Enrico Laeng

If x , y , and z are three positive integers such that x^{n}+y^{n}=z^{n} and the exponent n is also a positive integer, then n≤2 .

John Steele

In a previous issue issue of Parabola (Vol 29 No 2 p.2), I discussed the effect on time measurement of Einstein’s two postulates of Special Relativity.

It is fairly generally known, even amongst not very advanced students of mathematics, that in addition to the many ingenious constructions with straight edge and compasses which were discovered by the ancient Greeks, there were a number of similar construction problems which defied all their efforts...

**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 ).

**Q1035.** Find all positive integers n and m such that n is a factor of 4m−1 and m is a factor of 4n−1 .

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.

I. Woodhouse

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.

John Steele

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.

Laurent Borredon, Bruce Henry and Susan Wearne

Many of you have now learnt how to calculate the first derivative 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

d^{(1/2)f}/dx^{1/2}

of f(x)=1√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 ).

Jim Franklin

So, you’re a high school student with mathematical talent, and you face a decision on whether to develop it at university, or aim for a career like medicine or law that won’t use it.

Peter Merrotsy

The article “From the Archives... Impossible Constructions” (Anonymous, 1999. *Parabola*, 35 (1), pp. 12-18) reminded me of the method which a friend, who is a Draftsman, uses to construct what he thinks are regular figures in his drawings.

**Q1057. **Remember that a regular polygon has all sides equal and all angles equal.

Q1051. What is the fractional derivative

d^{(1/2)f}/dx^{1/2}

of f(x)=1/√x (see the article on fractional calculus in *Parabola*, Vol. 35, No. 2)?

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