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2022 Euclid Contest

Tuesday, April 5, 2022
(in North America and South America)

Wednesday, April 6, 2022
(outside of North American and South America)

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©2022 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 bepublished 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 organizationsfor 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 mustbe shown and justified in your written solutions, and specific marks may be allocated forthese 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. LightbulbWhat is the value of \(\dfrac{3^2-2^3}{2^{3}-3^{2}}\) ?

    2. LightbulbWhat is the value of \(\sqrt{\sqrt{81}+\sqrt{9}-\sqrt{64}}\) ?

    3. Full SolutionDetermine all real numbers \(x\) for which \(\dfrac{1}{\sqrt{x^2 + 7}} = \dfrac{1}{4}\).

    1. LightbulbFind the three ordered pairs of integers \((a,b)\) with \(1<a<b\) and \(ab=2022\).

    2. LightbulbSuppose that \(c\) and \(d\) are integers with \(c>0\) and \(d>0\) and \(\dfrac{2c+1}{2d+1}=\dfrac{1}{17}\). What is the smallest possible value of \(d\)?

    3. Full SolutionSuppose that \(p\), \(r\) and \(t\) are real numbers for which \((px+r)(x+5)=x^2+3x+t\) is true for all real numbers \(x\). Determine the value of \(t\).

    1. LightbulbA large water jug is \(\frac{1}{4}\) full of water. After 24 litres of water are added, the jug is \(\frac{5}{8}\) full. What is the volume of the jug, in litres?

    2. LightbulbStephanie starts with a large number of soccer balls. She gives \(\frac{2}{5}\) of them to Alphonso and \(\frac{6}{11}\) of them to Christine. The number of balls that she is left with is a multiple of 9. What is the smallest number of soccer balls with which Stephanie could have started?

    3. Full SolutionEach student in a math club is in either the Junior section or the Senior section.
      No student is in both sections.
      Of the Junior students, 60% are left-handed and 40% are right-handed.
      Of the Senior students, 10% are left-handed and 90% are right-handed.
      No student in the math club is both left-handed and right-handed.
      The total number of left-handed students is equal to the total number of right-handed students in the math club.
      Determine the percentage of math club members that are in the Junior section.

    1. LightbulbHexagon \(ABCDEF\) has vertices \(A(0,0)\), \(B(4,0)\), \(C(7,2)\), \(D(7,5)\), \(E(3,5)\), \(F(0,3)\). What is the area of hexagon \(ABCDEF\)?

    2. Full SolutionIn the diagram, \(\triangle PQS\) is right-angled at \(P\) and \(\triangle QRS\) is right-angled at \(Q\). Also, \(PQ=x\), \(QR=8\), \(RS=x+8\), and \(SP = x+3\) for some real number \(x\). Determine all possible values of the perimeter of quadrilateral \(PQRS\).

    1. LightbulbA list \(a_1,a_2,a_3,a_4\) of rational numbers is defined so that if one term is equal to \(r\), then the next term is equal to \(1 + \dfrac{1}{1+r}\). For example, if \(a_3=\dfrac{41}{29}\), then \(a_4 = 1 + \dfrac{1}{1 + (41/29)} = \dfrac{99}{70}\). If \(a_3=\dfrac{41}{29}\), what is the value of \(a_1\)?

    2. Full SolutionA hollow cylindrical tube has a radius of 10 mm and a height of 100 mm. The tube sits flat on one of its circular faces on a horizontal table. The tube is filled with water to a depth of \(h\) mm. A solid cylindrical rod has a radius of 2.5 mm and a height of 150 mm. The rod is inserted into the tube so that one of its circular faces sits flat on the bottom of the tube. The height of the water in the tube is now 64 mm. Determine the value of \(h\).

    1. LightbulbA function \(f\) has the property that \(\displaystyle{f \left( \frac{2x+1}{x} \right) = x+6}\) for all real values of \(x \neq 0\). What is the value of \(f(4)\)?

    2. Full SolutionDetermine all real numbers \(a\), \(b\) and \(c\) for which the graph of the function \(y=\log_a(x+b)+c\) passes through the points \(P(3,5)\), \(Q(5,4)\) and \(R(11,3)\).

