Volume 32
, Issue 2

1996

Once again we apologise for the lateness of this issue. Please bear with us while we sort out the bugs.

It is a well known fact in cricket that the new ball when bowled by a fast bowler will often swing in its flight on the way down the pitch to the batsman.

In high school we learn some interesting mathematics and develop some (potentially) very useful skills. But how exactly are these skills and techniques applied to the real world?

Solutions - Junior Division

If $x$ is a real number, $[x]$ denotes the largest integer less than or equal to $x$; for example, $[\pi]=3$.  Find all positive real numbers $x,y$ satisfying the equation $$[x]\,[y]=x+y\ .$$

SENIOR DIVISION

Equal first prize
MAH Alexandre,          North Sydney Boys’ High School.
STITT Daniel Ian,        Sydney Grammar School.

Q.975 For which real numbers $x$ is it true that
$$[5x] = [3x] + 2[x] + 1\ ?$$
Here $[x]$ denotes the greatest integer less than or equal to $x$; for example, $[\pi] = 3.$

Q.966 Prove that
$$\left(n\atop1\right)-{1\over2}\left(n\atop 2\right) +{1\over3}\left(n\atop 3\right)-\cdots\pm{1\over n}\left(n\atop n\right) =1+{1\over2}+{1\over3}+\cdots+{1\over n}\ ,$$