Problems

Are there any two-digit numbers which are the product of their digits?

In the triangle \(\triangle ABC\), the angle \(\angle ACB=60^{\circ}\), marked at the top. The angle bisectors \(AD\) and \(BE\) intersect at the point \(I\).

Find the angle \(\angle AIB\), marked in red.

Find, with proof, all integer solutions of \(a^3+b^3=9\).

Let \(x,x',y,y'\) be integers such that \(x+\sqrt{d}y=x'+\sqrt{d}y'\), where \(d\) is a number that is not a square. Show that \(x=x'\) and \(y=y'\).

Show that if \(u_1\) and \(u_2\) are solutions to Pell’s equation, then \(u_1u_2\) is also a solution to Pell’s equation. What can you conclude about the number of solutions, if there are any?

Find all integer solutions to \(x^2+y^2-1=4xy\).

In a bag we have \(99\) red balls and \(99\) blue balls. We take balls from the bag, two balls at a time:

• If the two balls are of the same colour, then we put in a red ball to the bag.

• If the two balls are of different colour, we return a blue ball to the bag.

Regardless, after each step, one ball is lost from the bag, so eventually there will be only one ball. What is the colour of this last ball?

You have an \(8\times 8\) chessboard coloured in the usual way. You can pick any two adjacent squares (i.e: any \(2\times 1\) or \(1\times 2\) section of the board) and flip the white tiles to black tiles and vice-versa. Is it possible to finish with \(63\) white pieces and \(1\) black piece?

We start with the point \((1,3)\) of the plane. We generate a sequence of points with the following rule: the \(x\)-coordinate of the new point is the arithmetic mean of the \(x\) and \(y\) coordinates of the previous point, and the \(y\)-coordinate of the new point is the harmonic mean of the \(x\) and \(y\) coordinates of the previous points. The harmonic mean of two numbers \(x\) and \(y\) is \(\frac{2}{\frac{1}{x} + \frac{1}{y}}\). Is the point \((3,2)\) in the sequence?

Four black dots are drawn on a whiteboard. On the dots we write the numbers \(10\), \(20\), \(30\), and \(40\) (one number on each dot). We then repeat the following move any number of times: choose one dot, decrease its number by \(3\), and increase the number on each of the other three dots by \(1\). After some number of moves, is it possible for all four dots to show the number \(25\) simultaneously?