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2015 Euclid Contest

Wednesday, April 15, 2015
(in North America and South America)

Thursday, April 16, 2015
(outside of North American and South America)

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©2015 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. LightbulbWhat is value of \(\dfrac{10^2-9^2}{10+9}\)?

    2. LightbulbIf \(\dfrac{x+1}{x+4}=4\), what is the value of \(3x+8\)?

    3. Full solutionIf \(f(x)=2x-1\), determine the value of \((f(3))^2+2(f(3))+1\).

    1. LightbulbIf \(\sqrt{a} + \sqrt{a} = 20\), what is the value of \(a\)?

    2. LightbulbTwo circles have the same centre. The radius of the smaller circle is 1. The area of the region between the circles is equal to the area of the smaller circle.

      The smaller circle is centered inside of the larger circle.

      What is the radius of the larger circle?

    3. Full solutionThere were 30 students in Dr. Brown’s class. The average mark of the students in the class was 80. After two students dropped the class, the average mark of the remaining students was 82. Determine the average mark of the two students who dropped the class.

    1. LightbulbIn the diagram, \(BD=4\) and point \(C\) is the midpoint of \(BD\). If point \(A\) is placed so that \(\triangle ABC\) is equilateral, what is the length of \(AD\)?

    2. Full solution\(\triangle MNP\) has vertices \(M(1,4)\), \(N(5,3)\), and \(P(5,c)\). Determine the sum of the two values of \(c\) for which the area of \(\triangle MNP\) is 14.

    1. LightbulbWhat are the \(x\)-intercepts and the \(y\)-intercept of the graph with equation \(y = (x-1)(x-2)(x-3)-(x-2)(x-3)(x-4)\)?

    2. Full solutionThe graphs of the equations \(y=x^3-x^2+3x-4\) and \(y=ax^2-x-4\) intersect at exactly two points. Determine all possible values of \(a\).

    1. LightbulbIn the diagram, \(\angle CAB = 90^\circ\). Point \(D\) lies on \(AB\) and point \(E\) lies on \(AC\) so that \(AB=AC=DE\), \(DB=9\), and \(EC=8\).

      Determine the length of \(DE\).

    2. Full solutionEllie has two lists, each consisting of 6 consecutive positive integers. The smallest integer in the first list is \(a\), the smallest integer in the second list is \(b\), and \(a<b\). She makes a third list which consists of the 36 integers formed by multiplying each number from the first list with each number from the second list. (This third list may include some repeated numbers.) If

      • the integer 49 appears in the third list,

      • there is no number in the third list that is a multiple of 64, and

      • there is at least one number in the third list that is larger than 75,

      determine all possible pairs \((a,b)\).

    1. LightbulbA circular disc is divided into 36 sectors. A number is written in each sector. When three consecutive sectors contain \(a\), \(b\) and \(c\) in that order, then \(b=ac\). If the number 2 is placed in one of the sectors and the number 3 is placed in one of the adjacent sectors, as shown, what is the sum of the 36 numbers on the disc?

      A few sample sectors are drawn in a circle (thin pie-shaped slices that originate from the circle's centre). Three consecutive sectors are labelled a, b, and c respectively. Two other consecutive sectors are labelled 2 and 3 respectively.

    2. Full solutionDetermine all values of \(x\) for which \(0<\dfrac{x^2-11}{x+1}<7\) .

    1. LightbulbIn the diagram, \(ACDF\) is a rectangle with \(AC=200\) and \(CD=50\). Point \(B\) lies close to \(A\) on \(AC\), and point \(E\) lies close to \(F\) on \(FD\). Also, \(\triangle FBD\) and \(\triangle AEC\) are congruent triangles which are right-angled at \(B\) and \(E\), respectively.

      The quadrilateral formed by the point of intersection of AE and FB, the point B, the point of intersection of BD and EC, and the point E, is shaded.

      What is the area of the shaded region?

