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2022 Gauss Contest
Grade 8

Wednesday, May 18, 2022
(in North America and South America)

Thursday, May 19, 2022
(outside of North American and South America)

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©2022 University of Waterloo

Instructions

Time: 1 hour

Calculating devices are allowed, provided that they do not have any of the following features: (i) internet access, (ii) the ability to communicate with other devices, (iii) information previously stored by students (such as formulas, programs, notes, etc.), (iv) a computer algebra system, (v) dynamic geometry software.

  1. Do not open the Contest booklet until you are told to do so.
  2. You may use rulers, compasses and paper for rough work.
  3. Be sure that you understand the coding system for your answer sheet. If you are not sure, ask your teacher to explain it.
  4. This is a multiple-choice test. Each question is followed by five possible answers marked A, B, C, D, and E. Only one of these is correct. When you have made your choice, enter the appropriate letter for that question on your answer sheet.
  5. Scoring:
    1. Each correct answer is worth 5 in Part A, 6 in Part B, and 8 in Part C.
    2. There is no penalty for an incorrect answer.
    3. Each unanswered question is worth 2, to a maximum of 10 unanswered questions.
  6. Diagrams are not drawn to scale. They are intended as aids only.
  7. When your supervisor instructs you to start, you will have sixty minutes of working time.
The name, school and location of some top-scoring students will be published on the Web site, cemc.uwaterloo.ca. On this website, you will also be able to find copies of past Contests and excellent resources for enrichment, problem solving and contest preparation.
Scoring:
  1. There is no penalty for an incorrect answer.
  2. Each unanswered question is worth 2, to a maximum of 10 unanswered questions.

Part A: Each correct answer is worth 5.

  1. The regular pentagon shown has a side length of 2 cm.

    A shape with 5 sides of equal length.

    The perimeter of the pentagon is

    1. \(2 \text{ cm}\)
    2. \(4 \text{ cm}\)
    3. \(6 \text{ cm}\)
    4. \(8 \text{ cm}\)
    5. \(10 \text{ cm}\)

  2. The faces of a cube are labelled with 1, 2, 3, 4, 5, and 6 dots. Three of the faces are shown.

    The visible faces of the die have 1, 3, and 5 dots.

    What is the total number of dots on the other three faces?

    1. \(6\)
    2. \(8\)
    3. \(10\)
    4. \(12\)
    5. \(15\)

  3. The equation that best represents "a number increased by five equals 15" is

    1. \(n-5=15\)
    2. \(n\div 5=15\)
    3. \(n+5=15\)
    4. \(n+15=5\)
    5. \(n\times 5=15\)

  4. The line graph shows the number of bobbleheads sold at a store each year.

    The horizontal axis of the line graph has the Year and the vertical axis has the Number of Bobble Heads. The year 2016 had 20 bobbleheads, 2017 had 35, 2018 had 40, 2019 had 38, 2020 had 60 and 2021 had 75.

    The sale of bobbleheads increased the most between

    1. \(\text{2016 and 2017}\)
    2. \(\text{2017 and 2018}\)
    3. \(\text{2018 and 2019}\)
    4. \(\text{2019 and 2020}\)
    5. \(\text{2020 and 2021}\)

  5. Starting at 72, Aryana counts down by 11s: \(72, 61, 50, \dots\). What is the last number greater than 0 that Aryana will count?

    1. \(4\)
    2. \(5\)
    3. \(6\)
    4. \(7\)
    5. \(8\)

  6. In the diagram, \(\angle ABC=90\degree\).

    Two lines divide angle ABC into three angles. The largest of these angles measures 44 degrees and the other two angles each measure x degrees.

    The value of \(x\) is

    1. \(68\)
    2. \(23\)
    3. \(56\)
    4. \(28\)
    5. \(26\)

  7. Which of the following values is closest to zero?

    1. \(-1\)
    2. \(\frac{5}{4}\)
    3. \(1^2\)
    4. \(-\frac{4}{5}\)
    5. \(0.9\)

  8. A jar contains 267 quarters. One quarter is worth $0.25. How many quarters must be added to the jar so that the total value of the quarters is $100.00?

    1. \(33\)
    2. \(53\)
    3. \(103\)
    4. \(133\)
    5. \(153\)

  9. A package of 8 greeting cards comes with 10 envelopes. Kirra has 7 cards but no envelopes. What is the smallest number of packages that Kirra needs to buy to have more envelopes than cards?

    1. \(3\)
    2. \(4\)
    3. \(5\)
    4. \(6\)
    5. \(7\)

  10. For the points in the diagram, which statement is true?

    The Cartesian plane with three points plotted. One point has coordinates (c, d) and is in the first quadrant close to the origin. Another point has coordinates (a, b) and also lies in the first quadrant. This point is to the right and above point (c, d). The final point has coordinates (e, f) and lies in the third quadrant.

    1. \(e > c\)
    2. \(b < d\)
    3. \(f > b\)
    4. \(a < e\)
    5. \(a > c\)

Part B: Each correct answer is worth 6.

  1. The 26 letters of the English alphabet are listed in an infinite, repeating loop:
    \(ABCDEFGHIJKLMNOPQRSTUVWXYZABC \dots\)
    What is the 258th letter in this sequence?

    1. \(V\)
    2. \(W\)
    3. \(X\)
    4. \(Y\)
    5. \(Z\)

  2. A public holiday is always celebrated on the third Wednesday of a certain month. In that month, the holiday cannot occur on which of the following days?

