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2022 Galois Contest
(Grade 10)

Tuesday, April 12, 2022
(in North America and South America)

Wednesday, April 13, 2022
(outside of North American and South America)

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©2022 University of Waterloo

Instructions

Time: 75 minutes

Number of Questions: 4
Each question is worth 10 marks.

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.

Parts of each question can be of two types:

  1. SHORT ANSWER parts indicated by Lightbulb
  2. FULL SOLUTION parts indicated by Full Solution

WRITE ALL ANSWERS IN THE ANSWER BOOKLET PROVIDED.

Do not discuss the problems or solutions from this contest online for the next 48 hours.
The name, grade, school and location, and score range of some top-scoring students will be published on our website, cemc.uwaterloo.ca. In addition, the name, grade, school and location, and score of some top-scoring students may be shared with other mathematical organizations for other recognition opportunities.
NOTE:
  1. Please read the instructions for the contest.
  2. Write all answers in the answer booklet provided.
  3. For questions marked Lightbulb, place your answer in the appropriate box in the answer booklet and show your work.
  4. For questions marked Full Solution, provide a well-organized solution in the answer booklet. Use mathematical statements and words to explain all of the steps of your solution. Work out some details in rough on a separate piece of paper before writing your finished solution.
  5. Diagrams are not drawn to scale. They are intended as aids only.
  6. While calculators may be used for numerical calculations, other mathematical steps must be shown and justified in your written solutions, and specific marks may be allocated for these steps. For example, while your calculator might be able to find the \(x\)-intercepts of the graph of an equation like \(y=x^{3} -x\), you should show the algebraic steps that you used to find these numbers, rather than simply writing these numbers down.

Questions

  1. Alice and Bello contributed to the cost of starting a new business. The ratio of Alice’s contribution to Bello’s contribution was \(3:8\).

    1. Lightbulb If the cost of starting the new business was $9240, what was Bello’s contribution to this starting cost?

    2. Lightbulb Alice and Bello divided up all profits in the first year of the business in the same ratio, \(3:8\). Alice’s share of the first year’s total profit was \(\$1881\). What was the total profit of the business for the first year?

    3. Full Solution In the second year, the business was changed so the share of that year’s profits for Alice and Bello was in the ratio of \(3:(8+x)\). If the profit for the second year was \(\$6400\) and Bello’s share of that profit was \(\$5440\), determine the value of \(x\).

  2. In the diagram shown, line \(L_1\) has equation \(y=\frac32x+k\), where \(k>0\), and \(L_1\) intersects the \(y\)-axis at \(P\). A second line, \(L_2\), is drawn through \(P\) perpendicular to \(L_1\), and intersects the \(x\)-axis at \(Q\). A third line, \(L_3\), is drawn through \(Q\) parallel to \(L_1\), and intersects the \(y\)-axis at \(R\).

    Three lines (L subscript 1, L subscript 2, and L subscript 3) and three points (P, Q, and R) plotted in the Cartesian plane. Line L subscript 1 goes up to the right and L subscript 2 goes down to the right. P is on the positive y-axis, Q is on the positive x-axis, and R is on the negative y-axis.

    1. Lightbulb What is the slope of \(L_2\)?

    2. Lightbulb Written in terms of \(k\), what is the \(x\)-coordinate of point \(Q\)?

    3. Full Solution If the area of \(\triangle PQR\) is 351, determine the value of \(k\).

  3. The prime factorization of 324 is \(2\times2\times3\times3\times3\times3\) or \(2^2\times 3^4\). Notice that 324 is a perfect square because it can be written in the form \((2\times 3^2)\times(2 \times 3^2)\).
    The prime factorization of 63 is \(3^2 \times 7\). Notice that 63 is not a perfect square, but \(63\times 7\) is a perfect square, because \(63\times 7 =3^2 \times 7^2 = (3 \times 7)\times(3 \times 7)\).

    1. Lightbulb The product \(84\times k\) is a perfect square. If \(k\) is a positive integer, what is the smallest possible value of \(k\)?

    2. Lightbulb The product \(572\times \ell\) is a perfect square. If \(\ell\) is a positive integer less than 6000, what is the greatest possible value of \(\ell\)?

    3. Full Solution Show that if \(m\) is a positive integer less than 200, then \(525\,000\times m\) cannot be a perfect square.

    4. Full Solution The list \(10, 10^3, 10^5, \ldots, 10^{99}\) contains the fifty powers of 10 with odd integer exponents from \(10^1\) to \(10^{99}\), inclusive. Show that the sum of every choice of three different powers of 10 from this list is not a perfect square.

  4. A Bauman string is a string of letters that satisfies the following two conditions.

    For example, \(AECD\) and \(BDCEC\) are Bauman strings of length 4 and length 5, respectively, and \(ABBC\) and \(DAEEE\) are not Bauman strings.

    1. Lightbulb How many Bauman strings of length 5 are there in which the first letter and the last letter are both \(A\)?

    2. Full Solution Determine the number of Bauman strings of length 6 that contain more than one \(B\).

    3. Full Solution Determine the number of Bauman strings of length 10 in which the first letter is \(C\) and the last letter is \(D\).

Further Information

For students...

Thank you for writing the Galois Contest!

Encourage your teacher to register you for the Canadian Intermediate Mathematics Contest or the Canadian Senior Mathematics Contest, which will be written in November.

Visit our website cemc.uwaterloo.ca to find

For teachers...

Visit our website cemc.uwaterloo.ca to