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2020 Euclid Contest

Tuesday, April 7, 2020
(in North America and South America)

Wednesday, April 8, 2020
(outside of North American and South America)

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©2020 University of Waterloo

Instructions

Time: \(2\frac{1}{2}\) hours

Number of Questions: 10
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.

    1. LightbulbIf \(x=11\), what is the value of \(\dfrac{3x + 6}{x + 2}\) ?

    2. LightbulbWhat is the \(y\)-intercept of the line that passes through \(A(-1, 5)\) and \(B(1, 7)\)?

    3. Full solutionThe lines with equations \(y = 3x + 7\), \(y = x + 9\), and \(y = mx + 17\) intersect at a single point. Determine the value of \(m\).

    1. LightbulbThe three-digit positive integer \(m\) is odd and has three distinct digits. If the hundreds digit of \(m\) equals the product of the tens digit and ones (units) digit of \(m\), what is \(m\)?

    2. LightbulbEleanor has 100 marbles, each of which is black or gold. The ratio of the number of black marbles to the number of gold marbles is \(1:4\). How many gold marbles should she add to change this ratio to \(1:6\)?

    3. Full solutionSuppose that \(n\) is a positive integer and that the value of \(\dfrac{n^2 + n + 15}{n}\) is an integer. Determine all possible values of \(n\).

    1. LightbulbConsider the rectangle \(ADFH\). Donna has a laser at \(C\). She points the laser beam at the point \(E\). The beam reflects off of \(DF\) at \(E\) and then off of \(FH\) at \(G\), as shown, arriving at point \(B\) on \(AD\).

      Four additional points are placed on the sides of the rectangle. This is outlined in the following description:

      If \(DE=EF=1\mbox{ m}\), what is the length of \(BD\), in metres?

    2. LightbulbAda starts with \(x = 10\) and \(y=2\), and applies the following process:

      Step 1: Add \(x\) and \(y\). Let \(x\) equal the result. The value of \(y\) does not change.

      Step 2: Multiply \(x\) and \(y\). Let \(x\) equal the result. The value of \(y\) does not change.

      Step 3: Add \(y\) and \(1\). Let \(y\) equal the result. The value of \(x\) does not change.

      Ada keeps track of the values of \(x\) and \(y\):

      \(x\) \(y\)
      Before Step 1 10 2
      After Step 1 12 2
      After Step 2 24 2
      After Step 3 24 3

      Continuing now with \(x = 24\) and \(y = 3\), Ada applies the process two more times. What is the final value of \(x\)?

    3. Full solutionDetermine all integers \(k\), with \(k \neq 0\), for which the parabola with equation \(y = kx^2 + 6x + k\) has two distinct \(x\)-intercepts.

    1. LightbulbThe positive integers \(a\) and \(b\) have no common divisor larger than 1. If the difference between \(b\) and \(a\) is 15 and \(\dfrac{5}{9} < \dfrac{a}{b} < \dfrac{4}{7}\), what is the value of \(\dfrac{a}{b}\)?

    2. Full solutionA geometric sequence has first term 10 and common ratio \(\frac{1}{2}\). An arithmetic sequence has first term 10 and common difference \(d\). The ratio of the 6th term in the geometric sequence to the 4th term in the geometric sequence equals the ratio of the 6th term in the arithmetic sequence to the 4th term in the arithmetic sequence. Determine all possible values of \(d\).

      (An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant, called the common difference. For example, \(3, 5, 7, 9\) are the first four terms of an arithmetic sequence.

      A geometric sequence is a sequence in which each term after the first is obtained from the previous term by multiplying it by a non-zero constant, called the common ratio. For example, 3, 6, 12 is a geometric sequence with three terms.)

    1. LightbulbFor each positive real number \(x\), define \(f(x)\) to be the number of prime numbers \(p\) that satisfy \(x \leq p \leq x+10\). What is the value of \(f(f(20))\)?

    2. Full solutionDetermine all triples \((x,y,z)\) of real numbers that satisfy the following system of equations: \[\begin{aligned} (x-1)(y-2) & = 0 \\ (x-3)(z+2) & = 0 \\ x + yz & = 9\end{aligned}\]

    1. LightbulbRectangle \(ABCD\) has \(AB=4\) and \(BC=6\). The semi-circles with diameters \(AE\) and \(FC\) each have radius \(r\), have centres \(S\) and \(T\), and touch at a single point \(P\), as shown.

      Two equally sized semicircles are drawn in the rectangle's interior. One diameter extends from A to point E on AD. The second semicircle's diameter extends from C to point F on BC.

      What is the value of \(r\)?

