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2021 Euclid Contest

Wednesday, April 7, 2021
(in North America and South America)

Thursday, April 8, 2021
(outside of North America and South America)

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©2021 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. LightbulbWhat is the value of \(a\) for which \((a-1)+(2a-3) = 14\)?

    2. LightbulbWhat are the two values of \(c\) for which \((c^2-c)+(2c-3) = 9\)?

    3. Full solutionDetermine all values of \(x\) for which \(\dfrac{1}{x^2}+\dfrac{3}{2x^2} = 10\).

    1. LightbulbWhat is the sum of the digits of the integer equal to \((10^{3} + 1)^2\)?

    2. LightbulbA bakery sells small and large cookies. Before a price increase, the price of each small cookie is $1.50 and the price of each large cookie is $2.00. The price of each small cookie is increased by 10% and the price of each large cookie is increased by 5%. What is the percentage increase in the total cost of a purchase of 2 small cookies and 1 large cookie?

    3. Full solutionQing is twice as old as Rayna. Qing is 4 years younger than Paolo. The average age of Paolo, Qing and Rayna is 13. Determine their ages.

    1. LightbulbIn the diagram, \(PQRS\) is a quadrilateral.

      P is on the positive y-axis and has coordinates (0, 12). Q is on the positive x-axis and has coordinates (5, 0). R is on the negative y-axis and has coordinates (0, negative 12). S is on the negative x-axis and has coordinates (negative 5, 0).

      What is its perimeter?

    2. LightbulbIn the diagram, \(A\) has coordinates \((0,8)\).

      A triangle with vertices A, B, and C. Vertex A lies on the positive y-axis and B and C lie in the first quadrant.

      Also, the midpoint of \(AB\) is \(M(3,9)\) and the midpoint of \(BC\) is \(N(7,6)\). What is the slope of \(AC\)?

    3. Full solutionThe parabola with equation \(y = -2x^2 + 4x + c\) has vertex \(V(1,18)\). The parabola intersects the \(y\)-axis at \(D\) and the \(x\)-axis at \(E\) and \(F\). Determine the area of \(\triangle DEF\).

    1. LightbulbIf \(3(8^x) + 5(8^x) = 2^{61}\), what is the value of the real number \(x\)?

    2. Full solutionFor some real numbers \(m\) and \(n\), the list \(3n^2\), \(m^2\), \(2(n+1)^2\) consists of three consecutive integers written in increasing order. Determine all possible values of \(m\).

    1. LightbulbChinara starts with the point \((3,5)\), and applies the following three-step process, which we call \(\mathcal{P}\):

      Step 1: Reflect the point in the \(x\)-axis.

      Step 2: Translate the resulting point 2 units upwards.

      Step 3: Reflect the resulting point in the \(y\)-axis.

      As she does this, the point \((3,5)\) moves to \((3,-5)\), then to \((3,-3)\), and then to \((-3,-3)\).

      Chinara then starts with a different point \(S_0\). She applies the three-step process \(\mathcal{P}\) to the point \(S_0\) and obtains the point \(S_1\). She then applies \(\mathcal{P}\) to \(S_1\) to obtain the point \(S_2\). She applies \(\mathcal{P}\) four more times, each time using the previous output of \(\mathcal{P}\) to be the new input, and eventually obtains the point \(S_6(-7,-1)\). What are the coordinates of the point \(S_0\)?

    2. Full solutionIn the diagram, \(ABDE\) is a rectangle, \(\triangle BCD\) is equilateral, and \(AD\) is parallel to \(BC\).

      Triangle BCD is beside rectangle ABDE, joined at common side BD.

      Also, \(AE=2x\) for some real number \(x\).

      1. Determine the length of \(AB\) in terms of \(x\).

      2. Determine positive integers \(r\) and \(s\) for which \(\dfrac{AC}{AD} = \sqrt{\dfrac{r}{s}}\).

    1. LightbulbSuppose that \(n>5\) and that the numbers \(t_1,t_2,t_3,\ldots,t_{n-2},t_{n-1},t_n\) form an arithmetic sequence with \(n\) terms. If \(t_3 = 5\), \(t_{n-2} = 95\), and the sum of all \(n\) terms is 1000, what is the value of \(n\)?

      (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.)

