Problems

A round necklace contains \(45\) beads of two different colours: red and blue. Show that it is possible to find two beads of the same colour next to each other.

Cut this figure into \(4\) identical shapes. (Note: you have to use the entire shape. Rotations and reflections count as identical shapes)

The diagram shows a \(3 \times 3\) square with one corner removed. Cut it into three pieces, not necessarily identical, which can be reassembled to make a square:

Find all possible non-zero digits \(A\) for which the following holds \((AA+AA+1) \times A = AAA\). (Recall \(AA\) means the two-digit number whose first and second digits are \(A\))

A square has been divided into \(4\) rectangles and a square. If the rectangle in the bottom left corner has dimensions \(1 \times 4\) and the one in the top right is \(2 \times 5\), what is the area of the small square in the middle?

There are \(25\) bugs sitting on the squares of a \(5 \times 5\) board, \(1\) at each square. When I clap my hands, each bug jumps to a square diagonally from where it was before. Show that after I clap my hands, at least \(5\) squares will be empty.

In a convex quadrilateral \(ABCD\), all the triangles \(\triangle ABC\), \(\triangle BCD\), \(\triangle CDA\) and \(\triangle DAB\) have equal perimeters. Show that \(ABCD\) is a rectangle.

Find the last two digits of the number \[33333333333333333347^4 - 11111111111111111147^4\]

Replace all stars with ”+” or ”\(\times\)” signs so the equation holds: \[1*2*3*4*5*6=100\] Extra brackets may be added if necessary. Please write down the expression into the answer box.

In how many ways can one change \(\pounds 2\) into coins worth \(50\)p, \(20\)p and \(10\)p? One does not necessarily need to use all available coin types, i.e. having \(5\) coins of \(20\)p and \(10\) coins of \(10\)p is allowed.