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2019 Galois Contest
(Grade 10)

Wednesday, April 10, 2019
(in North America and South America)

Thursday, April 11, 2019
(outside of North American and South America)

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©2019 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. The Galois Restaurant is in a region that adds 10% sales tax onto the price of food and drinks purchased at a restaurant. The prices listed on their menu do not include the sales tax.

    1. LightbulbFrom the menu, Becky orders a plate of lasagna listed for $7.50, a side salad listed for $5.00, and a lemonade listed for $3.00. After tax is included, how much is Becky’s total bill?

    2. LightbulbA burrito is listed on the menu for $6.00. After tax is included, what is the greatest number of burritos that Jackson can buy if he has $50.00?

    3. Full solutionOn the Galois Restaurant menu, hotdogs are listed at the regular price of $5.00. The restaurant has the following promotional deals:

      • On Mondays, if you buy a hotdog at the regular menu price of $5.00, then the price for a second hotdog is $4.50.

      • On Tuesdays, you pay half the tax on all hotdogs.

      Chase bought two hotdogs on Monday and then two hotdogs on Tuesday. After tax is included, determine on which day Chase spent less money.

  2. The hypotenuse of right-angled \(\triangle AOB\) lies on the line with equation \(y = -2x + 12\), as shown in Figure 1. The legs of \(\triangle AOB\) lie on the axes.

    Point A lies on the positive y-axis, point B lies on the positive x-axis, and point O is at the origin (0,0).

    1. LightbulbWhat is the area of \(\triangle AOB\)?

    2. Full solutionA second line passes through \(O\) and is perpendicular to the first line, as shown in Figure 2.

      OC meets AB at a right angle.

      The two lines intersect at \(C\). Determine the coordinates of \(C\).

    3. Full solutionThe second line passes through the point \(D\) in the first quadrant, as shown in Figure 3.

      OC extends to point D, point F lies above point A on the y-axis, and point E lies to the right of point B on the x-axis.

      Points \(E\) and \(F\) are positioned on the axes so that \(DEOF\) is a rectangle. If the area of \(DEOF\) is 1352, determine the coordinates of \(D\).

  3. If \(n\) is a positive integer, the notation \(n!\) (read “\(n\) factorial”) is used to represent the product of the integers from 1 to \(n\). That is, \(n! = n(n -1)(n - 2) \cdots (3)(2)(1)\). For example, \(5! = 5(4)(3)(2)(1)\) or \(5! = 120\).

    1. LightbulbWhat is the largest positive integer \(m\) for which \(2^m\) is a divisor of \(9!\)?

    2. LightbulbWhat is the smallest value of \(n\) for which \(n!\) is divisible by \(7^2\)?

    3. Full solutionExplain why there is no positive integer \(n\) for which \(n!\) is divisible by \(7^{7}\) but is not divisible by \(7^{8}\).

    4. Full solutionShow that there is exactly one positive integer \(n\) for which \[n! = 2^a\cdot 3^b \cdot 5^c \cdot 7^d \cdot 11^2 \cdot 13^2 \cdot 17 \cdot 19 \cdot 23, \mbox{ and}\] \[a+b+c+d=45\] for some positive integers \(a,b,c,d\).

  4. A positive integer is digit-balanced if each digit \(d\), with \(0\leq d\leq 9\), appears at most \(d\) times in the integer. For example, 13224 is digit-balanced, but 21232 is not.

    1. Full solutionExplain why a digit-balanced integer is not divisible by 10.

    2. LightbulbHow many 4-digit integers have all non-zero digits and are not digit-balanced?

    3. Full solutionDetermine all positive integers \(k\) for which there exist digit-balanced positive integers \(m\) and \(n\), where \(m+n=10^k\) and \(m\) and \(n\) each have \(k\) digits.

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