    2. Full solutionConsider an irregular convex pentagon \(ABCDE\). In the diagram, \(\triangle ABE\) is right-angled at \(A\), \(\triangle BCD\) is right-angled at \(C\), \(\angle ABC = 135^\circ\), and \(AB=AE=7\sqrt{2}\).

      ABCDE contains three triangles, triangle ABE, triangle BED, and triangle BCD.

      If \(DC=4x\), \(DB=8x\) and \(DE = 8x-6\) for some real number \(x\), determine all possible values of \(x\).

    1. LightbulbSuppose that the function \(g\) satisfies \(g(x) = 2x - 4\) for all real numbers \(x\) and that \(g^{-1}\) is the inverse function of \(g\). Suppose that the function \(f\) satisfies \(g(f(g^{-1}(x))) = 2x^2 + 16x + 26\) for all real numbers \(x\). What is the value of \(f(\pi)\)?

    2. Full solutionDetermine all pairs of angles \((x,y)\) with \(0^\circ \leq x < 180^\circ\) and \(0^\circ \leq y < 180^\circ\) that satisfy the following system of equations: \[\begin{aligned} \log_2 ( \sin x \cos y ) & = -\dfrac{3}{2} \\ \log_2 \left( \dfrac{\sin x}{\cos y} \right) & = \dfrac{1}{2}\end{aligned}\]

    1. Full solutionFour tennis players Alain, Bianca, Chen, and Dave take part in a tournament in which a total of three matches are played. First, two players are chosen randomly to play each other. The other two players also play each other. The winners of the two matches then play to decide the tournament champion. Alain, Bianca and Chen are equally matched (that is, when a match is played between any two of them, the probability that each player wins is \(\frac{1}{2}\)). When Dave plays each of Alain, Bianca and Chen, the probability that Dave wins is \(p\), for some real number \(p\). Determine the probability that Bianca wins the tournament, expressing your answer in the form \(\dfrac{ap^2+bp+c}{d}\) where \(a\), \(b\), \(c\), and \(d\) are integers.

    2. Full solutionThree microphones \(A\), \(B\) and \(C\) are placed on a line such that \(A\) is 1 km west of \(B\) and \(C\) is 2 km east of \(B\). A large explosion occurs at a point \(P\) not on this line. Each of the three microphones receives the sound. The sound travels at \(\frac{1}{3}\) km/s. Microphone \(B\) receives the sound first, microphone \(A\) receives the sound \(\frac{1}{2}\) s later, and microphone \(C\) receives it 1 s after microphone \(A\). Determine the distance from microphone \(B\) to the explosion at \(P\).

  9. Full solution

    1. An L shape is made by adjoining three congruent squares. The L is subdivided into four smaller L shapes, as shown.

      A description of the diagram follows.

      Each of the resulting L’s is subdivided in this same way. After the third round of subdivisions, how many L’s of the smallest size are there?

    2. After the third round of subdivisions, how many L’s of the smallest size are in the same orientation as the original L?

    3. Starting with the original L shape, 2020 rounds of subdivisions are made. Determine the number of L’s of the smallest size that are in the same orientation as the original L.

  10. Full solutionKerry has a list of \(n\) integers \(a_1,a_2,\ldots,a_n\) satisfying \(a_1 \leq a_2 \leq \ldots \leq a_n\). Kerry calculates the pairwise sums of all \(m = \frac{1}{2}n(n-1)\) possible pairs of integers in her list and orders these pairwise sums as \(s_1 \leq s_2 \leq \ldots \leq s_m\). For example, if Kerry’s list consists of the three integers \(1,2,4\), the three pairwise sums are \(3, 5, 6\).

    1. Suppose that \(n=4\) and that the 6 pairwise sums are \(s_1=8\), \(s_2=104\), \(s_3=106\), \(s_4=110\), \(s_5=112\), and \(s_6=208\). Determine two possible lists \(a_1,a_2,a_3,a_4\) that Kerry could have.

    2. Suppose that \(n=5\) and that the 10 pairwise sums are \(s_1,s_2,\ldots,s_{10}\). Prove that there is only one possibility for Kerry’s list \(a_1,a_2,a_3,a_4,a_5\).

    3. Suppose that \(n=16\). Prove that there are two different lists \(a_1,a_2,\ldots,a_{16}\) and \(b_1,b_2,\ldots,b_{16}\) that produce the same list of sums \(s_1,s_2,\ldots,s_{120}\).

Further Information

For students...

Thank you for writing the Euclid Contest!

If you are graduating from secondary school, good luck in your future endeavours! If you will be returning to secondary school next year, encourage your teacher to register you for the Canadian Senior Mathematics Contest, which will be written in November.

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For teachers...

Visit our website cemc.uwaterloo.ca to