2016 Euclid Contest

Tuesday, April 12, 2016
(in North America and South America)

Wednesday, April 13, 2016
(outside of North American and South America)

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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:

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
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, $π + 1$ and $1 - \sqrt{2}$ are simplified exact numbers.

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:

Please read the instructions for the contest.
Write all answers in the answer booklet provided.
For questions marked , place your answer in the appropriate box in the answer booklet and show your work.
For questions marked , 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.
Diagrams are not drawn to scale. They are intended as aids only.
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.

Questions

1. What is the average of the integers $5, 15, 25, 35, 45, 55$ ?
2. If $x^{2} = 2016$ , what is the value of $(x + 2) (x - 2)$ ?
3. In the diagram, points $P (7, 5)$ , $Q (a, 2 a)$ , and $R (12, 30)$ lie on a straight line.
  
  Determine the value of $a$ .
1. What are all values of $n$ for which $\frac{n}{9} = \frac{25}{n}$ ?
2. What are all values of $x$ for which $(x - 3) (x - 2) = 6$ ?
3. At Willard’s Grocery Store, the cost of 2 apples is the same as the cost of 3 bananas. Ross buys 6 apples and 12 bananas for a total cost of $6.30. Determine the cost of 1 apple.
1. In the diagram, point $B$ lies on the line segment $A C$ between $A$ and $C$ . Point $D$ lies above $A B$ to form the triangle $A D$ , and point $E$ lies above $B C$ to form the triangle $B C E$ . Point $F$ lies on $D B$ and point $G$ lies on $E B$ . Line segment $F G$ is constructed. Specific angles are labelled as follows: $∠ D A B = p °$ , $∠ A D B = q °$ , $∠ B F G = r °$ , $∠ B G F = s °$ , $∠ B E C = t °$ , and $∠ B C E = u °$ .
  
  What is the value of $p + q + r + s + t + u$ ?
2. Let $n$ be the integer equal to $10^{20} - 20$ . What is the sum of the digits of $n$ ?
3. A parabola intersects the $x$ -axis at $P (2, 0)$ and $Q (8, 0)$ . The vertex of the parabola is at $V$ , which is below the $x$ -axis. If the area of $△ V P Q$ is 12, determine the coordinates of $V$ .
1. Determine all angles $θ$ with $0^{\circ} \leq θ \leq 180^{\circ}$ and $\sin^{2} θ + 2 \cos^{2} θ = \frac{7}{4}$ .
2. The sum of the radii of two circles is 10 cm. The circumference of the larger circle is 3 cm greater than the circumference of the smaller circle. Determine the difference between the area of the larger circle and the area of the smaller circle.
1. Charlotte’s Convenience Centre buys a calculator for $ $p$ (where $p > 0$ ), raises its price by $n %$ , then reduces this new price by 20%. If the final price is 20% higher than $ $p$ , what is the value of $n$ ?
2. A function $f$ is defined so that if $n$ is an odd integer, then $f (n) = n - 1$ and if $n$ is an even integer, then $f (n) = n^{2} - 1$ . For example, if $n = 15$ , then $f (n) = 14$ and if $n = - 6$ , then $f (n) = 35$ , since 15 is an odd integer and $- 6$ is an even integer. Determine all integers $n$ for which $f (f (n)) = 3$ .
1. What is the smallest positive integer $x$ for which $\frac{1}{32} = \frac{x}{10^{y}}$ for some positive integer $y$ ?
2. Determine all possible values for the area of a right-angled triangle with one side length equal to 60 and with the property that its side lengths form an arithmetic sequence.
  
  (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, $3, 5, 7, 9$ are the first four terms of an arithmetic sequence.)
1. Amrita and Zhang cross a lake in a straight line with the help of a one-seat kayak. Each can paddle the kayak at 7 km/h and swim at 2 km/h. They start from the same point at the same time with Amrita paddling and Zhang swimming. After a while, Amrita stops the kayak and immediately starts swimming. Upon reaching the kayak (which has not moved since Amrita started swimming), Zhang gets in and immediately starts paddling. They arrive on the far side of the lake at the same time, 90 minutes after they began. Determine the amount of time during these 90 minutes that the kayak was not being paddled.
2. Determine all pairs $(x, y)$ of real numbers that satisfy the system of equations $\begin{aligned} x (\frac{1}{2} + y - 2 x^{2}) & = 0 \\ y (\frac{5}{2} + x - y) & = 0 \end{aligned}$
1. In the diagram, $A B C D$ is a parallelogram.
  
  Point $E$ is on $D C$ with $A E$ perpendicular to $D C$ , and point $F$ is on $C B$ with $A F$ perpendicular to $C B$ . If $A E = 20$ , $A F = 32$ , and $\cos (∠ E A F) = \frac{1}{3}$ , determine the exact value of the area of quadrilateral $A E C F$ .

Full solution Determine all real numbers $x > 0$ for which $\log_{4} x - \log_{x} 16 = \frac{7}{6} - \log_{x} 8$

The string $A A A B B B A A B B$ is a string of ten letters, each of which is $A$ or $B$ , that does not include the consecutive letters $A B B A$ .

The string $A A A B B A A A B B$ is a string of ten letters, each of which is $A$ or $B$ , that does include the consecutive letters $A B B A$ .

Determine, with justification, the total number of strings of ten letters, each of which is $A$ or $B$ , that do not include the consecutive letters $A B B A$ .
In the diagram, $A B C D$ is a square.

Points $E$ and $F$ are chosen on $A C$ so that $∠ E D F = 45^{\circ}$ . If $A E = x$ , $E F = y$ , and $F C = z$ , prove that $y^{2} = x^{2} + z^{2}$ .

Full solution Let $k$ be a positive integer with $k \geq 2$ . Two bags each contain $k$ balls, labelled with the positive integers from $1$ to $k$ . André removes one ball from each bag. (In each bag, each ball is equally likely to be chosen.) Define $P (k)$ to be the probability that the product of the numbers on the two balls that he chooses is divisible by $k$ .

Calculate $P (10)$ .
Determine, with justification, a polynomial $f (n)$ for which
- $P (n) \geq \frac{f (n)}{n^{2}}$ for all positive integers $n$ with $n \geq 2$ , and
- $P (n) = \frac{f (n)}{n^{2}}$ for infinitely many positive integers $n$ with $n \geq 2$ .
(A polynomial $f (x)$ is an algebraic expression of the form $f (x) = a_{m} x^{m} + a_{m - 1} x^{m - 1} + \dots + a_{1} x + a_{0}$ for some integer $m \geq 0$ and for some real numbers $a_{m}, a_{m - 1}, \dots, a_{1}, a_{0}$ .)
Prove there exists a positive integer $m$ for which $P (m) > \frac{2016}{m}$ .

Further Information

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