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2026 Galois Contest
(Grade 10)

Wednesday, April 1, 2026
(in North America and South America)

Thursday, April 2, 2026
(outside of North American and South America)

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©2026 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 Short answer
  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 Short answer, 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. A prime number is a positive integer greater than \(1\) whose only positive divisors are \(1\) and itself. For example, \(2\) is a prime number since it is greater than \(1\) and its only positive divisors are \(1\) and \(2\).

    1. Short answer What is the product of the three smallest prime numbers?

    2. Short answer There are two integers \(c\) with \(1\leq c\leq 20\) for which \(\dfrac{c-3}{4}\) is a prime number. What are these two possible values of the integer \(c\)?

    3. Full solution Determine all integers \(d\) with \(1\leq d\leq 10\) for which \(21d-77\) is a prime number.

  2. In the diagram, \(\triangle ABC\) has vertices \(A(3,4)\), \(B(2,1)\) and \(C(6,1)\).

    The Cartesian plane, with x and y each ranging from negative 9 to 9, and triangle ABC plotted in the first quadrant.

    1. Short answer What is the area of \(\triangle ABC\)?

    2. Short answer Point \(D\) is the image of point \(B\) after it is reflected in the \(y\)-axis. What is the area of \(\triangle ADC\)?

    3. Short answer Point \(E\) is the image of point \(A\) after it is reflected in the horizontal line \(y=-2\). What is the area of \(\triangle EBC\)?

    4. Full solution Point \(F\) is the image of point \(A\) after it is reflected in the horizontal line \(y=k\). Determine the two different values of \(k\) for which the area of \(\triangle FBC\) is equal to \(12\).

  3. For every positive integer \(a\), the units digits of \(a^1\), \(a^2\), \(a^3\), \(a^4\), \(a^5\), \(\ldots\) will form a repeating sequence. In each such sequence, the smallest number of consecutive units digits that repeat consecutively and indefinitely is at most \(4\). This number is called the cycle length. For example, when \(a=3\), \[3^1=3, \ 3^2=9, \ 3^3=27, \ 3^4=81, \ 3^5=243, \ 3^6=729, \ldots\] and the sequence of units digits is \(3\), \(9\), \(7\), \(1\), \(3\), \(9\), \(\dots\). In this example, the consecutive units digits that repeat are \(3\), \(9\), \(7\), \(1\), and so the cycle length is \(4\).

    1. Short answer What is the units digit of \(3^{43}\)?

    2. Full solution Determine the number of integers \(j\) with \(1\leq j\leq 2026\) for which \(4^{j}+8^{j}\) is a multiple of 10.

    3. Full solution Determine the number of integers \(k\) with \(1\leq k\leq 50\) for which \(2^k+3^k\) has the same units digit as \(8^{2026k}+9^{2026k}\).

    1. Short answer The set \(R\) contains the points \((0,0)\), \((0,1)\), \((0,4)\), \((0,9)\), \((0,16)\), and \((0,25)\).The midpoint of every pair of distinct points from \(R\) is plotted on the \(xy\)-plane. What is the number of distinct points that are plotted?

    2. Full solution The set \(S\) contains exactly \(20\) distinct points, each of the form \((n,n+2)\), where \(n\) is an integer and \(1\leq n\leq 20\). The midpoint of every pair of distinct points from \(S\) is plotted on the \(xy\)-plane. Determine the number of distinct points that are plotted.

    3. Full solution A set \(T\) contains exactly \(100\) points having distinct \(y\)-coordinates. The midpoint of every pair of distinct points from \(T\) is plotted on the \(xy\)-plane. Determine \(m\), the minimum number of distinct points that could be plotted. A complete solution must

      • include the value of \(m\),

      • describe a set of points \(T\) for which the number of distinct midpoints is \(m\), and

      • provide an explanation of why, for all possible choices for the set \(T\), fewer than \(m\) distinct midpoints is not possible.

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