    1. LightbulbA computer is programmed to choose an integer between 1 and 99, inclusive, so that the probability that it selects the integer \(x\) is equal to \(\log_{100}\left(1+\dfrac{1}{x}\right)\). Suppose that the probability that \(81 \leq x \leq 99\) is equal to 2 times the probability that \(x = n\) for some integer \(n\). What is the value of \(n\)?

    2. Full SolutionIn the diagram, \(\triangle ABD\) has \(C\) on \(BD\). Also, \(BC=2\), \(CD=1\), \(\dfrac{AC}{AD} = \dfrac{3}{4}\), and \(\cos(\angle ACD) = -\dfrac{3}{5}\). Determine the length of \(AB\).

    1. Full SolutionSuppose that \(a>\frac{1}{2}\) and that the parabola with equation \(y=ax^2+2\) has vertex \(V\). The parabola intersects the line with equation \(y=-x+4a\) at points \(B\) and \(C\), as shown.

      The parabola opens upward and lies entirely above the x-axis. V is on the positive y axis, B is in the second quadrant, and C is in the first quadrant. The line goes down to the right and passes through the positive x-axis to the right of point C.

      If the area of \(\triangle VBC\) is \(\frac{72}{5}\), determine the value of \(a\).

    2. Full SolutionConsider the following statement:

      There is a triangle that is not equilateral whose side lengths form a geometric sequence, and the measures of whose angles form an arithmetic sequence.

      Show that this statement is true by finding such a triangle or prove that it is false by demonstrating that there cannot be such a triangle.

  9. Full SolutionSuppose that \(m\) and \(n\) are positive integers with \(m \geq 2\). The \((m,n)\)-sawtooth sequence is a sequence of consecutive integers that starts with \(1\) and has \(n\) teeth, where each tooth starts with \(2\), goes up to \(m\) and back down to 1. For example, the \((3,4)\)-sawtooth sequence is

    The number 1 forms the bottom left corner of the sequence. From there, the numbers 2, 3, 2, and 1 are placed to form a tooth with 3 at the top of the tooth and the 1s forming the ends. This pattern is repeated three more times so that there are five 1s across the bottom edge of the figure, four 3s across the top edge, and eight 2s across the middle.

    The \((3,4)\)-sawtooth sequence includes 17 terms and the average of these terms is \(\frac{33}{17}\).

    1. Determine the sum of the terms in the \((4,2)\)-sawtooth sequence.

    2. For each positive integer \(m \geq 2\), determine a simplified expression for the sum of the terms in the \((m,3)\)-sawtooth sequence.

    3. Determine all pairs \((m,n)\) for which the sum of the terms in the \((m,n)\)-sawtooth sequence is 145.

    4. Prove that, for all pairs of positive integers \((m,n)\) with \(m \geq 2\), the average of the terms in the \((m,n)\)-sawtooth sequence is not an integer.

  10. Full SolutionAt Pizza by Alex, toppings are put on circular pizzas in a random way. Every topping is placed on a randomly chosen semi-circular half of the pizza and each topping’s semi-circle is chosen independently. For each topping, Alex starts by drawing a diameter whose angle with the horizontal is selected uniformly at random. This divides the pizza into two semi-circles. One of the two halves is then chosen at random to be covered by the topping.

    1. For a 2-topping pizza, determine the probability that at least \(\frac{1}{4}\) of the pizza is covered by both toppings.

    2. For a 3-topping pizza, determine the probability that some region of the pizza with non-zero area is covered by all 3 toppings. (The following diagram shows an example where no region is covered by all 3 toppings.)

      A circle is divided into 6 sectors. The first sector contains topping 1 and topping 3. Moving clockwise the second sector contains only topping 1. The third sector contains toppings 1 and 2. The fourth sector contains only topping 2. The fifth sector contains toppings 2 and 3. The sixth sector contains only topping 3.

    3. Suppose that \(N\) is a positive integer. For an \(N\)-topping pizza, determine the probability, in terms of \(N\), that some region of the pizza with non-zero area is covered by all \(N\) toppings.

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.

Visit our website cemc.uwaterloo.ca to find

For teachers...

Visit our website cemc.uwaterloo.ca to