    2. Full solutionSuppose that \(a\) and \(r\) are real numbers. A geometric sequence with first term \(a\) and common ratio \(r\) has 4 terms. The sum of this geometric sequence is \(6 + 6\sqrt{2}\). A second geometric sequence has the same first term \(a\) and the same common ratio \(r\), but has 8 terms. The sum of this second geometric sequence is \(30 + 30\sqrt{2}\). Determine all possible values for \(a\).

      (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, \(3\), \(-6\), \(12\), \(-24\) are the first four terms of a geometric sequence.)

    1. LightbulbA bag contains 3 green balls, 4 red balls, and no other balls. Victor removes balls randomly from the bag, one at a time, and places them on a table. Each ball in the bag is equally likely to be chosen each time that he removes a ball. He stops removing balls when there are two balls of the same colour on the table. What is the probability that, when he stops, there is at least 1 red ball and at least 1 green ball on the table?

    2. Full solutionSuppose that \(f(a) = 2a^2 - 3a + 1\) for all real numbers \(a\) and \(g(b) = \log_{\frac{1}{2}} b\) for all \(b >0\). Determine all \(\theta\) with \(0 \leq \theta \leq 2\pi\) for which \(f(g(\sin \theta)) = 0\).

    1. Full solutionFive distinct integers are to be chosen from the set \(\{1,2,3,4,5,6,7,8\}\) and placed in some order in the top row of boxes in the diagram.

      Fifteen boxes are arranged into five rows forming a triangular shape. Reading from top to bottom, there are five boxes in the first row, four boxes in the second row, three in the third row, two in the fourth row and one in the fifth row (the bottom box). Each box in the first three rows is connected to the two closest boxes in the row below it. Each of the two boxes in the fourth row is connected to the bottom box.

      Each box that is not in the top row then contains the product of the integers in the two boxes connected to it in the row directly above. Determine the number of ways in which the integers can be chosen and placed in the top row so that the integer in the bottom box is \(9\,953\,280\,000\).

    2. Full solutionProve that the integer \(\dfrac{(1!)(2!)(3!) \cdots (398!)(399!)(400!)}{200!}\) is a perfect square. (In this fraction, the numerator is the product of the factorials of the integers from 1 to 400, inclusive.)

  9. Full solution

    1. Suppose that \(a = 5\) and \(b = 4\). Determine all pairs of integers \((K,L)\) for which \(K^2 + 3L^2 = a^2 + b^2 - ab\).

    2. Prove that, for all integers \(K\) and \(L\), there is at least one pair of integers \((a,b)\) for which \(K^2 + 3L^2 = a^2 + b^2 - ab\).

    3. Prove that, for all integers \(a\) and \(b\), there is at least one pair of integers \((K,L)\) for which \(K^2 + 3L^2 = a^2 + b^2 - ab\).

  10. Full solution

    1. In the diagram, eleven circles of four different sizes are drawn.

      The circle labelled Z is the largest circle. The circle labelled Y is placed in the centre. The other 10 circles are placed side by side around circle Y. The labels of the circles, in order, are Z, followed by X, W, X, W, X, W, X, W, X, with the last circle labelled X touching the circle named Z.

      Each circle labelled \(W\) has radius 1, each circle labelled \(X\) has radius 2, the circle labelled \(Y\) has radius 4, and the circle labelled \(Z\) has radius \(r\). Each of the circles labelled \(W\) or \(X\) is tangent to three other circles. The circle labelled \(Y\) is tangent to all ten of the other circles. The circle labelled \(Z\) is tangent to three other circles. Determine positive integers \(s\) and \(t\) for which \(r = \dfrac{s}{t}\).

    2. Suppose that \(c\) is a positive integer. Define \(f(c)\) to be the number of pairs \((a,b)\) of positive integers with \(c<a<b\) for which two circles of radius \(a\), two circles of radius \(b\), and one circle of radius \(c\) can be drawn so that

      • each circle of radius \(a\) is tangent to both circles of radius \(b\) and to the circle of radius \(c\), and

      • each circle of radius \(b\) is tangent to both circles of radius \(a\) and to the circle of radius \(c\),

      as shown.

      The four circles of radius a or b are placed around the smallest circle of radius c in the order a, b, a, b.

      Determine all positive integers \(c\) for which \(f(c)\) is even.

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