    1. 16th
    2. 22nd
    3. 18th
    4. 19th
    5. 21st

  3. A circular spinner is divided into three sections. An arrow is attached to the centre of the spinner. The arrow is spun once. The probability that the arrow stops on the largest section is \(50\%\). The probability it stops on the next largest section is 1 in 3. The probability it stops on the smallest section is

    1. \(\frac{1}{4}\)
    2. \(\frac{2}{5}\)
    3. \(\frac{1}{6}\)
    4. \(\frac{2}{7}\)
    5. \(\frac{3}{10}\)

  4. A positive number is divisible by both 3 and 4. The tens digit is greater than the ones digit. How many positive two-digit numbers have this property?

    1. \(4\)
    2. \(5\)
    3. \(6\)
    4. \(7\)
    5. \(8\)

  5. A rectangular pool measures 20 m by 8 m. There is a 1 m wide walkway around the outside of the pool, as shown by the shaded region.

    The area of the walkway is

    1. \(56\text{ m}^2\)
    2. \(60\text{ m}^2\)
    3. \(29\text{ m}^2\)
    4. \(52\text{ m}^2\)
    5. \(50\text{ m}^2\)

  6. The results of asking 50 students if they participate in music or sports are shown in the Venn diagram.

    15 students participate in music but not sports, 20 participate in sports but not music, and 5 participate in both music and sports.

    What percentage of the 50 students do not participate in music and do not participate in sports?

    1. \(0\%\)
    2. \(80\%\)
    3. \(20\%\)
    4. \(70\%\)
    5. \(40\%\)

  7. There are \(\frac{2}{3}\) as many golf balls in Bin F as in Bin G. If there are a total of golf balls, how many fewer golf balls are in Bin F than in Bin G?

    1. \(15\)
    2. \(30\)
    3. \(50\)
    4. \(60\)
    5. \(90\)

  8. In the sequence shown, Figure 1 is formed using 7 squares. Each figure after Figure 1 has 5 more squares than the previous figure.

    Figure 2 is formed using 12 squares. Figure 3 is formed using 17 squares. The squares in each figure are arranged to look like a capital H.

    What figure has 2022 squares?

    1. \(\text{Figure } 400\)
    2. \(\text{Figure } 402\)
    3. \(\text{Figure } 404\)
    4. \(\text{Figure } 406\)
    5. \(\text{Figure } 408\)

  9. Mateo’s 300 km trip from Edmonton to Calgary passed through Red Deer. Mateo started in Edmonton at 7 a.m. and drove until stopping for a 40 minute break in Red Deer. Mateo arrived in Calgary at 11 a.m. Not including the break, what was his average speed for the trip?

    1. \(83 \text{ km/h}\)
    2. \(94\text{ km/h}\)
    3. \(90\text{ km/h}\)
    4. \(95\text{ km/h}\)
    5. \(64\text{ km/h}\)

  10. Equilateral triangle \(ABC\) has sides of length 4. The midpoint of \(BC\) is \(D\), and the midpoint of \(AD\) is \(E\). The value of \(EC^2\) is

    1. \(7\)
    2. \(6\)
    3. \(6.25\)
    4. \(8\)
    5. \(10\)

Part C: Each correct answer is worth 8.

  1. The positive factors of 6 are 1, 2, 3, and 6. There are two perfect squares less than 100 that have exactly five positive factors. What is the sum of these two perfect squares?

    1. \(177\)
    2. \(80\)
    3. \(145\)
    4. \(52\)
    5. \(97\)

  2. In the list \(p,q,r,s,t,u,v\), each letter represents a positive integer. The sum of the values of each group of three consecutive letters in the list is 35.If \(q+u=15\), then \(p+q+r+s+t+u+v\) is

    1. \(85\)
    2. \(70\)
    3. \(80\)
    4. \(90\)
    5. \(75\)

  3. The net shown is folded to form a cube.

    The net consists of six identical squares. Four squares are placed side by side to form the middle of the net. These squares, in order, are labelled E, A, B, and F. The remaining squares are placed on either side of square B and are labelled C and D.

    An ant walks from face to face on the cube, visiting each face exactly once. For example, \(ABCFED\) and \(ABCEFD\) are two possible orders of faces the ant visits. If the ant starts at \(A\), how many possible orders are there?

    1. \(24\)
    2. \(48\)
    3. \(32\)
    4. \(30\)
    5. \(40\)

  4. The number 385 is an example of a three-digit number for which one of the digits is the sum of the other two digits. How many numbers between 100 and 999 have this property?

    1. \(144\)
    2. \(126\)
    3. \(108\)
    4. \(234\)
    5. \(64\)

  5. Student A, Student B, and Student C have been hired to help scientists develop a new flavour of juice. There are 4200 samples to test. Each sample either contains blueberry or does not. Each student is asked to taste each sample and report whether or not they think it contains blueberry. Student A reports correctly on exactly 90% of the samples containing blueberry and reports correctly on exactly 88% of the samples that do not contain blueberry. The results for all three students are shown below.

    Student A Student B Student C
    Percentage correct on samples
    containing blueberry    		 90% 98% \((2m)\)%
    Percentage correct on samples
    not containing blueberry    		 88% 86% \((4m)\)%

    Student B reports 315 more samples as containing blueberry than Student A. For some positive integers \(m\), the total number of samples that the three students report as containing blueberry is equal to a multiple of 5 between 8000 and 9000. The sum of all such values of \(m\) is

    1. \(45\)
    2. \(36\)
    3. \(24\)
    4. \(27\)
    5. \(29\)