Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP
have equal areas. Prove X and Y divide the sides of PQRS in the
A 1 metre cube has one face on the ground and one face against a
wall. A 4 metre ladder leans against the wall and just touches the
cube. How high is the top of the ladder above the ground?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
Consider the kangaroo's starting position as the origin of coordinate axes, with East and North being the positive x and y directions, respectively, and one metre being one unit along the axes.
We begin by considering the first quadrant. If the kangaroo's end point has coordinates $(a,b)$, then a and b must be integers. Also, after $10$ jumps, it must be that $a+b \leq 10$. Hence his end points are bounded by the right-angled triangle with vertices at $(10,0)$, $(0,10)$ and $(0,0)$.
He can finish at any point on the hypotenuse of the triangle since all these points satisfy $a+b=10$ and so can be reached by a jumps East and b jumps North. But he can only end up at a point $(a,b)$ on the other two edges or inside the triangle if $a+b$ is even. (He can certainly reach all such points in $a+b\leq
10$ jumps and if $a+b$ is even, with $a+b< 10$, he can jump away and back again using up $2$ jumps, and can repeat this until he has made $10$ jumps, and so end up at $(a,b)$.)
By symmetry we see that the possible end points form a square of side $11$, and so there are $121$ of them, as shown in the diagram.
This problem is taken from the UKMT Mathematical Challenges.