2020 Euclid Contest

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

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

©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
   • worth 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.

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

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

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

2. 1. The 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. Eleanor 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. Suppose 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 \(n\).

3. 1. Consider 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 \(AD\).

      

      Hide/Reveal Description of Diagram for Question 3a

      Four additional points are placed on the sides of the rectangle as follow:

      • AD and HF are on the longer sides of the rectangle.
      • Place E at the centre of DF.
      • Place C on AD such that angle CED is 45 degrees.
      • Place G on FH such that angle GEF is 45 degrees.
      • Finally, place B on AD such that angle BGH is 45 degrees.
      • Note that the length of AD is such that both C and B lie between A and D.

      ---

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

   2. Ada 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 \(y\):

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

4. 1. The 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. A 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.)

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

6. 1. Rectangle \(ABCD\) 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.

      

      What is the value of \(r\)?

   2. Consider 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 \(AB=AE=7\sqrt{2}\).

      

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

7. 1. Suppose 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. Determine 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: \[\begin{aligned} \log_2 ( \sin x \cos y ) & = -\dfrac{3}{2} \\ \log_2 \left( \dfrac{\sin x}{\cos y} \right) & = \dfrac{1}{2}\end{aligned}\]

8. 1. Four 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. Three 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. 

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

      

      Hide/Reveal Description of Diagram for Question 9a

      The "Original L" looks like a two by two grid of squares with the top right square removed. The bend in the L is at the bottom left.

      The L "After 1 Round" is the original L shape divided into four smaller L shapes. Each smaller L is one-quarter the size of the original and two are in the same orientation as the original. The small L's are arranged as follows:

      • One small L, in the same orientation as original, makes up the bend in the bottom left.
      • Another small L, oriented so its bend is in the bottom right corner, is to the right of the first small L.
      • Another small L, oriented so its bend is in the top left corner, is above the first small L.
      • A final small L, in the same orientation as the original, completes the middle of the shape.

      ---

      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. Kerry 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 \(s_1,s_2,\ldots,s_{120}\).

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