2022 Euclid Contest

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

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

©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
   • worth 2 or 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 beinserted into your answer booklet. Write your name, school name, and question numberon 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.

Questions

1. 1. What is the value of ${\displaystyle \frac{3^2-2^3}{2^3-3^2}}$ ?

   2. What is the value of $\sqrt{\sqrt{81}+\sqrt{9}-\sqrt{64}}$ ?

   3. Determine all real numbers $x$ for which ${\displaystyle \frac{1}{\sqrt{x^2+7}}}={\displaystyle \frac{1}{4}}$.

2. 1. Find the three ordered pairs of integers $(a,b)$ with $1<a<b$ and $ab=2022$.

   2. Suppose that $c$ and $d$ are integers with $c>0$ and $d>0$ and ${\displaystyle \frac{2c+1}{2d+1}}={\displaystyle \frac{1}{17}}$. What is the smallest possible value of $d$?

   3. Suppose 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 $t$.

3. 1. A 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. Stephanie 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. Each 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.

4. 1. Hexagon $ABCDEF$ 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. In the diagram, $\mathrm{△}PQS$ is right-angled at $P$ and $\mathrm{△}QRS$ 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 $PQRS$.

      

5. 1. A 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 $1+{\displaystyle \frac{1}{1+r}}$. For example, if $a_3={\displaystyle \frac{41}{29}}$, then $a_4=1+{\displaystyle \frac{1}{1+(41/29)}}={\displaystyle \frac{99}{70}}$. If $a_3={\displaystyle \frac{41}{29}}$, what is the value of $a_1$?

   2. A 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 $h$.

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

   2. Determine 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 $R(11,3)$.

7. 1. A 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 ${\log }_{100}(1+{\displaystyle \frac{1}{x}})$. Suppose that the probability that $81\le x\le 99$ is equal to 2 times the probability that $x=n$ for some integer $n$. What is the value of $n$?

   2. In the diagram, $\mathrm{△}ABD$ has $C$ on $BD$. Also, $BC=2$, $CD=1$, ${\displaystyle \frac{AC}{AD}}={\displaystyle \frac{3}{4}}$, and $\cos (\mathrm{\angle }ACD)=-{\displaystyle \frac{3}{5}}$. Determine the length of $AB$.

      

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

      

      If the area of $\mathrm{△}VBC$ is $\frac{72}{5}$, determine the value of $a$.

   2. Consider 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. Suppose that $m$ and $n$ are positive integers with $m\ge 2$. 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 $(3,4)$-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 $m\ge 2$, 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 $m\ge 2$, the average of the terms in the $(m,n)$-sawtooth sequence is not an integer.

10. At 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.)

       

    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

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