2018 Galois Contest
(Grade 10)

Thursday, April 12, 2018
(in North America and South America)

Friday, April 13, 2018
(outside of North American and South America)

©2018 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. 1. Given that $x\ne 0$, simplify the expression ${\displaystyle \frac{12x^2}{3x}}$.

   2. What is the value of the expression ${\displaystyle \frac{12x^2}{3x}}$ when $x=5$?

   3. Given that $n=2m$ and $m\ne 0$, what is the value of the expression ${\displaystyle \frac{8mn}{3m^2}}$?

   4. If $q=6$, determine all positive integers $p$ for which $3\le {\displaystyle \frac{8p^{2}q}{5p{q}^2}}\le 4$.

2. Here are two facts about circles:

   • If points $A$, $B$, $C$ lie on a circle so that $\mathrm{\angle }ABC={90}^{\circ }$, then $AC$ is a diameter of the circle. This means that in Figure 1, $AC$ is a diameter of the circle.

   • If points $D$, $E$, $F$ lie on a circle so that $EF$ is a diameter, then $\mathrm{\angle }EDF={90}^{\circ }$. This means that in Figure 2, $\mathrm{\angle }EDF={90}^{\circ }$.

   

   1. In Figure 1 above, $AB=8$ and $BC=15$. What is the length of diameter $AC$?

   2. In Figure 2 above, $DE=24$ and the radius of the circle is 13. What is the length of $DF$?

   3. In Figure 3, points $P$, $Q$, $R$, and $S$ are on a circle with centre $O$. Also, $SQ$ is a diameter of the circle and $O$ is joined to $R$. If $SP=PQ$ and $\mathrm{\angle }RQP={80}^{\circ }$, determine the measure of $\mathrm{\angle }ROQ$ and the measure of $\mathrm{\angle }RSQ$.

      

3. Cylinder A has radius 12 and height 25. Cylinder B has radius 9 and height $h$. Cylinder A is filled with water to a depth of 19. Cylinder B is empty. Cylinder B is lowered to the bottom of Cylinder A, as shown. Depending on the value of $h$,

   1. some water may spill out of Cylinder A onto the ground (Figure 1), or

   2. some water may pour into Cylinder B (Figure 2), or

   3. (i) then (ii).

   

   

   The walls and bases of the two cylinders are thin enough that their width can be ignored.

   1. 

      Suppose that $h=30$. What is the volume of water that spills out of Cylinder A onto the ground?

   2. Suppose that $h=20$. Determine the volume of water that spills out of Cylinder A onto the ground and the depth of water in Cylinder B when it is on the bottom of Cylinder A.

   3. Determine the range of values of $h$ so that when Cylinder B is on the bottom of Cylinder A, there is some water in Cylinder B but it is not full.

4. For each positive integer $k$, we define $C(k)$ to be the number of ways in which $k$ can be written as the sum of one or more consecutive positive integers. For example, $C(21)=4$ because $21$ can be written as $$21,10+11,6+7+8,\text{and}1+2+3+4+5+6,$$ and there are no other lists of one or more consecutive positive integers whose sum is 21.

   1. What is the value of $C(45)$?

   2. The positive integer $m$ equals the sum of the positive integers from 4 to $n$, inclusive. Determine the values of $a$ and $b$, with $a<b$, for which $m=\frac{1}{2}(n+a)(n+b)$ for each positive integer $n\ge 4$.

   3. Determine the value of $C(2\times 3^4\times 5^6)$.

   4. Determine the smallest positive integer $k$ for which $C(k)=215$.

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