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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)

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©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 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. LightbulbGiven that \(x\neq 0\), simplify the expression \(\dfrac{12x^2}{3x}\).

    2. LightbulbWhat is the value of the expression \(\dfrac{12x^2}{3x}\) when \(x=5\)?

    3. LightbulbGiven that \(n=2m\) and \(m\neq 0\), what is the value of the expression \(\dfrac{8mn}{3m^2}\)?

    4. Full solutionIf \(q=6\), determine all positive integers \(p\) for which \(3\leq\dfrac{8p^2q}{5pq^2}\leq 4\).

  2. Here are two facts about circles:

    In Figure 1, points A,B, and C lie on a circle, and angle ABC is marked as right-angled. In Figure 2, points D,E, and F lie on a circle so that EF passes through the centre of the circle.

    1. LightbulbIn Figure 1 above, \(AB=8\) and \(BC=15\). What is the length of diameter \(AC\)?

    2. LightbulbIn Figure 2 above, \(DE=24\) and the radius of the circle is 13. What is the length of \(DF\)?

    3. Full solutionIn 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 \(\angle RQP=80^{\circ}\), determine the measure of \(\angle ROQ\) and the measure of \(\angle RSQ\).

  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).

    Figure 1. A cross sectional view of a smaller cylinder placed inside a larger cylinder, with the top of the smaller cylinder above the top of the larger cylinder. In the larger cylinder, the space not taken up by the smaller cylinder is completely filled with water. The smaller cylinder is empty. Arrows inside the smaller cylinder point downward. Arrows point from the inside of the larger cylinder to the outside.

    Figure 2. A cross sectional view of a smaller cylinder placed inside a larger cylinder, with the top of the smaller cylinder below the top of the larger cylinder. The larger cylinder is partially filled with water, up to the top of the smaller cylinder, and the smaller cylinder is empty. Arrows inside the smaller cylinder point downward. Arrows point from the inside of the larger cylinder to the inside of the smaller cylinder.

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

    1. Lightbulb

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

    2. Full solutionSuppose 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. Full solutionDetermine 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 \[21, \quad 10 + 11,\quad 6+7+8, \quad \mbox{and} \quad 1+2+3+4+5+6,\] and there are no other lists of one or more consecutive positive integers whose sum is 21.

    1. LightbulbWhat is the value of \(C(45)\)?

    2. Full solutionThe 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 \(n \geq 4\).

    3. Full solutionDetermine the value of \(C(2\times 3^4\times 5^6)\).

    4. Full solutionDetermine 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

For teachers...

Visit our website cemc.uwaterloo.ca to