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2016 Gauss Contest
Grade 7

Wednesday, May 11, 2016
(in North America and South America)

Thursday, May 12, 2016
(outside of North American and South America)

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©2015 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.
  5. Diagrams are not drawn to scale. They are intended as aids only.
  6. 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 value of \(333+33+3\) is

    1. \(396\)
    2. \(399\)
    3. \(669\)
    4. \(369\)
    5. \(963\)

  2. The graph shows the number of text messages received by Tanner in a given week.

    A bar graph titled Text Messages Received by Tanner. It shows Day of the Week on the horizontal axis and the Number of Messages on the vertical axis. The number of text messages received each day is given in the following list.

    On what day did Tanner receive the most text messages?

    1. \(\textrm{Friday}\)
    2. \(\textrm{Tuesday}\)
    3. \(\textrm{Thursday}\)
    4. \(\textrm{Saturday}\)
    5. \(\textrm{Wednesday}\)

  3. Which of the following is a multiple of 7?

    1. \(75\)
    2. \(76\)
    3. \(77\)
    4. \(78\)
    5. \(79\)

  4. Which of these fractions is larger than \(\frac{1}{2}\)?

    1. \(\frac{2}{5}\)
    2. \(\frac{3}{7}\)
    3. \(\frac{4}{7}\)
    4. \(\frac{3}{8}\)
    5. \(\frac{4}{9}\)

  5. A cube has exactly one face painted as shown.

    On the front face of a cube, the triangular region with vertices at the midpoint of the top edge, the top right vertex, and the midpoint of the right side is painted.

    The other five faces of the cube are not painted. If the cube is rolled, which of the following could be the same cube?

    1. On the front face, the triangular region with vertices at the top right vertex, the bottom right vertex, and bottom left vertex is painted.
    2. On the front face, the triangular region with vertices at the midpoint of the right edge, the bottom right vertex, and the midpoint of the bottom edge is painted.
    3. On the front face, the triangular region with vertices at the centre, the bottom right vertex, and the midpoint of the bottom edge is painted.
    4. On the front face, two triangular regions are painted, the first is the region with vertices at the midpoint of the top edge, the top right vertex, and the midpoint of the right side. The second is the region with vertices at the midpoint of the right edge, the bottom right vertex, and the midpoint of the bottom edge.
    5. On the front face, two triangular regions are painted. The first is the region with vertices at the midpoint of the right edge, the bottom right vertex, and the midpoint of the bottom edge. The second is the region with vertices at the midpoint of the top edge, the top left vertex, and the midpoint of the left edge.

  6. The measures of two angles of a triangle are \(25^{\circ}\) and \(70^{\circ}\). The measure of the third angle is

    1. \(85^{\circ}\)
    2. \(105^{\circ}\)
    3. \(65^{\circ}\)
    4. \(95^{\circ}\)
    5. \(75^{\circ}\)

  7. A box of fruit contains 20 apples, 10 oranges, and no other fruit. When a fruit is randomly chosen from the box, what is the probability that the fruit is an orange?

    1. \(\frac{1}{10}\)
    2. \(\frac{1}{20}\)
    3. \(\frac{1}{30}\)
    4. \(\frac{1}{3}\)
    5. \(\frac{2}{3}\)

  8. Alex pays $2.25 to take the bus. Sam pays $3.00 to take the bus. If they each take the bus 20 times, how much less would Alex pay than Sam in total?

    1. \(\$25\)
    2. \(\$10\)
    3. \(\$15\)
    4. \(\$45\)
    5. \(\$60\)

  9. Carrie is travelling at a constant speed of 85 km/h. If Carrie is halfway through a 510 km trip, how much longer will the trip take?

    1. \(5 \ \mbox{hours}\)
    2. \(425 \ \mbox{hours}\)
    3. \(12 \ \mbox{hours}\)
    4. \(1.5 \ \mbox{hours}\)
    5. \(3 \ \mbox{hours}\)

  10. Points \(P, Q\) and \(R\) are on a number line. \(Q\) is halfway between \(P\) and \(R\). If \(P\) is at \(-6\) and \(Q\) is at \(-1\), then \(R\) is at

    1. \(4\)
    2. \(-11\)
    3. \(3\)
    4. \(-7\)
    5. \(5\)

Part B: Each correct answer is worth 6.

  1. The diagram shown contains octagons and squares only.

    The octagons are arranged into four rows with five octagons in each row. Each square is surrounded by four octagons, one on each side of the square, and so there are three rows with four squares in each.

    The ratio of the number of octagons to the number of squares is

    1. \(1:1\)
    2. \(2:1\)
    3. \(25:12\)
    4. \(5:4\)
    5. \(5:3\)

  2. In the sum shown, \(P\) and \(Q\) each represent a digit.

    PQQ plus PPQ plus QQQ equals 876, written vertically.

    The value of \(P+Q\) is

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

  3. A larger cube has volume 64 cm\(^3\). A smaller cube has edges that are half the length of the edges of the larger cube. What is the volume of the smaller cube?

    1. \(24 \mbox{\,cm}^3\)
    2. \(48 \mbox{\,cm}^3\)
    3. \(8 \mbox{\,cm}^3\)
    4. \(16 \mbox{\,cm}^3\)
    5. \(27 \mbox{\,cm}^3\)

  4. Ahmed chooses two different items for a snack. His choices are an apple, an orange, a banana, and a granola bar. How many different pairs of snacks could he choose?

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

  5. Sophia did push-ups every day for 7 days. Each day after the first day, she did 5 more push-ups than the day before. In total she did 175 push-ups. How many push-ups did Sophia do on the last day?

