CEMC Banner

2025 Galois Contest
(Grade 10)

Thursday, April 3, 2025
(in North America and South America)

Friday, April 4, 2025
(outside of North American and South America)

University of Waterloo Logo

©2025 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. Members of the Art Club are between \(14\) and \(17\) years old, inclusive. The graph shows the number of students of each age in the club.

    A bar graph with the ages 14, 15, 16, and 17 on the horizontal axis. 1 student is 14 years old, 0 students are 15, 3 students are 16, and 2 students are 17.

    1. Lightbulb What is the total number of students in the Art Club?

    2. Full Solution Determine the mean (average) age of the students in the Art Club.

    3. Full Solution Determine the number of \(15\)-year old students that must join the Art Club so that the mean age of all the students in the club is exactly \(15.5\).

  2. In a magic square, the numbers in each of the three rows, each of the three columns, and each of the two diagonals have the same sum. This sum is called the magic constant. Each of the four figures shown below is a magic square.

    1. Lightbulb In Figure 1, the magic constant is \(18\). What is the value of \(n\)?

      Figure 1
      \(7\) \(2\)
      \(n\)
      \(3\)

    2. Lightbulb In Figure 2, what is the value of \(p\)?

      Figure 2
      \(8\) \(p\)
      \(9\) \(5\)
      \(4\)

    3. Lightbulb In Figure 3, what is the value of \(r\)?

      Figure 3
      \(13\) \(r\)
      \(7\) \(17\)
      \(\!r\!+\!1\!\) \(\!r\!+\!3\!\)

    4. Full Solution In Figure 4, determine the value of \(u\).

      Figure 4
      \(\!u\!+\!3\!\)
      \(12\)
      \(\!u\!+\!2\!\) \(\!u\!-\!5\!\) \(u\)

  3. Point \(A\) has coordinates \((10,15)\) and \(C\) has coordinates \((20,27)\). Rectangle \(ABCD\) has side \(AD\) parallel to the \(y\)-axis and side \(AB\) parallel to the \(x\)-axis.

    1. Lightbulb What is the area of rectangle \(ABCD\)?

    2. Full Solution The line with equation \(y=-\frac 32x+39\) intersects side \(AD\) at \(E\) and side \(AB\) at \(F\). Determine the area of the pentagon \(BCDEF\).

    3. Full Solution The line with equation \(y=mx+b\), with \(m<0\), intersects side \(AD\) at \(G\) and side \(AB\) at \(H\). Determine all ordered pairs of integers \((m,b)\) with \(b<50\) for which the area of \(\triangle GAH\) is equal to \(-\dfrac 8m\).

  4. To determine the number of positive divisors of \(N\), we take the exponents on the prime powers in the prime factorization of \(N\), add \(1\) to each of the exponents, and multiply the resulting numbers together. For example, the prime factorization of \(280\) is \(2^35^17^1\), and so \(280\) has \((3+1)(1+1)(1+1)=16\) positive divisors.

    1. Lightbulb How many ordered pairs of positive integers \((a,b)\) are there for which \(ab=400\) and \(1 < a \le b\)?

    2. Full Solution Determine the number of ordered triples of positive integers \((p,q,r)\) for which \(pqr=270\,000\).

    3. Full Solution Determine the number of ordered triples of positive integers \((x,y,z)\) for which \(xyz=270\,000\) and \(1 < x \le y \le z\).

Further Information

For students...

Thank you for writing the Galois 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