𝑖𝑎 walk?
(A) 2.0 (B) 2.2 (C) 2.8 (D) 3.4 (E) 4.4

8. Points A (11, 9) and B (2, –3) are vertices of ∆ABCwith AB = AC. The altitude from
A meets the opposite side at D (–1, 3). What are the coordinates of point C?
(A) (−8,9) (B) (−4,8) (C) (−4,9) (D) (−2,3) (E) (−1,0)

9. A radio program has a quiz consisting of 3 multiple-choice questions, each with 3 choices. A contestant
wins if he or she gets 2 or more of the questions right. The contestant answers randomly to each question.
What is the probability of winning?
1 1 2 7 1
(A) 27 (B) 9 (C) 9 (D) 27 (E) 2
10.The lines with equations 𝑎𝑥 − 2𝑦 = 𝑐 and 2𝑥 + 𝑏𝑦 = −𝑐 are perpendicular and intersect at
(1, – 5). What is 𝑐?
(A) −13 (B)−8 (C) 2 (D) 8 (E) 13

11.At 𝑇𝑦𝑝𝑖𝑐𝑜 High School, 60% of the students like dancing, and the rest dislike it. Of those who like
dancing, 80% say that they like it, and the rest say that they dislike it. Of those who dislike dancing, 90% say
that they dislike it, and the rest say that they like it. What fraction of students who say they dislike dancing
actually like it?
1
(A) 10% (B) 12% (C) 20% (D) 25% (E) 333%

12.Elmer's new car gives 50% better fuel efficiency. However, the new car uses diesel fuel, which is 20%
more expensive per liter than the gasoline the old car used. By what percent will Elmer save money if he uses
his new car instead of his old car for a long trip?
2 7 1 2
(A) 20% (B) 26 % (C) 27 % (D) 33 % (E) 66 %
3 9 3 3

13. There are 20 students participating in an after-school program offering classes in yoga, bridge, and
painting. Each student must take at least one of these three classes, but may take two or all three. There are 10
students taking yoga, 13 taking bridge, and 9 taking painting. There are 9 students taking at least two classes.
How many students are taking all three classes?
(A) 1 (B) 2 (C) 3 (D) 4 (E) 5
14. Rectangle 𝐴𝐵𝐶𝐷 has 𝐴𝐵 = 3 and 𝐵𝐶 = 4. Point 𝐸 is the foot of the perpendicular from 𝐵 to diagonal 𝐴𝐶.
What is the area of ∆𝐴𝐸𝐷?
42 28 54
(A) 1 (B) 25 (C) 15 (D) 2 (E) 25

15. The roots of the equation 𝑥 2 + 4𝑥 − 5 = 0 are also the roots of the equation 2𝑥 3 + 9𝑥 2 − 6𝑥 − 5 =
0. What is the third root of the second equation?

16. The numbers 𝑎, 𝑏, 𝑐 are the digits of a three digit number which satisfy 49𝑎 + 7𝑏 + 𝑐 = 286 What is the
three digit number (100𝑎 + 10𝑏 + 𝑐)?
17. The vertices of a right-angled triangle are on a circle of radius 𝑅 and the sides of the triangle are tangent to
another circle of radius 𝑟. If the lengths of the sides about the right angle are 16 and 30, determine the value of
𝑅 + 𝑟.

18. Determine the smallest positive integer, n, which satisfies the equation 𝑛3 + 2𝑛2 = 𝑏 where 𝑏 is the
square of an odd integer.

19. In a 14 team baseball league, each team played each of the other teams 10 times. At the end of the season,
the number of games won by each team differed from those won by the team that immediately followed it by
the same amount. Determine the greatest number of games the last place team could have won, assuming that
no ties were allowed.
20. Triangle 𝐴𝐵𝐶 is right angled at 𝐴. The circle with center 𝐴 and radius 𝐴𝐵 cuts 𝐵𝐶 and 𝐴𝐶 internally at 𝐷
and 𝐸 respectively. If 𝐵𝐷 = 20 and 𝐷𝐶 = 16, determine 𝐴𝐶 2

𝐴
𝐸
𝐵 𝐷 𝐶

21.Begin with any two-digit positive integer and multiply the two digits together. If the resulting product is a
two-digit number, then repeat the process. When this process is repeated, all two-digit numbers will eventually
become a single digit number. Once a product results in a single digit, the process stops.
For example,

Two-digit Step 1 Step 2 Step 3

number
97 9 × 7 = 63 6 × 3 = 18 1×8=8 The process stops at 8 after 3 steps.
48 4 × 8 = 32 3×2=6
The process stops at 6 after 2 steps.
50 5×0=0
The process stops at 0 after 1 step.

(a) Beginning with the number 68, determine the number of steps required for the process to stop.

(b) Determine all two-digit numbers for which the process stops at 8 after 2 steps.

(c) Determine all two-digit numbers for which the process stops at 4.

(d) Determine a two-digit number for which the process stops after 4 steps.
22. Ian buys a cup of tea every day at Jim 𝐵𝑜𝑟𝑡𝑜𝑛𝑠 for $1.72 with money from his coin jar. He starts the year
with 365 two-dollar (200¢) coins and no other coins in the jar. Ian makes payment and the cashier provides
change according to the following rules:
• Payment is only with money from the coin jar.
• The amount Ian offers the cashier is at least $1.72.
• The amount Ian offers the cashier is as close as possible to the price of the cup of tea.
• Change is given with the fewest number of coins.
• Change is placed into the coin jar.
• Possible coins that may be used have values of 1¢, 5¢, 10¢, 25¢, and 200¢.
(a) How much money will Ian have in the coin jar after 365 days?

(b) What is the maximum number of 25¢ coins that Ian could have in the coin jar at any one time?

(c) How many of each type of coin does Ian have in his coin jar after 277 days?