2017 Euclid Contest

Wednesday, April 6, 2017
(in North America and South America)

Friday, April 7, 2017
(outside of North American and South America)

©2017 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.

Questions

1. 1. There is one pair $(a,b)$ of positive integers for which $5a+3b=19$. What are the values of $a$ and $b$?

   2. How many positive integers $n$ satisfy $5<2^n<2017$?

   3. Jimmy bought 600 Euros at the rate of 1 Euro equals $1.50. He then converted his 600 Euros back into dollars at the rate of $1.00 equals 0.75 Euros. How many fewer dollars did Jimmy have after these two transactions than he had before these two transactions?

2. 1. What are all values of $x$ for which $x\ne 0$ and $x\ne 1$ and ${\displaystyle \frac{5}{x(x-1)}}={\displaystyle \frac{1}{x}}+{\displaystyle \frac{1}{x-1}}$ ?

   2. In a magic square, the numbers in each row, the numbers in each column, and the numbers on each diagonal have the same sum. In the magic square shown, what are the values of $a$, $b$ and $c$?

      $$\begin{array}{|ccc|}\hline 0& 20& a\\ c& 4& \\ & -12& b\\ \hline\end{array}$$

   3. 

      1. For what positive integer $n$ is ${100}^2-n^2=9559$ ?

      2. Determine one pair $(a,b)$ of positive integers for which $a>1$ and $b>1$ and $ab=9559$.

3. 1. In the diagram, $\mathrm{△}ABC$ is right-angled at $B$ and $\mathrm{△}ACD$ is right-angled at $A$. Also, $AB=3$, $BC=4$, and $CD=13$. What is the area of quadrilateral $ABCD$?

      

   2. Three identical rectangles $PQRS$, $WTUV$ and $XWVY$ are arranged, as shown, so that $RS$ lies along $TX$. The perimeter of each of the three rectangles is 21 cm. What is the perimeter of the whole shape?

      

   3. One of the faces of a rectangular prism has area $27{\text{ cm}}^2$. Another face has area $32{\text{ cm}}^2$. If the volume of the prism is $144{\text{ cm}}^3$, determine the surface area of the prism in ${\text{cm}}^2$.

4. 1. The equations $y=a(x-2)(x+4)$ and $y=2(x-h{)}^2+k$ represent the same parabola. What are the values of $a$, $h$ and $k$?

   2. In an arithmetic sequence with 5 terms, the sum of the squares of the first 3 terms equals the sum of the squares of the last 2 terms. If the first term is 5, determine all possible values of the fifth term.

      (An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant. For example, $3,5,7,9,11$ is an arithmetic sequence with five terms.)

5. 1. Dan was born in a year between 1300 and 1400. Steve was born in a year between 1400 and 1500. Each was born on April 6 in a year that is a perfect square. Each lived for 110 years. In what year while they were both alive were their ages both perfect squares on April 7?

   2. Determine all values of $k$ for which the points $A(1,2)$, $B(11,2)$ and $C(k,6)$ form a right-angled triangle.

6. 1. The diagram shows two hills that meet at $O$. One hill makes a ${30}^{\circ }$ angle with the horizontal and the other hill makes a ${45}^{\circ }$ angle with the horizontal. Points $A$ and $B$ are on the hills so that $OA=OB=20\text{ m}$. Vertical poles $BD$ and $AC$ are connected by a straight cable $CD$. If $AC=6\text{ m}$, what is the length of $BD$ for which $CD$ is as short as possible?

      

   2. If $\cos \theta =\tan \theta$, determine all possible values of $\sin \theta$, giving your answer(s) as simplified exact numbers.

7. 1. Linh is driving at 60 km/h on a long straight highway parallel to a train track. Every 10 minutes, she is passed by a train travelling in the same direction as she is. These trains depart from the station behind her every 3 minutes and all travel at the same constant speed. What is the constant speed of the trains, in km/h?

   2. Determine all pairs $(a,b)$ of real numbers that satisfy the following system of equations: $$\begin{array}{rl}\sqrt{a}+\sqrt{b}& =8\\ {\log }_{10}a+{\log }_{10}b& =2\end{array}$$ Give your answer(s) as pairs of simplified exact numbers.

8. 1. In the diagram, line segments $AC$ and $DF$ are tangent to the circle at $B$ and $E$, respectively. Also, $AF$ intersects the circle at $P$ and $R$, and intersects $BE$ at $Q$, as shown. If $\mathrm{\angle }CAF=35\mathrm{°}$, $\mathrm{\angle }DFA=30\mathrm{°}$, and $\mathrm{\angle }FPE=25\mathrm{°}$, determine the measure of $\mathrm{\angle }PEQ$.

      

   2. In the diagram, $ABCD$ and $PNCD$ are squares of side length 2, and $PNCD$ is perpendicular to $ABCD$. Point $M$ is chosen on the same side of $PNCD$ as $AB$ so that $\mathrm{△}PMN$ is parallel to $ABCD$, so that $\mathrm{\angle }PMN=90\mathrm{°}$, and so that $PM=MN$. Determine the volume of the convex solid $ABCDPMN$.

      

9. A permutation of a list of numbers is an ordered arrangement of the numbers in that list. For example, $3,2,4,1,6,5$ is a permutation of $1,2,3,4,5,6$. We can write this permutation as $a_1,a_2,a_3,a_4,a_5,a_6$, where $a_1=3,a_2=2,a_3=4,a_4=1,a_5=6$, and $a_6=5$.

   1. Determine the average value of $$|a_1-a_2|+|a_3-a_4|$$ over all permutations $a_1,a_2,a_3,a_4$ of $1,2,3,4$.

   2. Determine the average value of $$a_1-a_2+a_3-a_4+a_5-a_6+a_7$$ over all permutations $a_1,a_2,a_3,a_4,a_5,a_6,a_7$ of $1,2,3,4,5,6,7$.

   3. Determine the average value of $$\begin{array}{}\text{(}\ast \text{)}& |a_1-a_2|+|a_3-a_4|+\cdots +|a_{197}-a_{198}|+|a_{199}-a_{200}|\end{array}$$ over all permutations $a_1,a_2,a_3,...,a_{199},a_{200}$ of $1,2,3,4,\dots ,199,200$. (The sum labelled $(\ast )$ contains 100 terms of the form $|a_{2k-1}-a_{2k}|$.)

10. Consider a set $S$ that contains $m\ge 4$ elements, each of which is a positive integer and no two of which are equal. We call $S$ boring if it contains four distinct integers $a,b,c,d$ such that $a+b=c+d$. We call $S$ exciting if it is not boring. For example, $\{2,4,6,8,10\}$ is boring since $4+8=2+10$. Also, $\{1,5,10,25,50\}$ is exciting.

    1. Find an exciting subset of $\{1,2,3,4,5,6,7,8\}$ that contains exactly 5 elements.

    2. Prove that, if $S$ is an exciting set of $m\ge 4$ positive integers, then $S$ contains an integer greater than or equal to ${\displaystyle \frac{m^2-m}{4}}$.

    3. Define $\text{rem}(a,b)$ to be the remainder when the positive integer $a$ is divided by the positive integer $b$. For example, $\text{rem}(10,7)=3$, $\text{rem}(20,5)=0$, and $\text{rem}(3,4)=3$.

       Let $n$ be a positive integer with $n\ge 10$. For each positive integer $k$ with $1\le k\le n$, define $x_k=2n\cdot \text{rem}(k^2,n)+k$. Determine, with proof, all positive integers $n\ge 10$ for which the set $\{x_1,x_2,\dots ,x_{n-1},x_n\}$ of $n$ integers is exciting.

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