CEMC Banner

2015 Hypatia Contest
(Grade 11)

Thursday, April 16, 2015
(in North America and South America)

Friday, April 17, 2015
(outside of North American and South America)

University of Waterloo Logo

©2015 University of Waterloo

Instructions

Time:75 minutes

Number of Questions: 4
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. Each Hypatia Railway train has one engine car followed by some boxcars in a straight line. The distance between consecutive boxcars is 2 m. The distance between the engine car and the first boxcar is also 2 m. The engine car is 26 m in length and each boxcar is 15 m in length. The total length of a train is the distance from the front of the engine car to the end of the last boxcar.

    1. LightbulbWhat is the total length of a train with 10 boxcars?

    2. LightbulbA train has a total length of 2015 m. How many boxcars does the train have?

    3. Full solutionIn the diagram, a southbound train with 14 boxcars crosses the border between Canada and the United States at a speed of 1.6 m/s.

      A compass rose is drawn where the horizontal west-east axis serves as the border between Canada, to the north, and the United States, to the south. The train is drawn vertically within the north-east quadrant so that it is heading south.

      Determine the length of time in seconds during which a portion of the train is in Canada and a portion is in the United States at the same time.

  2. In the questions below, \(A,B,M, N, P,Q,\) and \(R\) are non-zero digits.

    1. Lightbulb A two-digit positive integer \(AB\) equals \(10A+B\). For example, \(37=10 \times 3 + 7\).

      If \(AB-BA=72\), what is the positive integer \(AB\)?

    2. Full solution A two-digit positive integer \(MN\) is given. Explain why it is not possible that \(MN-NM=80\).

    3. Full solution A three-digit positive integer \(PQR\) equals \(100P + 10Q+ R\). If \(P>R\), determine the number of possible values of \(PQR-RQP\).

  3. Consider \(n\) line segments, where each pair of line segments intersect at a different point, and not at an endpoint of any of the \(n\) line segments. Let \(T(n)\) be the sum of the number of intersection points and the number of endpoints of the line segments. For example, \(T(1) = 2\) and \(T(2) = 5.\) The diagram below illustrates that \(T(3)=9\).

    Three line segments. The first two segments intersect in the shape of an X. The third line segment is placed vertically so that it intersects the first and second in one place each.

    1. LightbulbWhat do \(T(4)\) and \(T(5)\) equal?

    2. LightbulbExpress \(T(n) -T(n-1)\) in terms of \(n\).

    3. Full solutionDetermine all possible values of \(n\) such that \(T(n) = 2015\).

  4. Let \(\gcd(a,b)\) represent the greatest common divisor of the two positive integers \(a\) and \(b.\) For example, \(\gcd(18,45) = 9\) since 9 is the largest positive integer that divides both 18 and 45.

    The function \(P(n)\) is defined to equal the sum of the \(n\) greatest common divisors, \(\gcd(1,n),\gcd(2,n), \dots, \gcd(n,n)\). For example:

    \[\begin{aligned} P(6)&=\gcd(1,6) + \gcd(2,6) + \gcd(3,6) + \gcd(4,6) + \gcd(5,6) + \gcd(6,6)\\ & = 1 + 2 + 3 + 2 + 1 + 6\\ & = 15\\\end{aligned}\]

    Note: You may use the fact that \(P(ab)=P(a)P(b)\) for all positive integers \(a\) and \(b\) with \(\gcd(a,b)=1\).

    1. Lightbulb What is the value of \(P(125)?\)

    2. Full solutionIf \(r\) and \(s\) are different prime numbers, prove that \(P(r^2s) = r(3r-2)(2s-1).\)

    3. Full solutionIf \(r\) and \(s\) are different prime numbers, prove that \(P(r^2s)\) can never be equal to a power of a prime number (that is, can never equal \(t^n\) for some prime number \(t\) and positive integer \(n\)).

    4. Full solutionDetermine, with justification, two positive integers \(m\) for which \(P(m)=243\).

Further Information

For students...

Thank you for writing the Hypatia Contest!

Encourage your teacher to register you for the Canadian Intermediate Mathematics Contest or 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