    2. Full solutionThe numbers \(a_1, a_2, a_3,\ldots\) form an arithmetic sequence with \(a_1 \neq a_2\). The three numbers \(a_1,a_2,a_6\) form a geometric sequence in that order. Determine all possible positive integers \(k\) for which the three numbers \(a_1,a_4,a_k\) also form a geometric sequence in that order.

      (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.

      A geometric sequence is a sequence in which each term after the first is obtained from the previous term by multiplying it by a non-zero constant. For example, 3, 6, 12 is a geometric sequence with three terms.)

    1. Full solutionFor some positive integers \(k\), the parabola with equation \(y=\dfrac{x^2}{k} - 5\) intersects the circle with equation \(x^2 + y^2 = 25\) at exactly three distinct points \(A\), \(B\) and \(C\). Determine all such positive integers \(k\) for which the area of \(\triangle ABC\) is an integer.

    2. Full solutionIn the diagram, \(\triangle XYZ\) is isosceles with \(XY=XZ=a\) and \(YZ=b\) where \(b<2a\). A larger circle of radius \(R\) is inscribed in the triangle (that is, the circle is drawn so that it touches all three sides of the triangle). A smaller circle of radius \(r\) is drawn so that it touches \(XY\), \(XZ\) and the larger circle.

      The smaller circle lies outside of the larger circle.

      Determine an expression for \(\dfrac{R}{r}\) in terms of \(a\) and \(b\).

  9. Full solutionConsider the following system of equations in which all logarithms have base 10: \[\begin{aligned} (\log x)(\log y)-3\log 5y-\log 8x &=a\\ (\log y)(\log z)-4\log 5y-\log 16z &=b\\ (\log z)(\log x)-4\log 8x-3\log 625z &=c\end{aligned}\]

    1. If \(a=-4\), \(b=4\), and \(c=-18\), solve the system of equations.

    2. Determine all triples \((a,b,c)\) of real numbers for which the system of equations has an infinite number of solutions \((x,y,z)\).

  10. Full solutionFor each positive integer \(n \geq 1\), let \(C_n\) be the set containing the \(n\) smallest positive integers; that is, \(C_n = \{1,2,\ldots,n-1,n\}\). For example, \(C_4 = \{1,2,3,4\}\). We call a set, \(F\), of subsets of \(C_n\) a Furoni family of \(C_n\) if no element of \(F\) is a subset of another element of \(F\).

    1. Consider \(A = \{ \{1,2\},\{1,3\},\{1,4\}\}\). Note that \(A\) is a Furoni family of \(C_4\). Determine the two Furoni families of \(C_4\) that contain all of the elements of \(A\) and to which no other subsets of \(C_4\) can be added to form a new (larger) Furoni family.

    2. Suppose that \(n\) is a positive integer and that \(F\) is a Furoni family of \(C_n\). For each non-negative integer \(k\), define \(a_k\) to be the number of elements of \(F\) that contain exactly \(k\) integers. Prove that \[\dfrac{a_0}{\displaystyle{\binom{n}{0}}} +\dfrac{a_1}{\displaystyle{\binom{n}{1}}} + \dfrac{a_2}{\displaystyle{\binom{n}{2}}} + \cdots + \dfrac{a_{n-1}}{\displaystyle{\binom{n}{n-1}}} + \dfrac{a_n}{\displaystyle{\binom{n}{n}}} \leq 1\] (The sum on the left side includes \(n+1\) terms.)

      (Note: If \(n\) is a positive integer and \(k\) is an integer with \(0 \leq k \leq n\), then \(\displaystyle\binom{n}{k} = \dfrac{n!}{k!(n-k)!}\) is the number of subsets of \(C_n\) that contain exactly \(k\) integers, where \(0!=1\) and, if \(m\) is a positive integer, \(m!\) represents the product of the integers from \(1\) to \(m\), inclusive.)

    3. For each positive integer \(n\), determine, with proof, the number of elements in the largest Furoni family of \(C_n\) (that is, the number of elements in the Furoni family that contains the maximum possible number of subsets of \(C_n\)).

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