Two rhombi $ABCD$ and $AXYZ$ are external to each other with a common vertex $A$. The vertices of the rhombi are labelled clockwise. If $\angle DAX=\angle BAZ$, prove that the centres of the rhombi and the midpoint of the segment $BZ$ form an isosceles triangle.
Find all 6-digit integers $n$ such that $n$ is a perfect square and that the number formed by the last 3 digits of $n$ is 1 more than the number formed by the first 3 digits of $n$.
Jack and Jill play the following game: Jack throws 3 dice and Jill can select some of them, possibly none, and turn each of them to the opposite side. Jill wins if the sum of the values on the dice is a multiple of 4. Can Jill always win? (Note the game is played with standard dice where the sum of the numbers on opposite sides is 7.)
Let $A,B,C,D$ be the 4 squares in the centre of a $4\times4$ grid of squares. These 4 squares form the centre of the grid. A frog can jump from one of the 4 squares in the centre to any square in the grid which shares a side. If the frog jumps out of the centre from a particular square for the first time, that square is called an exit square. Find the number of ways in which the frog can start from $A$ with $A$ being the exit square in $n$ jumps. Give your answer in terms of $n$.
Let $ABCD$ be a convex quadrilateral and $M$ be the intersection of the diagonals. Through $M$ draw a line meeting the side $AB$ at $P$ and the side $CD$ at $Q$. Find all the quadrilaterals so that there exists the segment $PQ$ that divides the triangles $ABM$ and $CDM$ into 4 similar triangles.