    1. \(55\)
    2. \(35\)
    3. \(50\)
    4. \(45\)
    5. \(40\)

  6. Each of \(\square, \triangle\) and \(\blacklozenge\) represents a non-zero number. If \(\square=\triangle+\triangle+\triangle\) and \(\square=\blacklozenge+\blacklozenge\), then \(\square+\blacklozenge+\triangle\) equals

    1. \(\square+\triangle\)
    2. \(\blacklozenge+\triangle+\triangle+\triangle+\triangle\)
    3. \(\blacklozenge+\blacklozenge+\square\)
    4. \(\triangle+\triangle+\triangle+\blacklozenge+\blacklozenge\)
    5. \(\blacklozenge+\blacklozenge+\blacklozenge+\triangle+\triangle\)

  7. Triangle \(T\) is reflected once.

    A coordinate plane with x and y axes. Triangle T is a right angled triangle. Starting at the vertex of the 90 degree angle, the shorter side goes up, while the longer side goes right. The hypotneuse then completes the triangle. There are 5 other triangles, all identical in shape and size to triangle T, that are labelled A through E. The orientation of each of those triangles is described in the following list.

    Which of the following triangles cannot be this reflection of triangle \(T\)?

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

  8. The mean (average) of a set of six numbers is 10. When the number 25 is removed from the set, the mean of the remaining numbers is

    1. \(6\)
    2. \(7\)
    3. \(8\)
    4. \(9\)
    5. \(10\)

  9. Suzy’s 5 m long ribbon has shaded and unshaded sections of equal length, as shown. Points \(A,B,C,D,E\) are equally spaced along the ribbon.

    A rectangle is divided into five sections, arranged side to side. From left to right, the first, third, and fifth sections are shaded while the second and fourth sections are unshaded. Points A, B, C, and D are equally spaced along the bottom of the fourth section, with A and D the start and end respectively. Point E is to the right of D.

    If Suzy wants a ribbon that is \(\frac{11}{15}\) of the size of this ribbon, at which point could she make a single vertical cut?

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

  10. In the diagram, four different integers from 1 to 9 inclusive are placed in the four boxes in the top row.

    There are three rows of boxes with two pairs of boxes in the top row, two boxes in the middle row, and one box in the bottom row. In the top row, a multiplication sign is between the left two boxes and an addition sign is between the right two boxes. An arrow points from the left two boxes in the top row to the left box in the middle row. An arrow points from the right two boxes in the top row to the right box in the middle row. A division sign is between the two boxes in the middle row. An arrow points from the two boxes in the middle row to the box in the bottom row.

    The integers in the left two boxes are multiplied and the integers in the right two boxes are added and these results are then divided, as shown. The final result is placed in the bottom box. Which of the following integers cannot appear in the bottom box?

    1. \(16\)
    2. \(24\)
    3. \(7\)
    4. \(20\)
    5. \(9\)

Part C: Each correct answer is worth 8.

  1. A 10 by 10 grid is created using 100 points, as shown.

    The points are arranged in 10 rows and 10 columns. Point P is located in the eighth row from the top and the seventh column from the left.

    Point \(P\) is given. One of the other 99 points is randomly chosen to be \(Q\). What is the probability that the line segment \(PQ\) is vertical or horizontal?

    1. \(\frac{2}{11}\)
    2. \(\frac{1}{5}\)
    3. \(\frac{1}{10}\)
    4. \(\frac{4}{25}\)
    5. \(\frac{5}{33}\)

  2. The eight vertices of a cube are randomly labelled with the integers from 1 to 8 inclusive. Judith looks at the labels of the four vertices of one of the faces of the cube. She lists these four labels in increasing order. After doing this for all six faces, she gets the following six lists: \((1, 2, 5, 8), (3, 4, 6, 7),\) \((2, 4, 5, 7), (1, 3, 6, 8), (2, 3, 7, 8)\), and \((1, 4, 5, 6)\). The label of the vertex of the cube that is farthest away from the vertex labelled 2 is

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

  3. Angie has a jar that contains 2 red marbles, 2 blue marbles, and no other marbles. She randomly draws 2 marbles from the jar. If the marbles are the same colour, she discards one and puts the other back into the jar. If the marbles are different colours, she discards the red marble and puts the blue marble back into the jar. She repeats this process a total of three times. What is the probability that the remaining marble is red?

    1. \(\frac 12\)
    2. \(\frac 14\)
    3. \(\frac 23\)
    4. \(\frac 13\)
    5. \(0\)

  4. How many of the five numbers \(101, 148, 200, 512, 621\) cannot be expressed as the sum of two or more consecutive positive integers?

    1. \(0\)
    2. \(1\)
    3. \(2\)
    4. \(3\)
    5. \(4\)

  5. In the triangle shown, the first diagonal line, \(1,2,3,4,\dots\), begins at 1 and each number after the first is one larger than the previous number.

    The first six rows of the triangle are given. Row 1 contains 1 number, the number 1. Row 2 contains two numbers, 2 and 2. Row 3 contains 3 numbers, 3, 4, and 3. Row 4 contains 4 numbers, 4, 6, 6, and 4. Row 5 contains 5 numbers, 5, 8, 9, 8, and 5. Row 6 contains 6 numbers, 6, 10, 12, 12, 10, and 6. Groups of three dots show the patterns continue.

    The second diagonal line, \(2,4,6,8,\dots\) begins at 2 and each number after the first is two larger than the previous number. The \(n^{th}\) diagonal line begins at \(n\) and each number after the first is \(n\) larger than the previous number. In which horizontal row does the number 2016 first appear?

    1. \(90\)
    2. \(94\)
    3. \(88\)
    4. \(91\)
    5. \(89\)