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)

©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
   • worth 2 or 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. 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. What is the total length of a train with 10 boxcars?

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

   3. In 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.

      

      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. 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. A two-digit positive integer $MN$ is given. Explain why it is not possible that $MN-NM=80$.

   3. 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 $T(3)=9$.

   

   1. What do $T(4)$ and $T(5)$ equal?

   2. Express $T(n)-T(n-1)$ in terms of $n$.

   3. Determine 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:

   $$\begin{array}{rl}P(6)& =gcd(1,6)+gcd(2,6)+gcd(3,6)+gcd(4,6)+gcd(5,6)+gcd(6,6)\\ & =1+2+3+2+1+6\\ & =15\end{array}$$

   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. What is the value of $P(125)?$

   2. If $r$ and $s$ are different prime numbers, prove that $P(r^{2}s)=r(3r-2)(2s-1).$

   3. If $r$ and $s$ are different prime numbers, prove that $P(r^{2}s)$ 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. Determine, 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

• Free copies of past contests
• Math Circles videos and handouts that will help you learn more mathematics and prepare for future contests
• Information about careers in and applications of mathematics and computer science

For teachers...

Visit our website cemc.uwaterloo.ca to

• Obtain information about future contests
• Look at our free online courseware for high school students
• Learn about our face-to-face workshops and our web resources
• Subscribe to our free Problem of the Week
• Investigate our online Master of Mathematics for Teachers
• Find your school's contest results