Volume 35
, Issue 2

1999 In our class, we were discussing applications to the discriminate of the quadratic function and came up with another approach to Q1037 (Vol 34 No 3) without calculus.

Suppose a certain lover of donuts (we will call him Homer), wants to put coloured icing on his donuts. Homer insists that each region of the donut is coloured in such a way that two regions that are next to each other have different colours.

Many of you have now learnt how to calculate the first derivative $\frac{df}{dx}$ for a wide range of functions such as $f(x)=x^{1/2}, f(x)=\sin(x), f(x)=1,$ etc.

If a number of copies of a shape can be fitted together to form a larger copy of the same shape, we call the shape a “replicating tile”, or a “rep–tile” for short.

Let $a, b$ be the sides and $c$ the hypotenuse of a right–angled triangle. If $a, b$ and $c$ are integers, show that

1. at least one of $a, b$ and $c$ is divisible by $5$,
2. if none of $a, b, c$ is divisible by $7$, then either $a + b$ or $a − b$ is divisible by $7$.

Q1051. What is the fractional derivative
$$\frac{d^{\frac{1}{2}f}}{dx^\frac{1}{2}}$$
of $f(x)=1/\sqrt{x}$ (see the article on fractional calculus in this issue of Parabola).

Q1043. An equilateral triangle $APQ$ is drawn so that $P,Q$ are on the sides $BC$ and $DC$ of a square $ABCD,$ with $|AP| =|AQ|.$ Show that the perimeter of $APQ$ is less than the perimeter of the triangle $ABD$ (unless $P$ is at $B$ and $Q$ is at $D$).