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Problem of the Month
Problem 3: Multiplication in two dimensions

December 2025

This month’s problem is inspired by Question B2(c) on the 2025 Canadian Senior Mathematics Contest. Here is the original question:

The points \(X\), \(Y\), and \(Z\) have coordinates \(X(0,0)\), \(Y(7,24)\), and \(Z(15,0)\). The point \(W\) is on the line segment \(YZ\) such that \(\angle WXZ = 3\angle WXY\). Determine the coordinates of \(W\).

Give it a try as a warm up before reading on!

One path to solving this problem is to introduce a point \(V\) on the line segment \(YZ\) so that \(\angle VXZ = \angle VXY\). If you do this, you find the coordinates of \(V\) are \((12,9)\). These are very nice numbers! In this month’s POTM, we will investigate how the points \(X\), \(Y\), and \(Z\) were chosen so that the coordinates of \(V\) (and eventually \(W\)) are nice rational numbers. This will involve a way to multiply points on the Cartesian plane.

  1. Given points \((a,b)\) and \((c,d)\) on the Cartesian plane, define their product as \[(a,b)*(c,d) = (ac - bd,ad + bc).\] So, for example, \((3,5)*(2,-1) = (3\cdot2 - 5\cdot(-1),3\cdot(-1) + 5\cdot 2) = (11,7)\).

    1. Find a point \((a,b)\) satisfying \((1,1)*(a,b) = (0,-2)\).

    2. Find a point \((a,b)\) with the property that \((a,d)*(c,d) = (c,d)\) for every point \((c,d)\) in the Cartesian plane.

  2. For a point \((a,b)\) in the Cartesian plane, denote by \((a,b)^k\) the point obtained by taking the product of \((a,b)\) with itself \(k\) times. Compute \(\left(\dfrac{1}{2},\dfrac{\sqrt 3}{2}\right)^{2025}\).

  3. Denote by \(O\) the origin \((0,0)\), and by \(E\) the point with coordinates \((1,0)\). Let \(A\) be a point in the Cartesian plane. Define \(|A|\) to be the distance from \(A\) to \(O\). Define \(\theta(A)\) to be the measure of the angle \(\angle EOA\), measured counterclockwise around \(O\) from the line segment \(OE\). For example, \(|(-\sqrt 2,-\sqrt 2)| = \sqrt{(-2)^2 + (-2)^2} = 2\) and \(\theta(-\sqrt 2,-\sqrt 2) = 225\degree\).

    1. Compute \(|(0,2)*(1,1)|\) and \(\theta((0,2)*(1,1))\).

    2. Let \(D_1\) and \(D_2\) be points with \(|D_1| = r_1\), \(|D_2| = r_2\), \(\theta(D_1) = \phi_1\) and \(\theta(D_2) = \phi_2\). Compute \(|D_1*D_2|\) and \(\theta(D_1*D_2)\) in terms of \(r_1,r_2,\phi_1\), and \(\phi_2\).

  4. The point \((2,3)\) satisfies the equation \(x^4 = (-119,-120)\) since \((2,3)^4 = (-119,-120)\). Find three other points satisfying \(x^4 = (-119,-120)\).

  5. Let \(Y\) have coordinates \((7,24)\). Find a point \(F\) in the Cartesian plane with integer coordinates so that \(OF^2 = OY\) and \(2\angle EOF = \angle EOY\).