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)

©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
   • worth 2 or 3 marks each
   • full marks are given for a correct answer which is placed in the box
   • part marks are awarded if relevant work is shown in the space provided
2. FULL SOLUTION parts indicated by
   • worth the remainder of the 10 marks for the question
   • must be written in the appropriate location in the answer booklet
   • marks awarded for completeness, clarity, and style of presentation
   • a correct solution poorly presented will not earn full marks

WRITE ALL ANSWERS IN THE ANSWER BOOKLET PROVIDED.

• Extra paper for your finished solutions supplied by your supervising teacher must be inserted into your answer booklet. Write your name, school name, and question number on any inserted pages.
• Express answers as simplified exact numbers except where otherwise indicated. For example, $\pi +1$ and $1-\sqrt{2}$ are simplified exact numbers.

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. From 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. A 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. On 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 $\mathrm{△}AOB$ lies on the line with equation $y=-2x+12$, as shown in Figure 1. The legs of $\mathrm{△}AOB$ lie on the axes.

   

   1. What is the area of $\mathrm{△}AOB$?

   2. A second line passes through $O$ and is perpendicular to the first line, as shown in Figure 2.

      

      The two lines intersect at $C$. Determine the coordinates of $C$.

   3. The second line passes through the point $D$ in the first quadrant, as shown in Figure 3.

      

      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. What is the largest positive integer $m$ for which $2^m$ is a divisor of $9!$?

   2. What is the smallest value of $n$ for which $n!$ is divisible by $7^2$?

   3. Explain why there is no positive integer $n$ for which $n!$ is divisible by $7^7$ but is not divisible by $7^8$.

   4. Show that there is exactly one positive integer $n$ for which $$n!=2^a\cdot 3^b\cdot 5^c\cdot 7^d\cdot {11}^2\cdot {13}^2\cdot 17\cdot 19\cdot 23,\text{ and}$$ $$a+b+c+d=45$$ for some positive integers $a,b,c,d$.

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

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

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

   3. Determine 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

• Free copies of past contests
• Math Circles videos and handouts that will help you learn more mathematics and prepare for future contests
• Information about careers in and applications of mathematics and computer science

For teachers...

Visit our website cemc.uwaterloo.ca to

• Obtain information about future contests
• Look at our free online courseware for high school students
• Learn about our face-to-face workshops and our web resources
• Subscribe to our free Problem of the Week
• Investigate our online Master of Mathematics for Teachers
• Find your school's contest results