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2017 Euclid Contest

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

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

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©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 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.

Questions

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

    2. LightbulbHow many positive integers \(n\) satisfy \(5 < 2^n < 2017\)?

    3. Full SolutionJimmy 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?

    1. LightbulbWhat are all values of \(x\) for which \(x \neq 0\) and \(x \neq 1\) and \(\dfrac{5}{x(x-1)} = \dfrac{1}{x}+\dfrac{1}{x-1}\) ?

    2. LightbulbIn 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 \(c\)?

      \[\large\begin{array}{|c|c|c|} \hline &&\\[-2mm] 0 & 20 & a \\ &&\\[-2mm]\hline&&\\[-2mm] c & 4 & \phantom{-12} \\&&\\[-2mm] \hline &&\\[-2mm] \phantom{-12} & -12 & b \\[3mm] \hline \end{array}\]

    3. Full Solution

      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 \(ab = 9559\).

    1. LightbulbIn the diagram, \(\triangle ABC\) is right-angled at \(B\) and \(\triangle ACD\) is right-angled at \(A\). Also, \(AB=3\), \(BC=4\), and \(CD=13\). What is the area of quadrilateral \(ABCD\)?

      So both triangles share the side AC.

    2. LightbulbThree 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?

      Label each rectangle counter-clockwise, starting at the upper-right corner. Rectangles WTUV and XWVY are adjacent and touching so that they share the side WV. Rectangle pQRS is stacked on the other two rectangles (so that RS lies along TX).

    3. Full SolutionOne of the faces of a rectangular prism has area \(27\mbox{ cm}^2\). Another face has area \(32\mbox{ cm}^2\). If the volume of the prism is \(144\mbox{ cm}^3\), determine the surface area of the prism in \(\mbox{cm}^2\).

    1. LightbulbThe 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. Full SolutionIn 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.)

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

    1. LightbulbThe 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 \(OA=OB=20\mbox{ m}\). Vertical poles \(BD\) and \(AC\) are connected by a straight cable \(CD\). If \(AC = 6\mbox{ m}\), what is the length of \(BD\) for which \(CD\) is as short as possible?

      Point O lies on a horizontal dashed line. Each hill is represented as a single sloped line segment. Starting at point O, the first line segment (hill) slopes up to the left at an angle of 30 degrees. Starting at point O again, the second line segment (hill) slopes up to the right at an angle of 45 degrees. Point A lies on the first line segment,a nd point B lies on the second line segment. Point C is above A and point D is above B.

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

    1. LightbulbLinh 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. Full SolutionDetermine all pairs \((a,b)\) of real numbers that satisfy the following system of equations: \[\begin{aligned} \sqrt{a}+\sqrt{b}&=8\\ \log_{10} a + \log_{10} b &=2\end{aligned}\] Give your answer(s) as pairs of simplified exact numbers.

    1. Full SolutionIn 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 \(\angle CAF = 35^\circ\), \(\angle DFA = 30^\circ\), and \(\angle FPE = 25^\circ\), determine the measure of \(\angle PEQ\).

      A circle is contained between two non-parallel tangent line segments. Segment AC is tangent on the left side of the circle, and segment DF is tangent on the right side of the circle.

    2. Full SolutionIn 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 \(\triangle PMN\) is parallel to \(ABCD\), so that \(\angle PMN = 90^\circ\), and so that \(PM=MN\). Determine the volume of the convex solid \(ABCDPMN\).

      The two squares are joined so that they share the side DC.

  9. Full SolutionA 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 \[|a_1 - a_2| + |a_3 - a_4| + \cdots + |a_{197} - a_{198}|+ |a_{199} - a_{200}| \qquad\qquad (*)\] over all permutations \(a_1, a_2, a_3, ... , a_{199}, a_{200}\) of \(1,2,3,4, \ldots,199,200\). (The sum labelled \((*)\) contains 100 terms of the form \(|a_{2k-1}-a_{2k}|\).)

  10. Full SolutionConsider a set \(S\) that contains \(m \geq 4\) 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 \(m \geq 4\) positive integers, then \(S\) contains an integer greater than or equal to \(\dfrac{m^2-m}{4}\).

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

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

Further Information

For students...

Thank you for writing the Euclid Contest!

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